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<html>

<head>

<link rel="stylesheet" type="text/css" href="style.css">

</head>

<body>

<div id="box">

<h1>
GF-Complete: A Comprehensive Open Source Library for Galois </br>
Field Arithmetic
</h1>

<h1> Version 1.02  </h1>

<h4>James S. Plank* &nbsp&nbsp&nbsp&nbsp&nbsp&nbsp Ethan L. Miller 
Kevin M. Greenan &nbsp&nbsp&nbsp&nbsp&nbsp&nbsp Benjamin A. Arnold<br>
John A. Burnum &nbsp&nbsp&nbsp&nbsp&nbsp&nbsp Adam W. Disney  &nbsp&nbsp&nbsp&nbsp&nbsp&nbsp
Allen C. McBride

</h4> <br>



<a href="">

https://bitbucket.org/jimplank/gf-complete

 </a><br><br>
<a href=""> 
http://web.eecs.utk.edu/~plank/plank/papers/GF-Complete-Manual-1.02.pdf


 </a> <br> <br> 







</div>


<div id="pages_paragraphs_2">

This is a user's manual for GF-Complete, version 1.02. This release supersedes version 0.1 and represents the first
major release of GF-Complete. To our knowledge, this library implements every Galois Field multiplication technique
applicable to erasure coding for storage, which is why we named it GF-Complete. The primary goal of this library is
to allow storage system researchers and implementors to utilize very fast Galois Field arithmetic for Reed-Solomon
coding and the like in their storage installations. The secondary goal is to allow those who want to explore different
ways to perform Galois Field arithmetic to be able to do so effectively.


<p>
If you wish to cite GF-Complete, please cite technical report UT-CS-13-716: [PMG<sup>+</sup>13].

</p>


<h2>If You Use This Library or Document </h2>



Please send me an email to let me know how it goes. Or send me an email just to let me know you are using the
library. One of the ways in which we are evaluated both internally and externally is by the impact of our work, and if
you have found this library and/or this document useful, we would like to be able to document it. Please send mail to
<em>plank@cs.utk.edu.</em> Please send bug reports to that address as well.



<p>
The library itself is protected by the New BSD License. It is free to use and modify within the bounds of this
license. To the authors' knowledge, none of the techniques implemented in this library have been patented, and the
authors are not pursing patents. </p> <br>

 </div>
<div id="footer"> 
 
<span id="footer_bar">&nbsp&nbsp&nbsp&nbsp.*plank@cs.utk.edu (University of Tennessee), el  </span> <em>m@cs.ucsc.edu </em>(UC Santa Cruz), <em>kmgreen2@gmail.com </em> (Box). This material
is based upon work supported by the National Science Foundation under grants CNS-0917396, IIP-0934401 and CSR-1016636, plus REU supplements
CNS-1034216, CSR-1128847 and CSR-1246277. Thanks to Jens Gregor for helping us wade through compilation issues, and for Will
Houston for his initial work on this library.

</div>

<b>Finding the Code </b>
<br><br>
This code is actively maintained on bitbucket:<a href=""> https://bitbucket.org/jimplank/gf-complete. </a> There are
previous versions on my UTK site as a technical report; however, that it too hard to maintain, so the main version is
on bitbucket.<br><br>


<b>Two Related Papers </b> <br><br>

This software acccompanies a large paper that describes these implementation techniques in detail [PGM13a]. We
will refer to this as <em> "The Paper." </em> You do not have to read The Paper to use the software. However, if you want to
start exploring the various implementations, then The Paper is where you'll want to go to learn about the techniques
in detail.



<p>This library implements the techniques described in the paper "Screaming Fast Galois Field Arithmetic Using Intel
SIMD Instructions," [PGM13b]. The Paper describes all of those techniques as well.
</p><br><br>

<b>If You Would Like HelpWith the Software </b><br><br>

Please contact the first author of this manual.<br><br>

<b>Changes from Revision 1.01</b>
<br><br>
The major change is that we are using autoconf to aid with compilation, thus obviating the need for the old <b>flag_tester</b>
code. Additionally, we have added a quick timing tool, and we have modified <b>gf_methods</b> so that it may be used to
run the timing tool and the unit tester.


















<br/>
CONTENT  <span class="aligning_page_number"> 3 </span> 
<h2>Contents </h2>
<div class="index">
1 <span class="aligning_numbers">Introduction </span> <span class="aligning_page_number"> 5 </span>
  <br><br> 
2 <span class="aligning_numbers">Files in the Library </span>	<span class="aligning_page_number"> 6  </span>  <br> </div>

<div class="sub_indices">
2.1 Header files in the directory <b>"include"</b>  . .. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .  . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . <span class="aligning_page_number"> 6 </span>  <br>
2.2 Source files in the <b>"src"</b> directory . . . . . . . . . . . . . . . . . . . . . . .  . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .  . . . . . . . . . . . . . . . . .<span class="aligning_page_number">   7  </span> <br>
2.3 Library tools files in the <b>"tools"</b> directory  . . . . . . . . . . ..  . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .  . . . . . . . . . . . . . . . . . .   <span class="aligning_page_number"> 7   </span> <br>
2.4 The unit tester in the <b>"test"</b> directory. . . . . . . . . . . . . . . . . .  . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .  . . . . . . . . . . . . . . . . <span class="aligning_page_number">  8  </span>  <br>
2.5 Example programs in the <b>"examples"</b> directory . . . .  . . . . . . . . . . . . . . . .  . . . . . . . . . . . . . . . . . . . . .  . . . . . . . . . . . . . . . . . .<span class="aligning_page_number"> 8  </span> 

</div>
<br>
<div class="index">

3 <span class="aligning_numbers">Compilation </span><span class="aligning_page_number">  8 </span>  <br> <br>
4 <span class="aligning_numbers">Some Tools and Examples to Get You Started </span><span class="aligning_page_number">  8 </span> <br><br>  </div> 



<div class="sub_indices">
4.1 Three Simple Command Line Tools: gf_mult, gf_div and gf_add . . . . . . . . . .  . . . .  . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . <span class="aligning_page_number"> 8</span>  <br>
4.2 Quick Starting Example #1: Simple multiplication and division . .  . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .  . . . . . . . . . . . . . . . <span class="aligning_page_number">   9  </span> <br>
4.3 Quick Starting Example #2: Multiplying a region by a constant    . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .  . . . . . . . . . . . . . . .  <span class="aligning_page_number"> 10   </span> <br>
4.4 Quick Starting Example #3: Using w = 64 . . . . . . . . . . . . .  . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .  . . . . . . . . . . . . . . . . <span class="aligning_page_number">  11  </span>  <br>
4.5 Quick Starting Example #4: Using w = 128. . . .  . . . . . . . . . . . . . . . .  . . . . . . . . . . . . . . . . . . . . .  . . . . . . . . . . . . . . . . . . . . . <span class="aligning_page_number"> 11  </span> 
</div>
<br>


<div class="index">
5 <span class="aligning_numbers"> Important Information on Alignment when Multiplying Regions </span><span class="aligning_page_number"> 12</span> <br><br>

6 <span class="aligning_numbers"> The Defaults</span><span class="aligning_page_number"> 13 </span> <br>

</div>

<div class="sub_indices">
6.1 Changing the Defaults . .  . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .  . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .  . . . . . . . .<span class="aligning_page_number">   14  </span> <br>


<ul style="list-style-type:none;">
<li>6.1.1 Changing the Components of a Galois Field with <b> create_gf_from_argv() </b>   . . . . . . . . . . . . . . . .  . . . . . . . . . . . . . . . .  <span class="aligning_page_number"> 15   </span> <br>
</li>
<li>
6.1.2 Changing the Polynomial. . . . . . . . . . . . .  . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .  . . . . . . . . . . . . . . . . . . . . . . . . <span class="aligning_page_number">  16  </span>  <br>
</li>
<li>
6.1.3 Changing the Multiplication Technique. . . .  . . . . . . . . . . . . . . . .  . . . . . . . . . . . . . . . . . . . . .  . . . . . . . . . . . . . . . . . . .<span class="aligning_page_number"> 17  </span> 
</li>


<li>
6.1.4 Changing the Division Technique . . . . . .  . . . . . . . . . . . . . . . . . . . . .  . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .  <span class="aligning_page_number"> 19  </span> 
</li>


<li>
6.1.5 Changing the Region Technique. . . .  . . . . . . . . . . . . . . . .  . . . . . . . . . . . . . . . . . . . . .  . . . . . . . . . . . . . . . . . . . .  . . . ..<span class="aligning_page_number"> 19  </span> 
</li>
</ul>
6.2 Determining Supported Techniques with <b>gf_methods</b> . . . . . . . . . .  . . . .  . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . <span class="aligning_page_number"> 20</span>  <br>

6.3 Testing with <b>gf_unit, gf_time,</b> and <b>time_tool.sh </b>. . . . . . . . . .  . . . .  . . . . . . . . . . . . . . . . . . . . . . . . . . .  . . . . . . . . . . . . . . .  . . <span class="aligning_page_number"> 21</span>

<ul style="list-style-type:none;">
<li>
6.3.1 <b>time_tool.sh</b> . . . . . .  . . . . . . . . . . . . . . . . . . . . .  . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .. . . . . . . .  <span class="aligning_page_number"> 22 </span> 
</li>

<li>
6.3.2 An example of <b>gf_methods</b> and <b>time_tool.sh</b> . . . . . .  . . . . . . . . . . . . . . . . . . . . .  . . . . . . . . . . . . . . . .. . . . . . . .  . .. .  .<span class="aligning_page_number"> 23  </span> 
</li>

</ul>

6.4 Calling <b>gf_init_hard()</b> . . . . . . . . . .  . . . .  . . . . . . . . . . . . . . . . . . . . . . . . . . . .. . . . . . . . .. . . . . . . . .. . . . . . . .  .. . . . . . . .  . . .  <span class="aligning_page_number"> 24</span>  <br>

6.5 <b>gf_size()</b> . . . . . . . . . .  . . . .  . . . . . . . . . . . . . . . . . . . . . . . . . . . .. . . . . . . .  .. . . . . . . . .. . . . . . . .  .. . . . . . . . .. . . . . . . . . . ..  .  <span class="aligning_page_number"> 26</span>  <br><br>
</div>


<div class="index">
8 <span class="aligning_numbers">  Further Information on Options and Algorithms </span><span class="aligning_page_number">   26 </span> </div> <br><br> </div>
<div class="sub_indices">
7.1 Inlining Single Multiplication and Division for Speed   . . . . . . . . . . . . . . .  . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .  . . . <span class="aligning_page_number"> 26 </span> <br>
7.2 Using different techniques for single and region multiplication . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .  .  <span class="aligning_page_number"> 27 </span> <br>
7.3 General <em>w</em> . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .  . . . . . . . . . . . . . .  . <span class="aligning_page_number"> 28  </span><br>

7.4 Arguments to <b>"SPLIT"</b> . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .  . . . . . . . . . . . . . . . . . <span class="aligning_page_number"> 28</span>  <br>
7.5 Arguments to <b>"GROUP"</b> . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .  . . . . .  . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .  <span class="aligning_page_number">29 </span> <br>
7.6 Considerations with <b>"COMPOSITE"</b> . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .  . . . . .  . . . . . . . . . . . . . . . . . . . . . . . . . . . .  .  <span class="aligning_page_number">30 </span> <br>
7.7 <b>"CARRY_FREE"</b> and the Primitive Polynomial  . . . . . . . . . . . . . . .  . . . . .  . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .  . .   <span class="aligning_page_number">31 </span> <br>
7.8 More on Primitive Polynomials . .  . . . . . . . . . . . . . . . . . . .  . . . . .  . . . . . . . . . . . . . . . . . . . . . . . . . . . .  . . . . . . . ..   . . . . . . . . .  <span class="aligning_page_number">31 </span> <br>


<ul style="list-style-type:none;">
<li>
7.8.1 Primitive Polynomials that are not Primitive . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .  . . . . . . . . . . . . . . . . . . .  . . .  <span class="aligning_page_number"> 31</span>  <br>

</li>
<li>7.8.2 Default Polynomials for Composite Fields . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .  . . . . . . . . . . . . . . . . . . . . . . .  <span class="aligning_page_number"> 32</span>  <br>

</li>
</ul>

</div>











<br/>
CONTENT  <span class="aligning_page_number"> 4 </span> 

<div class="sub_indices">
<ul style="list-style-type:none">
<li> 7.8.3 The Program <b>gf_poly</b> for Verifying Irreducibility of Polynomials </span><span class="aligning_page_number">  33 </span> 
</li>
</ul>


7.9<span class="aligning_numbers"><b>"ALTMAP"</b> considerations and <b>extract_word()</b> </span><span class="aligning_page_number">  34 </span>  
<ul style="list-style-type:none">
<li>

7.9.1 Alternate mappings with <b>"SPLIT"</b> . . . . . . . . . .  . . . .  . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .<span class="aligning_page_number"> 34</span>  <br>
</li>
<li>
7.9.2 Alternate mappings with <b>"COMPOSITE"</b> . .  . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .  . . . . . . . . . . . . . . . . . . . . . . . . <span class="aligning_page_number">   36  </span> <br>
</li>
<li>
7.9.3 The mapping of <b>"CAUCHY"</b>    . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .  . . . . . . . . . . . . . . . . . .. . . . . .. . . . . . . . . .  . ..  <span class="aligning_page_number"> 37   </span> <br>
</li>
</ul>
</div>


8 <span class="aligning_numbers"><b>Thread Safety </b></span><span class="aligning_page_number">  37 </span> <br><br>  </div> 

9 <span class="aligning_numbers"><b>Listing of Procedures</b> </span><span class="aligning_page_number">  37 </span> <br><br>  </div> 

10 <span class="aligning_numbers"><b>Troubleshooting</b> </span><span class="aligning_page_number">  38 </span> <br><br>  </div> 
11 <span class="aligning_numbers"><b>Timings</b> </span><span class="aligning_page_number">  41 </span> <br><br>  </div> 

<div class="sub_indices">
11.1 Multiply() . . . . . . . . . .  . . . .  . . . . . . . . . . . . . . . . . . . . . . . . . . . .. . . . . . . . .. . . . . . . . .. . . . . . . . .. . . . . . . . . . . . .  . . . .. . . . <span class="aligning_page_number"> 42</span>  <br>
11.2 Divide() . .  . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .  . . . . . . . . . . . . . . .. . . . . . . . .. . . . . . . . .. . . . . . . . .. . . . . . . . . . . .. . . . . <span class="aligning_page_number">   42  </span> <br>
11.3 Multiply Region()    . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .  . . . . . . . . . . . . . . .. . . . . . . . .. . . . . . . . .. . . . . . . . . . . . .  . . . . .  <span class="aligning_page_number"> 43   </span> <br>
</div>






<br/>
INTRODUCTION  <span class="aligning_page_number"> 5 </span> 


<h3>1 Introduction </h3>

Galois Field arithmetic forms the backbone of erasure-coded storage systems, most famously the Reed-Solomon
erasure code. A Galois Field is defined over w-bit words and is termed <em>GF(2<sup>w</sup>).</em> As such, the elements of a Galois
Field are the integers 0, 1, . . ., 2<sup>w</sup> - 1. Galois Field arithmetic defines addition and multiplication over these closed
sets of integers in such a way that they work as you would hope they would work. Specifically, every number has a
unique multiplicative inverse. Moreover, there is a value, typically the value 2, which has the property that you can
enumerate all of the non-zero elements of the field by taking that value to successively higher powers.


<p>Addition in a Galois Field is equal to the bitwise exclusive-or operation. That's nice and convenient. Multiplication
is a little more complex, and there are many, many ways to implement it. The Paper describes them all, and the
following references providemore supporting material: [Anv09, GMS08, LHy08, LD00, LBOX12, Pla97]. The intent
of this library is to implement all of the techniques. That way, their performancemay be compared, and their tradeoffs
may be analyzed. <p>




<ol>

When used for erasure codes, there are typically five important operations:<br>
<li> <b>Adding two numbers in </b> GF(2<sup>w</sup>). That's bitwise exclusive-or. </li>
<li> <b>Multiplying two numbers in</b> GF(2<sup>w</sup>). Erasure codes are usually based on matrices in GF(2<sup>w</sup>), and constructing
these matrices requires both addition and multiplication.</li>
<li> <b>Dividing two numbers in </b>GF(2<sup>w</sup>). Sometimes you need to divide to construct matrices (for example, Cauchy
Reed-Solomon codes [BKK<sup>+</sup>95, Rab89]). More often, though, you use division to invert matrices for decoding.
Sometimes it is easier to find a number's inverse than it is to divide. In that case, you can divide by multiplying
by an inverse. </li>

<li><b>adding two regions of numbers in</b> GF(2<sup>w</sup>), which will be explained along with... </li>
<li> <b>Mutiplying a region of numbers in </b>GF(2<sup>w</sup>) by a constant in GF(2<sup>w</sup>). Erasure coding typically boils down
to performing dot products in GF(2<sup>w</sup>). For example, you may define a coding disk using the equation: </li><br>




<center>c<em><sub>0</sub></em>= d<em><sub>0</sub></em> + 2d<em><sub>1</sub></em> + 4d<em><sub>2</sub></em> + 8d<em><sub>3</sub></em>.</sup> </center><br>

That looks like three multiplications and three additions However, the way ' implemented in a disk system
looks as in Figure 1. Large regions of disks are partitioned into w-bit words in GF(2<sup>w</sup>). In the example, let us
suppose that <em>w</em> = 8, and therefore that words are bytes. Then the regions pictured are 1 KB from each disk.
The bytes on disk Di are labeled d<sub>i,0,</sub> d<sub>i,1, . . . ,</sub> d<sub>i,1023,</sub> and the equation above is replicated 1024 times. For
0 &#8804 j < 1024:
<br><br>
<center>c<em><sub>0,j</sub></em> = d<em><sub>0,j</sub></em> + 2d<em><sub>1,j</sub></em> + 4d<em><sub>2,j</sub></em> + 8d<em><sub>3,j</sub></em> . </center>
<br>


While it's possible to implement each of these 1024 equations independently, using the single multiplication
and addition operations above, it is often much more efficient to aggregate. For example, most computer architectures
support bitwise exclusive-or of 64 and 128 bit words. Thus, it makes much more sense to add regions
of numbers in 64 or 128 bit chunks rather than as words in GF(2<sup>w</sup>). Multiplying a region by a constant can
leverage similar optimizations. </ol>


<p>GF-Complete supports multiplication and division of single values for all values of <em>w</em> &#8804 32, plus <em>w</em> = 64 and <em>w</em> =
128. It also supports adding two regions of memory (for any value of <em>w</em>, since addition equals XOR), and multiplying
a region by a constant in <em>GF(2<sup>4</sup>), GF(2<sup>8</sup>), GF(2<sup>16</sup>), GF(2<sup>32</sup>), GF(2<sup>64</sup>) and GF(2<sup>128</sup>).</em> These values are chosen
because words in GF(2<sup>w</sup>) fit into machine words with these values of <em>w.</em> Other values of w don't lend themselves
to efficient multiplication of regions by constants (although see the <b>"CAUCHY"</b> option in section 6.1.5 for a way to
multiply regions for other values of <em>w</em>).</p>






<br/>

2 &nbsp &nbsp  <em>  FILES IN THE LIBRARY     </em>   <span id="index_number">6  </span> <br><br><br>



<div class="image-cell_1"> </div>  <br><br><br>

Figure 1: An example of adding two regions of numbers, and multiplying a region of numbers by a constant
in <em>GF(2<sup>w</sup>) </em>. In this example, <em>w</em> = 8, and each disk is holding a 1KB region. The same coding equation -
c<sub>0,j</sub></b> = d<sub>0,j</sub> + ad<sub>1,j</sub> + a<sup>2</sup>d<sub>2,j</sub> + a<sup>3</sup>d<sub>3,j</sub> is applied 1024 times. However, rather than executing this equation 1024
times, it is more efficient to implement this with three region-constant multiplications and three region-region additions.

<h3>2 &nbsp&nbsp&nbsp Files in the Library </h3>
This section provides an overview of the files that compose GF-Complete. They are partitioned among multiple
directories.

<h4> <b>2.1 &nbsp&nbsp&nbsp Header files in the directory  "include"</b> </h4>

The following header files are part of GF-Complete.
<ul>
<li><b>gf_complete.h:</b> This is the header file that applications should include. It defines the gf_t type, which holds
all of the data that you need to perform the various operations in GF(2<sup>w</sup>). It also defines all of the arithmetic
operations. For an application to use this library, you should include gf_complete.h and then compile with the
library src/libgf_complete.la. </li><br>

<li><b>gf_method.h:</b> If you are wanting to modify the implementation techniques from the defaults, this file provides
a "helper" function so that you can do it from the Unix command line.
</li><br>

<li><b>gf_general.h:</b> This file has helper routines for doing basic Galois Field operations with any legal value of <em>w.</em>
The problem is that <em>w </em> &#8804 32, <em>w </em> = 64 and <em> w </em> = 128 all have different data types, which is a pain. The procedures
in this file try to alleviate that pain. They are used in <b>gf_mult, gf_unit</b> and <b>gf_time.</b> I'm guessing that most
applications won't use them, as most applications use <em>w</em> &#8804 32. </li><br>

<li><b>gf_rand.h:</b> I've learned that <b>srand48()</b> and its kin are not supported in all C installations. Therefore, this file
defines some randomnumber generators to help test the programs. The randomnumber generator is the "Mother
</li>

</ul>







<br/>

2 &nbsp &nbsp  <em>  FILES IN THE LIBRARY     </em>   <span id="index_number">7  </span> <br><br><br>
<ul>

of All" random number generator [Mar94] which we've selected because it has no patent issues. <b>gf_unit</b> and
<b>gf_time</b> use these random number generators.<br><br>
<li><b>gf_int.h:</b> This is an internal header file that the various source files use. This is <em>not</em> intended for applications to
include.</li><br>
<li><b>config.xx</b> and <b>stamp-h1</b> are created by autoconf, and should be ignored by applications. </li>
</ul>

<h3>2.2 &nbsp &nbsp <b> Source files in the "src" directory" </b> </h3>
<ul>
The following C files compose <b>gf_complete.a,</b> and they are in the direcoty src. You shouldn't have to mess with these
files, but we include them in case you have to:<br><br>
<li><b> gf_.c:</b> This implements all of the procedures in both <b>gf_complete.h</b> and <b>gf_int.h.</b> </li><br>
<li><b> gf_w4.c:</b> Procedures specific to <em>w </em> = 4. </li><br>
<li> <b>gf_w8.c:</b> Procedures specific to <em>w </em> = 8</li><br>
<li> <b>gf_w16.c:</b> Procedures specific to <em>w </em> = 16</li><br>
<li> <b>gf_w32.c:</b> Procedures specific to <em>w </em> = 32</li><br>
<li><b>gf_w64.c:</b> Procedures specific to <em>w </em> = 64</li><br>
<li> <b>gf_w128.c:</b> Procedures specific to <em>w </em> = 128</li><br>
<li> <b>gf_wgen.c:</b> Procedures specific to other values of <em>w </em> between 1 and 31</li><br>
<li> <b>gf_general.c:</b> Procedures that let you manipulate general values, regardless of whether <em>w </em> &#8804 32, <em>w </em> = 64
or <em>w </em> = 128. (I.e. the procedures defined in <b>gf_ general.h</b>)</li><br>
<li> <b>gf_method.c:</b> Procedures to help you switch between the various implementation techniques. (I.e. the procedures
defined in <b>gf_method.h</b>)</li><br>
<li> <b>gf_ rand.c:</b>"The Mother of all" random number generator. (I.e. the procedures defined in <b>gf_rand.h</b>)</li><br> </ul>

<h3>2.3 &nbsp &nbsp Library tools files in the "tools" directory </h3>

<ul>
The following are tools to help you with Galois Field arithmetic, and with the library. They are explained in greater
detail elsewhere in this manual.<br><br>
<li> <b>gf_mult.c, gf_ div.c</b> and <b>gf_ add:</b> Command line tools to do multiplication, division and addition by single numbers</li><br>
<li> <b>gf_time.c:</b> A program that times the procedures for given values of <em>w </em> and implementation options</li><br>
<li> <b>time_tool.sh:</b> A shell script that helps perform rough timings of the various multiplication, division and region
operations in GF-Complete</li><br>
<li> <b>gf_methods.c:</b> A program that enumerates most of the implementation methods supported by GF-Complete</li><br>
<li> <b> gf_poly.c:</b> A program to identify irreducible polynomials in regular and composite Galois Fields</li><br>

</ul>








<br/>

3 &nbsp &nbsp  <em>  COMPILATION     </em>   <span id="index_number">8  </span> <br><br><br>


<h3>2.4 &nbsp &nbsp The unit tester in the "test" directory </h3>

The test directory contains the proram <b>gf_unit.c,</b> which performs a battery of unit tests on GF-Complete. This is
explained in more detail in section 6.3.


<h3>2.5&nbsp &nbsp Example programs in the "examples" directory </h3>

There are seven example programs to help you understand various facets of GF-Complete. They are in the files
<b>gf_example x.c </b> in the <b>examples</b> directory. They are explained in sections 4.2 through 4.5, and section 7.9.<br><br>

<h2>3 &nbsp &nbsp Compilation </h2>

<em>From revision 1.02 forward, we are using autoconf. The old "flag tester" directory is now gone, as it is no longer in
use. </em><br><br>
To compile and install, you should do the standard operations that you do with most open source Unix code:<br><br>

UNIX> ./configure <br>
... <br>
UNIX> make <br>
... <br>
UNIX> sudo make install <br><br>


<p>If you perform the <b>install,</b> then the header, source, tool, and library files will be moved to system locations. In
particular, you may then compile the library by linking with the flag <b>-lgf_complete,</b> and you may use the tools from a
global executable directory (like <b>/usr/local/bin</b>). </p>

<p>
If you don't perform the install, then the header and tool files will be in their respective directories, and the library
will be in <b>src/libgf_complete.la.</b> </p>
<p>
If your system supports the various Intel SIMD instructions, the compiler will find them, and GF-Complete will
use them by default. </p>



<h2>4 &nbsp &nbsp Some Tools and Examples to Get You Started </h2> 
<h3>4.1 Three Simple Command Line Tools: gf_mult, gf_div and gf_add </h3>


Before delving into the library, it may be helpful to explore Galois Field arithmetic with the command line tools:
<b>gf_mult, gf_div </b> and <b>gf_add.</b> These perform multiplication, division and addition on elements in <em>GF(2<sup>w</sup>).</em> If these are
not installed on your system, then you may find them in the tools directory. Their syntax is:
<ul>
<li><b>gf_mult a b</b> <em>w </em> - Multiplies a and b in <em> GF(2<sup>w</sup>)</em>. </li><br>
<li> <b>gf_div a b </b><em>w </em> - Divides a by b in GF(2<em><sup>w </sup></em>). </li><br>
<li><b>gf_add a b </b> <em>w </em> - Adds a and b in GF(2<em><sup>w </sup> </em>). </li><br>

You may use any value of <em>w </em> from 1 to 32, plus 64 and 128. By default, the values are read and printed in decimal;
however, if you append an 'h' to <em>w </em>, then <em>a, b </em> and the result will be printed in hexadecimal. For <em>w </em> = 128, the 'h' is
mandatory, and all values will be printed in hexadecimal.







<br/>

4 &nbsp &nbsp  <em>   SOME TOOLS AND EXAMPLES TO GET YOU STARTED 9     </em>   <span id="index_number">9  </span> <br><br><br>


<p>Try them out on some examples like the ones below. You of course don't need to know that, for example, 5 * 4 = 7
in <em>GF(2<sup>4 </sup>) </em>; however, once you know that, you know that 7/
5 = 4 and 7/4 = 5. You should be able to verify the <b>gf_add</b>
statements below in your head. As for the other <b>gf_mult's</b>, you can simply verify that division and multiplication work
with each other as you hope they would. </p>
<br><br>
<div id="number_spacing">

UNIX> gf_mult 5 4 4  <br>
7 <br>
UNIX> gf_div 7 5 4 <br>
4 <br>
UNIX> gf_div 7 4 4 <br>
5   <br>
UNIX> gf_mult 8000 2 16h <br>
100b  <br>
UNIX> gf_add f0f0f0f0f0f0f0f0 1313131313131313 64h <br>
e3e3e3e3e3e3e3e3 <br>
UNIX> gf_mult f0f0f0f0f0f0f0f0 1313131313131313 64h <br>
8da08da08da08da0 <br>
UNIX> gf_div 8da08da08da08da0 1313131313131313 64h <br>
f0f0f0f0f0f0f0f0  <br>
UNIX> gf_add f0f0f0f0f0f0f0f01313131313131313 1313131313131313f0f0f0f0f0f0f0f0 128h <br>
e3e3e3e3e3e3e3e3e3e3e3e3e3e3e3e3 <br>
UNIX> gf_mult f0f0f0f0f0f0f0f01313131313131313 1313131313131313f0f0f0f0f0f0f0f0 128h <br>
786278627862784982d782d782d7816e <br>
UNIX> gf_div 786278627862784982d782d782d7816e f0f0f0f0f0f0f0f01313131313131313 128h <br>
1313131313131313f0f0f0f0f0f0f0f0 <br>
UNIX> <br><br>

</div>


Don't bother trying to read the source code of these programs yet. Start with some simpler examples  like the ones
below. <br><br>

<h3>4.2 Quick Starting Example #1: Simple multiplication and division </h3>

The source files for these examples are in the examples directory.
<p>These two examples are intended for those who just want to use the library without getting too complex. The
first example is <b>gf_example 1,</b> and it takes one command line argument - w, which must be between 1 and 32. It
generates two random non-zero numbers in <em>GF(2<sup>w </sup>) </em> and multiplies them. After doing that, it divides the product by
each number. </p>
<p>
To perform multiplication and division in <em>GF(2<sup>w </sup>) </em>, you must declare an instance of the gf_t type, and then initialize
it for <em>GF(2<sup>w </sup>) </em> by calling <b>gf_init_easy().</b> This is done in <b>gf_example 1.c</b> with the following lines: </p><br><br>

gf_t gf; <br><br>r
... <br><br>
if (!gf_init_easy(&gf, w)) { <br>
fprintf(stderr, "Couldn't initialize GF structure.\n"); <br>
exit(0); <br>
}  <br>






<br/>

4 &nbsp &nbsp  <em>   SOME TOOLS AND EXAMPLES TO GET YOU STARTED      </em>   <span id="index_number">10  </span> <br><br><br>

<p>Once <b>gf</b> is initialized, you may use it for multiplication and division with the function pointers <b>multiply.w32</b> and
<b>divide.w32.</b> These work for any element of <em>GF(2<sup>w</sup>)</em> so long as w &#8804 32. </p> <br><br>

<div id="number_spacing">
<div style="padding-left:54px">
c = gf.multiply.w32(&gf, a, b);<br>
printf("%u * %u = %u\n", a, b, c);<br><br>
printf("%u / %u = %u\n", c, a, gf.divide.w32(&gf, c, a));<br>
printf("%u / %u = %u\n", c, b, gf.divide.w32(&gf, c, b));<br>


</div> </div>
<br><br>
Go ahead and test this program out. You can use <b>gf_mult</b> and <b>gf_div</b> to verify the results:<br><br>

<div id="number_spacing">
UNIX> gf_example_1 4 <br>
12 * 4 = 5  <br>
5 / 12 = 4  <br>
5 / 4 = 12  <br>
UNIX> gf_mult 12 4 4 <br>
5  <br>
UNIX> gf_example_1 16 <br>
14411 * 60911 = 44568 <br>
44568 / 14411 = 60911 <br>
44568 / 60911 = 14411  <br>
UNIX> gf_mult 14411 60911 16 <br>
44568 <br>
UNIX>  <br><br>
</div>

<b>gf_init_easy()</b> (and <b>later_gf_init_hard()</b>) do call <b>malloc()</b> to implement internal structures. To release memory, call
<b>gf_free().</b> Please see section 6.4 to see how to call <b>gf_init_hard()</b> in such a way that it doesn't call <b>malloc().</b> <br><br>



<h3>4.3 &nbsp &nbsp &nbspQuick Starting Example #2: Multiplying a region by a constant </h3>


The program <b>gf_example</b> 2 expands on <b>gf_example</b> 1. If <em>w</em> is equal to 4, 8, 16 or 32, it performs a region multiply
operation. It allocates two sixteen byte regions, <b>r1</b> and <b>r2,</b> and then multiples <b>r1</b> by a and puts the result in <b>r2</b> using
the <b>multiply_region.w32</b> function pointer: <br><br>

<div style="padding-left:52px">
gf.multiply_region.w32 (&gf, r1, r2, a, 16, 0); <br><br>
</div>

That last argument specifies whether to simply place the product into r2 or to XOR it with the contents that are already
in r2. Zero means to place the product there. When we run it, it prints the results of the <b>multiply_region.w32</b> in
hexadecimal. Again, you can verify it using <b>gf_mult</b>:<br><br>
<div id="number_spacing">
UNIX> gf_example_2 4 <br>
12 * 2 = 11 <br>
11 / 12 = 2 <br>
11 / 2 = 12 <br><br>
multiply_region by 0xc (12) <br><br>
R1 (the source): 0 2 d 9 d 6 8 a 8 d b 3 5 c 1 8 8 e b 0 6 1 5 a 2 c 4 b 3 9 3 6 <br>
R2 (the product): 0 b 3 6 3 e a 1 a 3 d 7 9 f c a a 4 d 0 e c 9 1 b f 5 d 7 6 7 e <br>

</div>










<br/>

4 &nbsp &nbsp  <em>   SOME TOOLS AND EXAMPLES TO GET YOU STARTED      </em>   <span id="index_number">11  </span> <br><br><br>

<div id="number_spacing">
<table cellpadding="6">
<tr><td>UNIX></td> <td colspan="4"> gf_example_2 16 </td> </tr>

<tr>

<td>49598</td> <td> * </td> <td> 35999</td> <td> = </td> <td>19867 </td> </tr>

<tr><td>19867 </td><td>/ </td> <td> 49598 </td> <td> =  </td> <td>35999 </td> </tr>
<tr><td>19867</td><td> /</td> <td> 35999 </td> <td> = </td> <td> 49598 </td> </tr>  </table><br>


&nbsp multiply_region by 0xc1be (49598) <br><br>


<table cellpadding="6" >
<tr>
<td>R1 (the source):</td> <td> 8c9f </td> <td> b30e </td> <td> 5bf3 </td> <td> 7cbb </td> <td>16a9 </td> <td> 105d </td> <td> 9368 </td> <td> 4bbe </td> </tr>
<td>R2 (the product):</td> <td> 4d9b</td> <td> 992d </td> <td> 02f2 </td> <td> c95c </td> <td> 228e </td> <td> ec82 </td> <td> 324e </td> <td> 35e4 </td></tr>
</table>
</div>
<div id="number_spacing">
<div style="padding-left:9px">
UNIX> gf_mult c1be 8c9f 16h<br>
4d9b <br>
UNIX> gf_mult c1be b30e 16h <br>
992d <br>
UNIX> <br><br>
</div>
</div>

<h3>4.4 &nbsp &nbsp &nbsp Quick Starting Example #3: Using <em>w </em>= 64 </h3>
The program in <b>gf_example 3.c </b> is identical to the previous program, except it uses <em> GF(2<sup>64 </sup>). </em> Now <em>a, b</em> and <em> c </em> are
<b>uint64 t'</b>s, and you have to use the function pointers that have <b>w64</b> extensions so that the larger types may be employed.
<br><br>
<div id="number_spacing">

UNIX> gf_example_31 
<table cellpadding="6">
<tr>

<td>a9af3adef0d23242 </td> <td> * </td> <td> 61fd8433b25fe7cd</td> <td> = </td> <td>bf5acdde4c41ee0c </td> </tr>

<td>bf5acdde4c41ee0c </td> <td> / </td> <td> a9af3adef0d23242 </td> <td> = </td> <td>61fd8433b25fe7cd </td> </tr>
<td>bf5acdde4c41ee0c </td> <td> / </td> <td> 61fd8433b25fe7cd  </td> <td>= </td> <td>a9af3adef0d23242 </td> </tr>
</table><br><br>

&nbsp multiply_region by a9af3adef0d23242<br><br>
<table cellpadding="6" >
<tr>
<td>R1 (the source): </td> <td> 61fd8433b25fe7cd </td> <td>272d5d4b19ca44b7 </td> <td> 3870bf7e63c3451a </td> <td> 08992149b3e2f8b7 </td> </tr>
<tr><td>R2 (the product): </td> <td> bf5acdde4c41ee0c </td> <td> ad2d786c6e4d66b7 </td> <td> 43a7d857503fd261 </td> <td> d3d29c7be46b1f7c </td> </tr>
</table>

<div style="padding-left:9px">

UNIX> gf_mult a9af3adef0d23242 61fd8433b25fe7cd 64h <br>
bf5acdde4c41ee0c<br>
UNIX><br><br>
</div>
</div>
<h3>4.5 &nbsp &nbsp &nbsp Quick Starting Example #4: Using <em>w </em>= 128 </h3>
Finally, the program in <b>gf_example_4.c</b> uses  <em>GF(2<sup>128</sup>).</em> Since there is not universal support for uint128 t, the library
represents 128-bit numbers as arrays of two uint64 t's. The function pointers for multiplication, division and region
multiplication now accept the return values as arguments:<br><br>

gf.multiply.w128(&gf, a, b, c); <br><br>

Again, we can use <b>gf_mult </b> and <b>gf_div </b>to verify the results:<br><br>
<div id="number_spacing">
<div style="padding-left:9px">
UNIX> gf_example_4 </div>
<table cellpadding="6" >
<tr>

<td>e252d9c145c0bf29b85b21a1ae2921fa </td> <td> * </td> <td> b23044e7f45daf4d70695fb7bf249432 </td> <td> = </td> </tr>
<tr><td>7883669ef3001d7fabf83784d52eb414 </td> </tr>

</table>

</div>








<br/>

4 &nbsp &nbsp  <em>   IMPORTANT INFORMATION ON ALIGNMENT WHEN MULTIPLYING REGIONS      </em>   <span id="index_number">12  </span> <br><br><br>

<div id="number_spacing">
multiply_region by e252d9c145c0bf29b85b21a1ae2921fa <br>
R1 (the source): f4f56f08fa92494c5faa57ddcd874149 b4c06a61adbbec2f4b0ffc68e43008cb <br>
R2 (the product): b1e34d34b031660676965b868b892043 382f12719ffe3978385f5d97540a13a1 <br>
UNIX> gf_mult e252d9c145c0bf29b85b21a1ae2921fa f4f56f08fa92494c5faa57ddcd874149 128h <br>
b1e34d34b031660676965b868b892043 <br>
UNIX> gf_div 382f12719ffe3978385f5d97540a13a1 b4c06a61adbbec2f4b0ffc68e43008cb 128h<br>
e252d9c145c0bf29b85b21a1ae2921fa<br>
UNIX><br><br>

</div>


<h2>5 &nbsp &nbsp &nbspImportant Information on Alignment when Multiplying Regions </h2>



In order to make multiplication of regions fast, we often employ 64 and 128 bit instructions. This has ramifications
for pointer alignment, because we want to avoid bus errors, and because on many machines, loading and manipulating
aligned quantities is much faster than unalinged quantities.<br><br>


When you perform multiply_region.wxx(<em>gf, source, dest, value, size, add </em>), there are three requirements:
<ol>
<li>
 The pointers <em>source</em> and <em>dest </em> must be aligned for <em>w</em>-bit words. For <em>w </em> = 4 and <em>w </em> = 8, there is no restriction;
however for <em>w </em> = 16, the pointers must be multiples of 2, for <em>w </em> = 32, they must be multiples of 4, and for
<em>w </em> &#1013; {64, 128}, they must be multiples of 8. </li><br>

<li> The <em>size</em> must be a multiple of &#91; <em>w /
</em> 
8 .&#93;
 With <em>w </em> = 4 and <em>w </em> = 8, <em>w/ </em>
8  = 1 and there is no restriction. The other
sizes must be multiples of <em>w </em>/
8  because you have to be multiplying whole elements of <em> GF(2<sup>w </sup>) </em>. </li><br>

<li> The <b>source</b> and <b>dest</b> pointers must be aligned identically with respect to each other for the implementation
chosen. This is subtle, and we explain it in detail in the next few paragraphs. However, if you'd rather not figure
it out, the following recommendation will <em>always </em> work in GF-Complete: </li>

</ol>



<div style="padding-left:100px">
<b>If you want to be safe, make sure that source and dest are both multiples of 16. That is not a
strict requirement, but it will always work! </b> <br><br>
</div>


If you want to relax the above recommendation, please read further.
<p>When performing <b>multiply_region.wxx() </b>, the implementation is typically optimized for a region of bytes whose
size must be a multiple of a variable <em>s </em> ,, and which must be aligned to a multiple of another variable <em>t </em>. For example,
when doing <b>multiply_region.w32() </b> in <em> GF(2<sup>16 </sup>) </em> with SSE enabled, the implementation is optimized for regions of
32 bytes, which must be aligned on a 16-byte quantity. Thus, <em>s </em> = 32 and <em>t</em> = 16. However, we don't want <b>multiply_
region.w32() </b> to be too restrictive, so instead of requiring <em>source</em> and <em> dest </em> to be aligned to 16-byte regions, we
require that (<em>source </em> mod 16) equal (<em>dest</em> mod 16). Or, in general, that (<em>source</em> mod t) equal (<em>dest</em> mod <em>t</em>). </p>


<p>
Then, <b>multiply_region.wxx()</b> proceeds in three phases. In the first phase,<b> multiply.wxx()</b> is called on successive
words until (<em>source</em> mod <em>t</em>) equals zero. The second phase then performs the optimized region multiplication on
chunks of <em> s  </em>bytes, until the remaining part of the region is less than s bytes. At that point, the third phase calls
<em>multiply.wxx() </em> on the last part of the region. </p>

A detailed example helps to illustrate. Suppose we make the following call in <em>GF(2<sup>16</sup>) </em> with SSE enabled:<br><br>
<center><b>multiply region.w32(gf, 0x10006, 0x20006, a, 274, 0)</b> </center>







<br/>

2 &nbsp &nbsp  <em>  FILES IN THE LIBRARY     </em>   <span id="index_number">13  </span> <br><br><br>



<div class="image-cell_2"> </div>  <br><br><br>

Figure 2: Example of multiplying a region of 274 bytes in GF(216) when (source mod 16) = (dest mod 16) = 6. The
alignment parameters are s = 32 and t = 16. The multiplication is in three phases, which correspond to the initial
unaligned region (10 bytes), the aligned region of s-byte chunks (256 bytes), and the final leftover region (8 bytes).


<p>First, note that <em>source</em> and <em>dest</em> are aligned on two-byte quantities, which they must be in <em>GF(2<sup>16</sup>).</em> Second, note
that size is a multiple of &#91; 16/
8 &#93 = 2. And last, note that (<em>source</em> mod 16) equals (<em>dest</em> mod 16). We illustrate the three
phases of region multiplication in Figure 2. Because (<em>source</em> mod 16) = 6, there are 10 bytes of unaligned words that
are multiplied with five calls to <b>multiply.w32()</b> in the first phase. The second phase multiplies 256 bytes (eight chunks
of <em>s</em> = 32 bytes) using the SSE instructions. That leaves 8 bytes remaining for the third phase.
</p>

<p>
When we describe the defaults and the various implementation options, we specify s and t as "alignment parameters."
</p>
<p>
One of the advanced region options is using an alternate mapping of words to memory ("ALTMAP"). These interact
in a more subtle manner with alignment. Please see Section 7.9 for details.
</p>

<h3> 6 &nbsp &nbspThe Defaults </h3>


GF-Complete implements a wide variety of techniques for multiplication, division and region multiplication. We have
set the defaults with three considerations in mind:
<ol>
<li>
<b>Speed:</b> Obviously, we want the implementations to be fast. Therefore, we choose the fastest implementations
that don’t violate the other considerations. The compilation environment is considered. For example, if SSE is
enabled, region multiplication in <em> GF(2<sup>4 </sup>) </em> employs a single multiplication table. If SSE is not enabled, then a
"double" table is employed that performs table lookup two bytes at a time. </li><br>
<li>
<b>Memory Consumption:</b> We try to keep the memory footprint of GF-Complete low. For example, the fastest
way to perform <b>multiply.w32()</b> in <em>GF(2<sup>32</sup>) </em> is to employ 1.75 MB of multiplication tables (see Section 7.4
below). We do not include this as a default, however, because we want to keep the default memory consumption
of GF-Complete low.
</li>

</ul>






<br/>

6 &nbsp &nbsp  <em>  THE DEFAULTS     </em>   <span id="index_number">14  </span> <br><br><br>

<ul>

3. &nbsp <b>Compatibility with "standard" implementations:</b> While there is no <em>de facto</em> standard of Galois Field arithmetic,
most libraries implement the same fields. For that reason, we have not selected composite fields, alternate
polynomials or memory layouts for the defaults, even though these would be faster. Again, see section 7.7 for
more information.

</ul>

<p>Table 1 shows the default methods used for each power-of-two word size, their alignment parameters <em>s</em> and <em> t,</em> their
memory consumption and their rough performance. The performance tests are on an Intel Core i7-3770 running at
3.40 GHz, and are included solely to give a flavor of performance on a standard microprocessor. Some processors
will be faster with some techniques and others will be slower, so we only put numbers in so that you can ballpark it.
For other values of <em>w</em> between 1 and 31, we use table lookup when w &#8804 8, discrete logarithms when w &#8804 16 and
"Bytwo<sub>p</sub>" for w &#8804 32. </p>
<br><br>
<center> With SSE 
<div id="data1">
<table cellpadding="6" cellspacing="0">
<tr>
<th>w </th><th class="double_border" >Memory <br> Usage </br> </th><th>multiply() <br> Implementation</th><th>Performance <br>(Mega Ops / s) </th><th>multiply region() <br> Implementation </th>
<th>s </th> <th>t </th> <th> Performance <br>(MB/s)</th>
</tr>
<tr>
<td>4 </td><td class="double_border"><1K </td><td>Table</td><td>501</td><td>Table</td>
<td>16 </td><td>16 </td> <td>11,659</td> </tr>

<tr>
<td>8 </td><td class="double_border">136K </td><td>Table</td><td>501</td><td>Split Table (8,4)</td>
<td>16 </td><td>16 </td> <td>11,824</td> </tr>

<tr>
<td>16 </td><td class="double_border">896K </td><td>Log</td><td>260</td><td>Split Table (16,4)</td>
<td>32 </td><td>16 </td> <td>7,749</td> </tr>

<tr>
<td>32 </td><td class="double_border"><1K </td><td>Carry-Free</td><td>48</td><td>Split Table (32,4)</td>
<td>64 </td><td>16 </td> <td>5,011</td> </tr>

<tr>
<td>64 </td><td class="double_border">2K </td><td>Carry-Free</td><td>84</td><td>Split Table (64,4)</td>
<td>128 </td><td>16 </td> <td>2,402</td> </tr>

<tr>
<td>128 </td><td class="double_border">64K </td><td>Carry-Free</td><td>48</td><td>Split Table (128,4)</td>
<td>16 </td><td>16 </td> <td>833</td> </tr>
</table></div>


<div id="data1">
<center>Without SE </center>
<table cellpadding="6" cellspacing="0">
<tr>
<th>w </th><th>Memory <br> Usage </br> </th><th>multiply() <br> Implementation</th><th>Performance <br>(Mega Ops / s) </th><th>multiply region() <br> Implementation </th>
<th>s </th> <th>t </th> <th> Performance <br>(MB/s)</th>
</tr>
<tr>
<td>4 </td><td>4K </td><td>Table</td><td>501</td><td>Double Table</td>
<td>16 </td><td>16 </td> <td>11,659</td> </tr>

<tr>
<td>8 </td><td>128K </td><td>Table</td><td>501</td><td>Table</td>
<td>1 </td><td>1 </td> <td>1,397</td> </tr>

<tr>
<td>16 </td><td>896K </td><td>Log</td><td>266</td><td>Split Table (16,8)</td>
<td>32 </td><td>16 </td> <td>2,135</td> </tr>

<tr>
<td>32 </td><td>4K </td><td>Bytwo<sub>p</sub></td><td>19</td><td>Split Table (32,4)</td>
<td>4 </td><td>4 </td> <td>1,149</td> </tr>

<tr>
<td>64 </td><td>16K </td><td>Bytwo<sub>p</sub></td><td>9</td><td>Split Table (64,4)</td>
<td>8 </td><td>8 </td> <td>987</td> </tr>

<tr>
<td>128 </td><td>64K </td><td>Bytwo<sub>p</sub></td><td>1.4</td><td>Split Table (128,4)</td>
<td>16 </td><td>8 </td> <td>833</td> </tr>
</table>
</div>
</center>
<br><br>
Table 1: The default implementations, memory consumption and rough performance when w is a power of two. The
variables s and t are alignment variables described in Section 5.
<p>
A few comments on Table 1 are in order. First, with SSE, the performance of <b>multiply()</b> is faster when <em> w </em> = 64
than when<em> w </em> = 32. That is because the primitive polynomial for <em> w  </em>= 32, that has historically been used in Galois
Field implementations, is sub-ideal for using carry-free multiplication (PCLMUL). You can change this polynomial
(see section 7.7) so that the performance matches <em>w </em> = 64. </p>
<p>
The region operations for <em> w  </em>= 4 and <em>w </em>= 8 without SSE have been selected to have a low memory footprint. There
are better options that consume more memory, or that only work on large memory regions (see section 6.1.5).
</p>

There are times that you may want to stray from the defaults. For example:
<ul>
<li>
You may want better performance.
</li>

</ul>










<br/>

6 &nbsp &nbsp  <em>  THE DEFAULTS     </em>   <span id="index_number">15  </span> <br><br><br>

<ul>
<li>You may want a lower memory footprint.</li>
<li>You may want to use a different Galois Field or even a ring.</li>
<li>You only care about multiplying a region by the value two.</li>

</ul>


<p>
Our command line tools allow you to deviate from the defaults, and we have two C functions <b>-gf_init_hard()</b>
and <b>create_gf_from_argv()</b> that can be called from application code to override the default methods. There are six
command-line tools that can be used to explore the many techniques implemented in GF-Complete: </p>

<ul><br>

<li> <b>gf_methods</b> is a tool that enumerates most of the possible command-line arguments that can be sent to the other
tools</li><br>
<li> <b>gf_mult</b> and <b>gf_div</b> are explained above. You may change the multiplication and division technique in these
tools if you desire</li><br>
<li> <b>gf_unit</b> performs unit tests on a set of techniques to verify correctness</li><br>
<li> <b> gf_time measures </b> the performance of a particular set of techniques</li><br>
<li> <b>time_tool.sh </b> makes some quick calls to <b>gf_time</b> so that you may gauge rough performance.</li><br>
<li> <b>gf_poly</b> tests the irreducibility of polynomials in a Galois Field</li><br>
</ul>


<p>To change the default behavior in application code, you need to call <b>gf_init_hard()</b> rather than <b>gf_init_easy().</b>
Alternatively, you can use <b>create_g_from_argv(),</b> included from <b>gf_method.h,</b> which uses an <b>argv</b>-style array of
strings to specify the options that you want. The procedure in <b>gf_method.c</b> parses the array and makes the proper
<b>gf_init_hard()</b> procedure call. This is the technique used to parse the command line in <b> gf_mult, gf_div, gf_unit </b><em>et al.</em> </p>


<h2>6.1.1 Changing the Components of a Galois Field with create <b>gf_from_argv()</b> </h2>
There are five main components to every Galois Field instance:
<ul>
<li> <em>w </em> </li>
<li> Multiplication technique </li>
<li> Division technique  </li>
<li> Region technique(s) </li>
<li> Polynomial </li>
</ul>

<p>The procedures <b>gf_init_hard()</b> and <b> create_gf_from_argv()</b> allow you to specify these parameters when you create
your Galois Field instance. We focus first on <b>create_gf_from_argv(),</b> because that is how the tools allow you to specify
the components. The prototype of <b>create_gf_from_argv()</b> is as follows: </p><br>

<div id="number_spacing">
int create_gf_from_argv(gf_t *gf, int w, int argc, char **argv, int starting);<br><br> </div>

You pass it a pointer to a gf_t, which it will initialize. You specify the word size with the parameter <em><b>w,</b></em> and then you
pass it an <b>argc/argv</b> pair as in any C or C++ program. You also specify a <b>starting</b> argument, which is where in <b>argv</b>
the specifications begin. If it successfully parses <b>argc</b> and <b>argv,</b> then it creates the <b>gf_t</b> using <b>gf_init_hard()</b> (described
below in section 6.4). It returns one past the last index of <b>argv</b> that it considered when creating the <b>gf_t.</b> If it fails, then
it returns zero, and the <b>gf_t</b> is unmodified.



<p>For example, <b>gf_mult.c</b> calls create gf_from_argv() by simply passing <b>argc</b> and <b>argv</b> from its <b>main()</b> declaration,
and setting starting to 4.</p>








<br/>

6 &nbsp &nbsp  <em>  THE DEFAULTS     </em>   <span id="index_number">16  </span> <br><br><br>

<p>
To choose defaults, <b>argv[starting]</b> should equal "-". Otherwise, you specify the component that you are changing
with "-m" for multiplication technique, "-d" for division technique, "-r" for region technique, and "-p" for the
polynomial. You may change multiple components. You end your specification with a single dash. For example, the
following call multiplies 6 and 5 in <em>GF(2<sup>4</sup>)</em> with polynomial 0x19 using the "SHIFT" technique for multiplication
(we'll explain these parameters later):
</p><br><br>

<div id="number_spacing">
UNIX> ./gf_mult 6 5 4 -p 0x19 -m SHIFT -<br>
7 <br>
UNIX> <br><br>
</div>

<p>If <b>create_gf_from_argv()</b> fails, then you can call the procedure <b>gf_error(),</b> which prints out the reason why <b>create_
gf_from_argv()</b> failed. </p>


<h2>6.1.2 Changing the Polynomial </h2>

Galois Fields are typically implemented by representing numbers as polynomials with binary coefficients, and then
using the properties of polynomials to define addition and multiplication. You do not need to understand any of that to
use this library. However, if you want to learn more about polynomial representations and how they construct fields,
please refer to The Paper.

<p>Multiplication is based on a special polynomial that we will refer to here as the "defining polynomial." This
polynomial has binary coefficients and is of degree <em> w.</em> You may change the polynomial with "-p" and then a number
in hexadecimal (the leading "0x" is optional). It is assumed that the <em>w</em>-th bit of the polynomial is set - you may include
it or omit it. For example, if you wish to set the polynomial for GF(2<sup>16</sup>) to x<sup>16</sup> + x<sup>5</sup> + x<sup>3</sup> + x<sup>2</sup> + 1, rather than its
default of x<sup>16</sup> + x<sup>12</sup> + x<sup>3</sup> + x + 1, you may say "-p 0x1002d," "-p 1002d," "-p 0x2d" or "-p 2d."
We discuss changing the polynomial for three reasons in other sections: </p>
<ul>
<li>Leveraging carry-free multiplication (section 7.7). </li>
<li>Defining composite fields (section 7.6). </li>
<li>Implementing rings (section 7.8.1). </li>

</ul>

<p>
Some words about nomenclature with respect to the polynomial. A Galois Field requires the polynomial to be
<em>irreducible </em>.. That means that it cannot be factored. For example, when the coefficients are binary, the polynomial x<sup>5</sup>+
x<sup>4</sup>+x+1 may be factored as (x<sup>4</sup>+1)(x+1). Therefore it is not irreducible and cannot be used to define a Galois Field.
It may, however, be used to define a ring. Please see section 7.8.1 for a discussion of ring support in GF-Complete. </p>
<p>
There is a subset of irreducible polynomials called primitive. These have an important property that one may enumerate
all of the elements of the field by raising 2 to successive posers. All of the default polynomials in GF-Complete 
are primitive. However, so long as a polynomial is irreducible, it defines a Galois Field. Please see section 7.7 for a
further discussion of the polynomial. </p>

<p>
One thing that we want to stress here is that changing the polynomial changes the field, so fields with different
polynomialsmay not be used interchangeably. So long as the polynomial is irreducible, it generates a Galois Field that
is isomorphic to all other Galois Fields; however the multiplication and division of elements will differ. For example,
the polynomials 0x13 (the default) and 0x19 in <em>GF(2<sup>4</sup>) </em> are both irreducible, so both generate valid Galois Fields.
However, their multiplication differs: </p><br>

<div id="number_spacing">
UNIX> gf_mult 8 2 4 -p 0x13 - <br>
3 <br>
UNIX> gf_mult 8 2 4 -p 0x19 - <br>
9 <br>
</div>









<br/>

6 &nbsp &nbsp  <em>  THE DEFAULTS     </em>   <span id="index_number">17  </span> <br><br><br>

<div id="number_spacing">
UNIX> gf_div 3 8 4 -p 0x13 -<br>
2 <br>
UNIX> gf_div 9 8 4 -p 0x19 - <br>
2 <br>
UNIX> <br>

</div>


<h3>6.1.3 &nbsp &nbsp Changing the Multiplication Technique </h3>
The following list describes the multiplication techinques that may be changed with "-m". We keep the description
here brief. Please refer to The Paper for detailed descriptions of these techniques.<br><br>


<li><b> "TABLE:" </b> Multiplication and division are implemented with tables. The tables consume quite a bit of memory
(2<sup>w</sup> &#215 2 <sup>w</sup> &#215  <sup>w</sup>/
8  bytes), so they are most useful when <em>w</em> is small. Please see <b>"SSE," "LAZY," "DOUBLE"</b> and

<b>"QUAD"</b> under region techniques below for further modifications to <b>"TABLE"</b> to perform <b>multiply_region()</b></li><br>


<li> <b>"LOG:"</b> This employs discrete (or "Zeph") logarithm <b>tables</b> to implement multiplication and division. The
memory usage is roughly (3 &#215 2<sup>w</sup> &#215 w /
8  bytes), so they are most useful when w is small, but they tolerate
larger <em>w</em> than <b>"TABLE."</b> If the polynomial is not primitive (see section 6.1.2), then you cannot use <b>"LOG"</b> as
an implementation. In that case,<b> gf_init_hard()</b> or <b>create_gf_from_argv()</b> will fail</li><br>


<li><b> "LOG_ZERO:"</b> Discrete logarithm tables which include extra room for zero entries. This more than doubles
the memory consumption to remove an <b>if</b> statement (please see [GMS08] or The Paper for more description). It
doesn’t really make a huge deal of difference in performance</li><br>

<li> <b>"LOG_ZERO_EXT:"</b> This expends even more memory to remove another <b>if</b> statement. Again, please see The
Paper for an explanation. As with <b>"LOG_ZERO,"</b> the performance difference is negligible</li><br>

<li> <b>"SHIFT:"</b> Implementation straight from the definition of Galois Field multiplication, by shifting and XOR-ing,
then reducing the product using the polynomial. This is <em>slooooooooow,</em> so we don’t recommend you use it</li><br>


<li> <b>"CARRY_FREE:"</b> This is identical to <b>"SHIFT,"</b> however it leverages the SSE instruction PCLMUL to perform
carry-freemultiplications in single instructions. As such, it is the fastest way to perform multiplication for large
values of <em>w</em> when that instruction is available. Its performance depends on the polynomial used. See The Paper
for details, and see section 7.7 below for the speedups available when <em>w </em>= 16 and <em>w</em> = 32 if you use a different
polynomial than the default one</li><br>


<li> <b>"BYTWO_p:"</b> This implements multiplication by successively multiplying the product by two and selectively
XOR-ing the multiplicand. See The Paper for more detail. It can leverage Anvin’s optimization that multiplies
64 and 128 bits of numbers in <em>GF(2<sup>w</sup>) </em> by two with just a few instructions. The SSE version requires SSE2</li><br>


<li> <b>"BYTWO_b:"</b> This implements multiplication by successively multiplying the multiplicand by two and selectively
XOR-ing it into the product. It can also leverage Anvin's optimization, and it has the feature that when
you're multiplying a region by a very small constant (like 2), it can terminate the multiplication early. As such,
if you are multiplying regions of bytes by two (as in the Linux RAID-6 Reed-Solomon code [Anv09]), this is
the fastest of the techniques, regardless of the value of <em>w.</em> The SSE version requires SSE2</li><br>


<li> <b>"SPLIT:"</b> Split multiplication tables (like the LR tables in [GMS08], or the SIMD tables for w &#8804 8 in [LHy08,
Anv09, PGM13b]). This argument must be followed by two more arguments, w<sub>a</sub> and w<sub>b</sub>, which are the index
sizes of the sub-tables. This implementation reduces the size of the table from <b>"TABLE,"</b> but requires multiple
</li><br>






<br/>

6 &nbsp &nbsp  <em>  THE DEFAULTS     </em>   <span id="index_number">18 </span> <br><br><br>
<ul>
table lookups. For example, the following multiplies 100 and 200 in <em>GF(2<sup>8</sup>) </em> using two 4K tables, as opposed 
to one 64K table when you use <b>"TABLE:"</b><br><br>
<div id="number_spacing">
UNIX> ./gf_mult 100 200 8 -m SPLIT 8 4 - <br>
79<br>
UNIX><br><br>
</div>

See section 7.4 for additional information on the arguments to <b>"SPLIT."</b> The SSE version typically requires
SSSE3.<br><br>


<li> <b>"GROUP:"</b> This implements the "left-to-right comb" technique [LBOX12]. I'm afraid we don't like that name,
so we call it <b>"GROUP,"</b> because it performs table lookup on groups of bits for shifting (left) and reducing (right).
It takes two additional arguments - g<sub>s,</sub> which is the number of bits you use while shifting (left) and g<sub>r</sub>, which
is the number of bits you use while reducing (right). Increasing these arguments can you higher computational
speed, but requires more memory. SSE version exists only for <em> w </em> = 128 and it requires SSE4. For more
description on the arguments g<sub>s</sub> and g<sub>r</sub>, see section 7.5. For a full description of <b>"GROUP"</b> algorithm, please
see The Paper.
</li><br>

<li> <b>"COMPOSITE:"</b> This allows you to perform operations on a composite Galois Field, <em> GF((2<sup>l</sup>)<sup>k</sup>)</em> as described
in [GMS08], [LBOX12] and The Paper. The field size <em>w </em> is equal to <em>lk.</em> It takes one argument, which is <em>k,</em> and
then a specification of the base field. Currently, the only value of <em>k</em> that is supported is two. However, that may
change in a future revision of the library. </li><br>


In order to specify the base field, put appropriate flags after specifying <em>k.</em> The single dash ends the base field,
and after that, you may continue making specifications for the composite field. This process can be continued
for multiple layers of <b>"COMPOSITE."</b> As an example, the following multiplies 1000000 and 2000000
in <em>GF((2<sup>16</sup>)<sup>2</sup>),</em> where the base field uses <b>BYTWO_p</b> for multiplication: <br><br>
<center>./gf_mult 1000000 2000000 32 -m COMPOSITE 2 <span style="color:red">-m BYTWO_p - -</span> </center><br>

In the above example, the red text applies to the base field, and the black text applies to the composite field.
Composite fields have two defining polynomials - one for the composite field, and one for the base field. Thus, if
you want to change polynomials, you should change both. The polynomial for the composite field must be of the
form x<sup>2</sup>+sx+1, where s is an element of <em>GF(2<sup>k</sup>).</em> To change it, you specify s (in hexadecimal)with "-p." In the
example below, we multiply 20000 and 30000 in <em>GF((2<sup>8</sup>)<sup>2</sup>) </em>, setting s to three, and using x<sup>8</sup>+x<sup>4</sup>+x<sup>3</sup>+x<sup>2</sup>+1
as the polynomial for the base field: <br><br>

<center>./gf_mult 20000 30000 16 -m COMPOSITE 2 <span style="color:red">-p 0x11d </span> - -p 0x3 - </center> <br><br>

If you use composite fields, you should consider using <b>"ALTMAP"</b> as well. The reason is that the region
operations will go much faster. Please see section 7.6.<br><br>
As with changing the polynomial, when you use a composite field, <em> GF((2<sup>l</sup>)<sup>k</sup>)</em>, you are using a different field
than the "standard" field for <em> GF((2<sup>l</sup>)<sup>k</sup>)</em>. All Galois Fields are isomorphic to each other, so they all have the
desired properties; however, the fields themselves change when you use composite fields.<br><br>
</ul>
<p>
With the exception of <b>"COMPOSITE"</b>, only one multiplication technique can be provided for a given Galois
Field instance. Composite fields may use composite fields as their base fields, in which case the specification will be
recursive. </p>








<br/>

6 &nbsp &nbsp  <em>  THE DEFAULTS     </em>   <span id="index_number">19 </span> <br><br><br>

<h3>6.1.4 &nbsp &nbsp &nbsp Changing the Division Technique </h3>

There are two techniques for division that may be set with "-d". If "-d" is not specified, then appropriate defaults
are employed. For example, when the multiplication technique is <b>"TABLE,"</b> a table is created for division as well as
multiplication. When <b>"LOG"</b> is specified, the logarithm tables are used for division. With <b>"COMPOSITE,"</b> a special
variant of Euclid's algorithm is employed that performs division using multiplication and division in the base field.
Otherwise, Euclid's algorithm is used. Please see The Paper for a description of Euclid's algorithm applied to Galois
Fields.

<p>If you use "-d", you must also specify the multiplication technique with "-m." </p>
<p>To force Euclid's algorithm instead of the defaults, you may specify it with "-d EUCLID." If instead, you would
rather convert elements of a Galois Field to a binary matrix and find an element's inverse by inverting the matrix,
then specify "-d MATRIX." In all of our tests, <b>"MATRIX"</b> is slower than <b>"EUCLID." "MATRIX" </b> is also not defined
for <em>w </em> > 32.
</p>


<h3>6.1.5  &nbsp&nbsp&nbsp Changing the Region Technique </h3>
The following are the region multiplication options ("-r"):
<ul>
<li>
<b>"SSE:"</b> Use SSE instructions. Initialization will fail if the instructions aren't supported. Table 2 details the
multiplication techniques which can leverage SSE instructions and which versions of SSE are required. </li><br>

<center>
<div id="data1">
<table cellpadding="6" cellspacing="0" style="text-align:center;font-size:19px">
<tr>
<th>Multiplication <br> Technique</th><th>multiply() </th><th>multiply_region() </th><th>SSE Version </th><th>Comments</th>

</tr>
<tr>
<td><b>"TABLE"</b></td><td >- </td><td>Yes</td><td>SSSE3</td><td>Only for <em>GF(2<sup>4</sup>). </em></td>

<tr>
<td><b>"SPLIT"</b></td><td>-</td><td>Yes</td><td>SSSE3</td><td>Only when the second argument equals 4.</td>

<tr>
<td><b>"SPLIT"</b></td><td>- </td><td>Yes</td><td>SSE4</td><td>When <em>w </em> = 64 and not using <b>"ALTMAP".</b></td>

<tr>
<td><b>"BYTWO_p"</b></td><td>- </td><td>Yes</td><td>SSE2</td><td></td>

<tr>
<td><b>"BYTWO_p"</b></td><td>- </td><td>Yes</td><td>SSE2</td><td></td>

</table></div> <br><br>
Table 2: Multiplication techniques which can leverage SSE instructions when they are available.
</center> <br><br>












<li> <b>"NOSSE:"</b> Force non-SSE version </li><br>

<li> <b> "DOUBLE:"</b> Use a table that is indexed on two words rather than one. This applies only to <em>w  </em> = 4, where
the table is indexed on bytes rather than 4-bit quantities, and to <em>w </em> = 8, where the table is indexed on shorts
rather than bytes. In each case, the table lookup performs two multiplications at a time, which makes region
multiplication faster. It doubles the size of the lookup table. </li><br>

<li> <b>"QUAD:"</b> Use a table that is indexed on four words rather than two or one. This only applies to <em>w </em> = 4, where
the table is indexed on shorts. The "Quad" table may be lazily created or created ahead of time (the default). If
the latter, then it consumes 2<sup>4</sup> &#215 2<sup>16</sup> &#215 2 = 2 MB of memory. </li><br>

<li> <b> "LAZY:"</b> Typically it's clear whether tables are constructed upon initialization or lazily when a region operation
is performed. There are two times where it is ambiguous: <b>"QUAD"</b> when <em>w </em> = 4 and <b>"DOUBLE"</b> when <em>w </em> = 8.
If you don't specify anything, these tables are created upon initialization, consuming a lot of memory. If you
specify <b>"LAZY,"</b> then the necessary row of the table is created lazily when you call <b>"multiply_region().</b>
</li>

</ul>











<br/>

6 &nbsp &nbsp  <em>  THE DEFAULTS     </em>   <span id="index_number">20 </span> <br><br><br>
<ul>

<li> <b>"ALTMAP:"</b> Use an alternate mapping, where words are split across different subregions of memory. There
are two places where this matters. The first is when implementing "<b>SPLIT</b> <em>w </em> 4" using SSE when <em>w </em> > 8. In
these cases, each byte of the word is stored in a different 128-bit vector, which allows the implementation to
better leverage 16-byte table lookups. See section 7.4 for examples, and The Paper or [PGM13b] for detailed
explanations.<br><br> </li>

The second place where it matters is when using <b>"COMPOSITE."</b> In this case, it is advantageous to split each
memory region into two chunks, and to store half of each word in a different chunk. This allows us to call
<b>region_multiply() </b> recursively on the base field, which is <em>much </em> faster than the alternative. See Section 7.6 for
examples, and The Paper for an explanation.<br><br>

It is important to note that with <b>"ALTMAP,"</b> the words are not "converted" from a standard mapping to an
alternate mapping and back again. They are assumed to always be in the alternate mapping. This typically
doesn't matter, so long as you always use the same <b>"ALTMAP"</b> calls. Please see section 7.9 for further details
on <b>"ALTMAP,"</b> especially with respect to alignment.<br><br>

<li> <b>"CAUCHY:"</b> Break memory into <em>w </em> subregions and perform only XOR's as in Cauchy Reed-Solomon coding
[BKK<sup>+</sup>95] (also described in The Paper). This works for <em>any</em> value of <em>w </em> &#8804 32, even those that are not
powers of two. If SSE2 is available, then XOR's work 128 bits at a time. For <b>"CAUCHY"</b> to work correctly,
<em>size</em> must be a multiple of <em>w </em>.</li> </ul>



<p>It is possible to combine region multiplication options. This is fully supported as long as <b>gf_methods</b> has the combination
listed. If multiple region options are required, they should be specified independently (as flags for <b>gf_init_hard()</b>
and independent options for command-line tools and <b>create_gf_from_argv()).</b> </p>


<h3>6.2  &nbsp&nbsp&nbspDetermining Supported Techniques with gf_methods </h3>


The program <b>gf_methods</b> prints a list of supported methods on standard output. It is called as follows:<br><br>
<div id="number_spacing">
<center>./gf_methods <em>w </em> -BADC -LUMDRB <br><br> </center> </div>

The first argument is <em>w </em>, which may be any legal value of <em>w </em>. The second argument has the following flags: <br><br>
<ul>

<li> <b>"B:"</b> This only prints out "basic" methods that are useful for the given value of <em>w </em>. It omits <b>"SHIFT"</b> and other
methods that are never really going to be useful.</li><br>

<li> <b> "A:"</b> In constrast, this specifies to print "all" methods. </li><br>

<li> <b>"D:"</b> This includes the <b>"EUCLID"</b> and <b>"MATRIX"</b> methods for division. By default, they are not included. </li><br>

<li> <b>"C:"</b> This includes the <b>"CAUCHY"</b> methods for region multiplication. By default, it is not included.</li> <br>
</ul>
<p>
You may specify multiple of these as the second argument. If you include both <b>"B"</b> and <b>"A,"</b> then it uses the last
one specified. </p>
<p>
The last argument determines the output format of <b>gf_methods.</b> If it is <b>"L,"</b> then it simply lists methods. If it
is <b>"U,"</b> then the output contains <b>gf_unit</b> commands for each of the methods. For the others, the output contains
<b>gf_time_tool.sh</b> commands for <b>M </b>ultiplication,<b>D</b>ivision,<b>R</b>egion multiplications with multiple buffer sizes, and the
<b>B</b>est region multiplication. </p>
<p>
<b>gf_methods</b> enumerates combinations of flags, and calls <b>create_gf_from_argv()</b> to see if the combinations are
supported. Although it enumerates a large number of combinations, it doesn't enumerate all possible parameters for
<b>"SPLIT," "GROUP"</b> or <b>"COMPOSITE."</b> </p>

<p>Some examples of calling <b>gf_methods</b> are shown below in section 6.3.2. </p>







<br/>

6 &nbsp &nbsp  <em>  THE DEFAULTS     </em>   <span id="index_number">21 </span> <br><br><br>


<h3>6.3 Testing with <b>gf_unit </b>, <b>gf_time </b>, and time_tool.sh </h3>



<b>gf_unit </b> and <b>gf_time </b> may be used to verify that a combination of arguments works correctly and efficiently on your
platform. If you plan to stray from the defaults, it is probably best to run both tools to ensure there are no issues with
your environment. <b>gf_unit </b> will run a set of unit tests based on the arguments provided to the tool, and <b>gf_time </b> will
time Galois Field methods based on the provided arguments.<br>
The usage of gf_ unit is:<br><br>
<div id="number_spacing">
<b>gf_unit </b> w tests seed method<br><br> </div>
The usage of gf_ time is:<br><br>
<div id="number_spacing">
<b>gf_time </b> w tests seed buffer-size iterations method<br><br>
</div>



The seed is an integer- negative one uses the current time. The tests are specified by a listing of characters. The
following tests are supported (All are supported by <b>gf_time.</b> Only ', 'S' and 'R' are supported by <b>gf_unit</b>):<br><br>

<ul>
<li> <b>'M':</b> Single multiplications</li><br>
<li> <b> 'D':</b> Single divisions</li><br>
<li> <b> 'I':</b> Single inverses</li><br>
<li> <b>'G': </b> Region multiplication of a buffer by a random constant</li><br>
<li> <b>'0': </b> Region multiplication of a buffer by zero (does nothing and<b>bzero()</b>)</li><br>
<li> <b>'1': </b> Region multiplication of a buffer by one (does <b>memcpy()</b> and <b>XOR</b>)</li><br>
<li> <b>'2': </b> Region multiplication of a buffer by two – sometimes this is faster than general multiplication</li><br>
<li> <b>'S':</b> All three single tests</li><br>
<li> <b>'R':</b> All four region tests</li><br>
<li> <b>'A':</b> All seven tests</li><br>
</ul>





<p>Here are some examples of calling <b>gf_unit</b> and <b>gf_time</b> to verify that <b>"-m SPLIT 32 4 -r ALTMAP -"</b> works
in <em>GF(2<sup>32</sup>),</em> and to get a feel for its performance. First, we go to the test directory and call <b>gf_unit:</b> </p><br><br>


<div id="number_spacing">
UNIX> cd test <br>
UNIX> ./gf_unit 32 A -1 -m SPLIT 32 4 -r ALTMAP - <br>
Args: 32 A -1 -m SPLIT 32 4 -r ALTMAP - / size (bytes): 684 <br>
UNIX> <br><br>
</div>

<b>gf_unit</b> reports on the arguments and how may bytes the <b>gf_t</b> consumes. If it discovers any problems or inconsistencies
with multiplication, division or region multiplication, it will report them. Here, there are no problems.
Next, we move to the <b>tools</b> directory and run performance tests on a 10K buffer, with 10,000 iterations of each test:<br><br>


UNIX> cd ../tools <br>
UNIX> ./gf_time 32 A -1 10240 10000 -m SPLIT 32 4 -r ALTMAP -<br>
Seed: 1388435794 <br>
<div id="number_spacing">
<table cellpadding="0" cellspacing="25" style="font-size:19px,font-family: 'Roboto Condensed', sans-serif;
">

<tr>

<td>Multiply:</td> <td>4.090548 s</td> <td> Mops: </td> <td> 24.414 </td> <td>5.968 Mega-ops/s </td> </tr>
<tr><td>Divide:</td> <td> 37.794962 s </td> <td>Mops: </td> <td> 24.414 </td> <td>0.646 Mega-ops/s </td> </tr>
<tr><td>Inverse:</td> <td> 33.709875 s </td> <td> Mops: </td> <td> 24.414 </td> <td> 0.724 Mega-ops/s </td> </tr>
<tr><td>Region-Random: XOR: 0 </td> <td> 0.035210 s </td> <td> MB:</td> <td> 97.656 </td> <td> 2773.527 MB/s </td></tr>
<tr><td>Region-Random: XOR: 1 </td> <td> 0.036081 s</td> <td> MB:</td> <td> 97.656 </td> <td>2706.578 MB/s </td></tr>
<tr><td>Region-By-Zero:XOR: 0 </td> <td> 0.003199 s </tD> <td> MB: </td> <td>97.656 </td> <td> 30523.884 MB/s </td> </tr>
<tr><td>Region-By-Zero: XOR: 1 </td> <td> 0.000626 s  </td> <td>MB: </td> <td> 97.656 </td> <td> 156038.095 MB/s </td></tr>

</table>
</div>










<br/>

6 &nbsp &nbsp  <em>  THE DEFAULTS     </em>   <span id="index_number">22 </span> <br><br><br>

<div id="number_spacing">
<table cellpadding="0" cellspacing="10" style="font-family: 'Roboto Condensed', sans-serif;
">

<tr>
<td>Region-By-One: XOR: 0</td> <td> 0.003810 s</td> <td> MB:</td> <td> 97.656 </td> <td> 25628.832 MB/s </td>
<tr><td>Region-By-One: XOR: 1 </td> <td> 0.008363 s </td> <td> MB:</td> <td> 97.656 </tD> <td>11677.500 MB/s </td></tr>

<tr><td>Region-By-Two: XOR: 0 </td> <td>0.032942 s  </td> <td>MB: </td> <td> 97.656 </td> <td> 2964.486 MB/s </td> </tr>
<tr><td>Region-By-Two: XOR: 1 </td> <td> 0.033488 s </td> <td> MB: </td> <td> 97.656 </td> <td> 2916.153 MB/s </td> </tr> </table>
</div>
UNIX><br><br>

<p>The first column of output displays the name of the test performed. Region tests will test with and without the XOR
flag being set (see Section 4.3 for an example). The second column displays the total time the test took to complete
measured in seconds (s). The third column displays the size of the test measured in millions of operations (Mops) for
single tests and in Megabytes (MB) for the region tests. The final column displays the speed of the tests calculated
from the second and third columns, and is where you should look to get an idea of a method's performance.</p>
<p>
If the output of <b>gf_unit</b> and <b>gf_time</b> are to your satisfaction, you can incorporate the method into application code
using create <b>gf_from_argv()</b> or <b>gf_init hard().</b></p>
<p>
The performance of "Region-By-Zero" and "Region-By-One" will not change from test to test, as all methods make
the same calls for these. "Region-By-Zero" with "XOR: 1" does nothing except set up the tests. Therefore, you may
use it as a control.</p>

<h3>6.3.1 &nbsp &nbsp &nbsp time_tool.sh </h3> 

Finally, the shell script <b>time_tool.sh</b> makes a bunch of calls to <b>gf_time</b> to give a rough estimate of performance. It is
called as follows:<br><br>
usage sh time_tool.sh M|D|R|B w method<br><br>


<p>The values for the first argument are <b>MDRB,</b> for <b>M</b>ultiplication, <b>D</b>ivision,<b>R</b>egion multiplications with multiple
buffer sizes, and the <b>B</b>est region multiplication. For the example above, let's call <b>time_tool.sh</b> to get a rough idea of
performance: </p><br><br>

<div id="number_spacing">
UNIX> sh time_tool.sh M 32 -m SPLIT 32 4 -r ALTMAP - <br>
M speed (MB/s): 6.03 W-Method: 32 -m SPLIT 32 4 -r ALTMAP - <br>
UNIX> sh time_tool.sh D 32 -m SPLIT 32 4 -r ALTMAP - <br>
D speed (MB/s): 0.65 W-Method: 32 -m SPLIT 32 4 -r ALTMAP - <br>
UNIX> sh time_tool.sh R 32 -m SPLIT 32 4 -r ALTMAP - <br>

<table cellpadding="0" cellspacing="10" style="font-family: 'Roboto Condensed', sans-serif;
">

<tr>
<td>Region Buffer-Size:</td> <td> 16K (MB/s):</td> <td>3082.91</td><td> W-Method: 32 </td> <td>-m SPLIT 32 4 </td> <td>-r ALTMAP -</td> </tr>
<tr><td>Region Buffer-Size:</td> <td>32K (MB/s): </td> <td>3529.07 </td><td> W-Method: 32 </td> <td>-m SPLIT 32 4 </td> <td>-r ALTMAP -</td> </tr>
<tr><td>Region Buffer-Size:</td> <td>64K (MB/s): </td> <td> 3749.94</td><td> W-Method: 32 </td> <td>-m SPLIT 32 4 </td> <td>-r ALTMAP -</td> </tr>
<tr><td>Region Buffer-Size:</td> <td>128K (MB/s):</td> <td>3861.27 </td> <td>W-Method: 32 </td> <td>-m SPLIT 32 4 </td> <td>-r ALTMAP -</td> </tr>
<tr><td>Region Buffer-Size:</td> <td>512K (MB/s):</td> <td>3820.82 </td><td> W-Method: 32 </td> <td>-m SPLIT 32 4 </td> <td>-r ALTMAP -</td> </tr>
<tr><td>Region Buffer-Size:</td> <td>1M (MB/s):</td> <td>3737.41 </td><td> W-Method: 32 </td> <td>-m SPLIT 32 4 </td> <td>-r ALTMAP -</td>  </tr>
<tr><td>Region Buffer-Size:</td> <td>2M (MB/s):</td> <td>3002.90 </td><td> W-Method: 32 </td> <td>-m SPLIT 32 4 </td> <td>-r ALTMAP -</td> </tr>
<tr><td>Region Buffer-Size:</td> <td>4M (MB/s): </td><td>2760.77</td><td> W-Method: 32 </td> <td>-m SPLIT 32 4 </td> <td>-r ALTMAP -</td> </tr>
<tr><td>Region Best (MB/s):</td><td> 3861.27</td><td> W-Method: 32 </td> <td>-m SPLIT 32 4 </td> <td>-r ALTMAP -</td> </tr>
</table>

UNIX> sh time_tool.sh B 32 -m SPLIT 32 4 -r ALTMAP - <br>
Region Best (MB/s): 3929.09  W-Method: 32  -m SPLIT 32 4 -r ALTMAP -</br>
UNIX><br><br>
</div>
<p>
We say that <b>time_tool.sh </b>is "rough" because it tries to limit each test to 5 ms or less. Thus, the time granularity
is fine, which means that the numbers may not be as precise as they could be were the time granularity to be course.
When in doubt, you should make your own calls to <b>gf_time</b> with a lot of iterations, so that startup costs and roundoff
error may be minimized. </p>








<br/>

6 &nbsp &nbsp  <em>  THE DEFAULTS     </em>   <span id="index_number">23 </span> <br><br><br>

<h3>6.3.2 &nbsp &nbsp &nbsp An example of gf_methods and time_tool.sh </h3><br><br>
Let's give an example of how some of these components fit together. Suppose we want to explore the basic techniques
in <em>GF(2<sup>32</sup>).</em> First, let's take a look at what <b>gf_methods</b> suggests as "basic" methods: <br><br>
<div id="number_spacing">
UNIX> gf_methods 32 -B -L <br>
w=32: - <br>
w=32: -m GROUP 4 8 - <br>
w=32: -m SPLIT 32 4 - <br>
w=32: -m SPLIT 32 4 -r ALTMAP - <br>
w=32: -m SPLIT 32 8 - <br>
w=32: -m SPLIT 8 8 - <br>
w=32: -m COMPOSITE 2 - - <br>
w=32: -m COMPOSITE 2 - -r ALTMAP - <br>
UNIX> <br><br>
</div>


<p>

You'll note, this is on my old Macbook Pro, which doesn't support (PCLMUL), so <b>"CARRY_FREE"</b> is not included
as an option. Now, let's run the unit tester on these to make sure they work, and to see their memory consumption: </p><br><br>

<div id="number_spacing">
UNIX> gf_methods 32 -B -U <br>
../test/gf_unit 32 A -1 - <br>
../test/gf_unit 32 A -1 -m GROUP 4 8 - <br>
../test/gf_unit 32 A -1 -m SPLIT 32 4 - <br>
../test/gf_unit 32 A -1 -m SPLIT 32 4 -r ALTMAP - <br>
../test/gf_unit 32 A -1 -m SPLIT 32 8 - <br>
../test/gf_unit 32 A -1 -m SPLIT 8 8 - <br>
../test/gf_unit 32 A -1 -m COMPOSITE 2 - - <br>
../test/gf_unit 32 A -1 -m COMPOSITE 2 - -r ALTMAP - <br>
UNIX> gf_methods 32 -B -U | sh <br>
Args: 32 A -1 - / size (bytes): 684 <br>
Args: 32 A -1 -m GROUP 4 8 - / size (bytes): 1296 <br>
Args: 32 A -1 -m SPLIT 32 4 - / size (bytes): 684 <br>
Args: 32 A -1 -m SPLIT 32 4 -r ALTMAP - / size (bytes): 684 <br>
Args: 32 A -1 -m SPLIT 32 8 - / size (bytes): 4268 <br>
Args: 32 A -1 -m SPLIT 8 8 - / size (bytes): 1839276 <br>
Args: 32 A -1 -m COMPOSITE 2 - - / size (bytes): 524648 <br>
Args: 32 A -1 -m COMPOSITE 2 - -r ALTMAP - / size (bytes): 524648 <br>
UNIX> <br> <br>
</div>
<p>
As anticipated, <b>"SPLIT 8 8"</b> consumes quite a bit of memory! Now, let's see how well they perform with both
single multiplications and region multiplications: </p> <br><br>
<div id="number_spacing">
UNIX> gf_methods 32 -B -M <br>
sh time_tool.sh M 32 - <br>
sh time_tool.sh M 32 -m GROUP 4 8  - <br>
sh time_tool.sh M 32 -m SPLIT 32 4 - <br>
sh time_tool.sh M 32 -m SPLIT 32 4 -r ALTMAP -<br>
sh time_tool.sh M 32 -m SPLIT 32 8 - <br>
sh time_tool.sh M 32 -m SPLIT 8 8 - <br>

</div>








<br/>

6 &nbsp &nbsp  <em>  THE DEFAULTS     </em>   <span id="index_number">24 </span> <br><br><br>

<div id="number_spacing">
sh time_tool.sh M 32 -m COMPOSITE 2 - <br>
sh time_tool.sh M 32 -m COMPOSITE 2 - -r ALTMAP <br>
UNIX> gf_methods 32 -B -M | sh
M speed (MB/s): 5.90 W-Method: 32 <br>
M speed (MB/s): 14.09 W-Method: 32 -m GROUP 4 8 <br>
M speed (MB/s): 5.60 W-Method: 32 -m SPLIT 32 4 <br>
M speed (MB/s): 5.19 W-Method: 32 -m SPLIT 32 4 -r ALTMAP <br>
M speed (MB/s): 5.98 W-Method: 32 -m SPLIT 32 8 <br>
M speed (MB/s): 22.10 W-Method: 32 -m SPLIT 8 8 <br>
M speed (MB/s): 34.98 W-Method: 32 -m COMPOSITE 2 - <br>
M speed (MB/s): 34.16 W-Method: 32 -m COMPOSITE 2 - -r ALTMAP <br>
UNIX> gf_methods 32 -B -B | sh
Region Best (MB/s): 2746.76 W-Method: 32 <br>
Region Best (MB/s): 177.06 W-Method: 32 -m GROUP 4 8 <br>
Region Best (MB/s): 2818.75 W-Method: 32 -m SPLIT 32 4 <br>
Region Best (MB/s): 3818.21 W-Method: 32 -m SPLIT 32 4 -r ALTMAP <br>
Region Best (MB/s): 728.68 W-Method: 32 -m SPLIT 32 8 <br>
Region Best (MB/s): 730.97 W-Method: 32 -m SPLIT 8 8 <br>
Region Best (MB/s): 190.20 W-Method: 32 -m COMPOSITE 2 - <br>
Region Best (MB/s): 1837.99 W-Method: 32 -m COMPOSITE 2 - -r ALTMAP <br>
UNIX>
</div>
<p>
The default is quite a bit slower than the best performing methods for both single and region multiplication. So
why are the defaults the way that they are? As detailed at the beginning of this chapter, we strive for lower memory
consumption, so we don't use <b>"SPLIT 8 8,"</b> which consumes 1.75MB.We don't implement alternate fields by default,
which is why we don't use <b>"COMPOSITE."</b> Finally, we don't implement alternate mappings of memory by default,
which is why we don't use "<b>-m SPLIT 32 4 -r ALTMAP -.</b>"</p>

<p>Of course, you may change these defaults if you please.</p>
<p>
<b>Test question:</b> Given the numbers above, it would appear that <b>"COMPOSITE"</b> yields the fastest performance of
single multiplication, while "SPLIT 32 4" yields the fastest performance of region multiplication. Should I use two
gf_t's in my application – one for single multiplication that uses <b>"COMPOSITE,"</b> and one for region multiplication
that uses <b>"SPLIT 32 4?"</b></p>
<p>
The answer to this is "no." Why? Because composite fields are different from the "standard" fields, and if you mix
these two <b>gf_t</b>'s, then you are using different fields for single multiplication and region multiplication. Please read
section 7.2 for a little more information on this.</p>

<h3>6.4 &nbsp &nbsp &nbspCalling gf_init_hard()</h3>

We recommend that you use <b>create_gf_from_argv()</b> instead of <b>gf_init_hard().</b> However, there are extra things that
you can do with <b>gf_init_hard().</b> Here's the prototype:<br><br>
<div id="number_spacing">
int gf_init_hard(gf_t *gf<br>
<div style="padding-left:100px">
int w<br>
int mult_type<br>
int region_type<br>
int divide_type<br>
uint64_t prim_poly<br>
int arg1<br>
int arg2<br>
</div>
</div>








<br/>

6 &nbsp &nbsp  <em>  THE DEFAULTS     </em>   <span id="index_number">25 </span> <br><br><br>
<div id="number_spacing">
<div style="padding-left:100px">
GFP base_gf, <br>
void *scratch_memory); </div><br><br>


The arguments mult type, region type and divide type allow for the same specifications as above, except the
types are integer constants defined in gf_complete.h: <br><br>
typedef enum {GF_MULT_DEFAULT,<br>
<div style="padding-left:124px">
GF_MULT_SHIFT<br>
GF_MULT_CARRY_FREE<br>
GF_MULT_GROUP<br>
GF_MULT_BYTWO_p<br>
GF_MULT_BYTWO_b<br>
GF_MULT_TABLE<br>
GF_MULT_LOG_TABLE<br>
GF_MULT_LOG_ZERO<br>
GF_MULT_LOG_ZERO_EXT<br>
GF_MULT_SPLIT_TABLE<br>
GF_MULT_COMPOSITE } gf_mult_type_t;<br><br>

</div>

#define GF_REGION_DEFAULT (0x0)<br>
#define GF_REGION_DOUBLE_TABLE (0x1) <br>
#define GF_REGION_QUAD_TABLE (0x2) <br>
#define GF_REGION_LAZY (0x4) <br>
#define GF_REGION_SSE (0x8) <br>
#define GF_REGION_NOSSE (0x10) <br>
#define GF_REGION_ALTMAP (0x20) <br>
#define GF_REGION_CAUCHY (0x40) <br><br>
typedef enum { GF_DIVIDE_DEFAULT<br>
<div style="padding-left:130px">GF_DIVIDE_MATRIX<br>
GF_DIVIDE_EUCLID } gf_division_type_t;<br><br>
</div>
</div>
<p>
You can mix the region types with bitwise or. The arguments to <b>GF_MULT_GROUP,GF_MULT_SPLIT_TABLE</b>
and <b>GF_MULT_COMPOSITE</b> are specified in arg1 and arg2. <b>GF_MULT_COMPOSITE</b> also takes a base field
in <b>base_gf.</b> The base field is itself a <b>gf_t,</b> which should have been created previously with <b>create_gf_fro_argv(),</b>
<b>gf_init_easy()</b> or <b>gf_init_hard().</b> Note that this <b>base_gf</b> has its own <b>base_gf</b> member and can be a composite field
itself.</p>
<p>
You can specify an alternate polynomial in <b>prim_poly.</b> For <em>w </em>&#8804 32, the leftmost one (the one in bit position <em>w</em>) is
optional. If you omit it, it will be added for you. For <em>w </em> = 64, there's no room for that one, so you have to leave it off.
For <em>w </em>= 128, your polynomial can only use the bottom-most 64 bits. Fortunately, the standard polynomial only uses
those bits. If you set <b>prim_poly</b> to zero, the library selects the "standard" polynomial.
</p>
<p>
Finally, <b>scratch_memory</b> is there in case you don't want <b>gf_init_hard()</b> to call <b>malloc()</b>. Youmay call <b>gf_scratch_size()</b>
to find out how much extra memory each technique uses, and then you may pass it a pointer for it to use in <b>scratc_memory.</b>
If you set scratch memory to NULL, then the extra memory is allocated for you with <b>malloc().</b> If you use <b>gf_init_easy()</b>
or <b>create_gf_from_argv(),</b> or you use <b>gf_init_hard()</b> and set <b>scratch_memory</b> to <b>NULL,</b> then you should call <b>gf_free()</b>
to free memory. If you use <b>gf_init_hard()</b> and use your own <b>scratch_memory</b> you can still call <b>gf_free(),</b> and it will
not do anything.</p>
<p>
Both <b>gf_init_hard()</b> and <b>gf_scratch_size()</b> return zero if the arguments don't specify a valid <b>gf_t.</b> When that happens,
you can call <b>gf_error()</b> to print why the call failed.</p>








<br/>


6 &nbsp &nbsp  <em>  FURTHER INFORMATION ON OPTIONS AND ALGORITHMS     </em>   <span id="index_number">26  </span> <br><br><br>


<p>We'll give you one example of calling <b>gf_ init_hard().</b> Suppose you want to make a <b>gf_ init_hard()</b> call to be
equivalent to "-m SPLIT 16 4 -r SSE -r ALTMAP -" and you want to allocate the scratch space yourself. Then you'd
do the following:</p><br><br>

<div id="number_spacing">
gf_t gf; <br>
void *scratch; <br>
int size; <br>
size = gf_scratch_size(16, GF_MULT_SPLIT_TABLE,<br>
GF_REGION_SSE | GF_REGION_ALTMAP,<br>
GF_DIVIDE_DEFAULT,<br>
16, 4); <br>
if (size == 0) { gf_error(); exit(1); } /* It failed. That shouldn’t happen */<br>
scratch = (void *) malloc(size); <br>
if (scratch == NULL) { perror("malloc"); exit(1); } <br>
if (!gf_init_hard(&gf, 16, GF_MULT_SPLIT_TABLE, <br>
GF_REGION_SSE | GF_REGION_ALTMAP, <br>
GF_DIVIDE_DEFAULT,<br>
0, 16, 4, NULL, scratch)) { <br>
gf_error(); <br>
exit(1); <br>
} <br>

</div>


<h3>6.5 &nbsp   &nbsp   gf_size() </h3>

You can call <b>gf_size(gf_t *gf)</b> to learn the memory consumption of the <b>gf_t.</b> It returns all memory consumed by the
<b>gf_t,</b> including the <b>gf_t</b> itself, any scratch memory required by the gf_ t, and the memory consumed by the sub-field
if the field is <b>"COMPOSITE."</b> If you provided your own memory to <b>gf_init_hard(),</b> it does not report the size of
this memory, but what the size should be, as determined by <b>gf_scratch size(). gf_ unit() </b> prints out the return value of
<b>gf_size()</b> on the given field.


<h2>7 &nbsp Further Information on Options and Algorithms </h2>
<h3>
7.1 &nbsp Inlining Single Multiplication and Division for Speed </h3>

Obviously, procedure calls are more expensive than single instructions, and the mechanics of multiplication in <b>"TABLE"</b>
and <b>"LOG"</b> are pretty simple. For that reason, we support inlining for <b>"TABLE"</b> when <em>w </em> = 4 and <em>w </em> = 8, and
for <b>"LOG"</b> when <em>w </em> = 16. We elaborate below.
<p>
When <em>w </em> = 4, you may inline multiplication and division as follows. The following procedures return pointers to
the multiplication and division tables respectively: </p> <br><br>

<div id="number_spacing">
uint8_t *gf_w4_get_mult_table(gf_t * gf);<br>
uint8_t *gf_w4_get_div_table(gf_t * gf);<br><br>
</div>
<p>The macro <b>Gf_W4_INLINE_MULTDIV </b>(<em>table, a, b</em>) then multiplies or divides <em>a </em> by <em>b</em> using the given table. This
of course only works if the multiplication technique is <b>"TABLE,"</b> which is the default for <em>w </em> = 4. If the multiplication
technique is not <b>"TABLE,"</b> then <b>gf_w4_get_mult_table()</b> will return <b>NULL.</b></p>








<br/>


6 &nbsp &nbsp  <em>  FURTHER INFORMATION ON OPTIONS AND ALGORITHMS     </em>   <span id="index_number">27  </span> <br><br><br>




<p>When <em>w </em> = 8, the procedures <b>gf_w8_et_mult_table()</b> and <b>gf_ w8_get_div_table(),</b> and the macro </p>

<b>GF_W8_INLINE_MULTDIV </b>(<em>table, a, b</em>) work identically to the <em>w </em> = 4 case.

<p>When <em>w </em> = 16, the following procedures return pointers to the logarithm table, and the two inverse logarithm tables
respectively: </p><br>

<div id="number_spacing">
uint16_t *gf_w16_get_log_table(gf_t * gf); <br>
uint16_t *gf_w16_get_mult_alog_table(gf_t * gf);<br>
uint16_t *gf_w16_get_div_alog_table(gf_t * gf);<br>

</div>
<br>
<p>
The first inverse logarithm table works for multiplication, and the second works for division. They actually point
to the same table, but to different places in the table. You may then use the macro <b>GF_W16_INLINE_MULT</b>(<em>log,
alog, a, b </em>) to multiply <em>a</em> and <em>b</em>, and the macro <b>GF_W16_INLINE_DIV </b>(<em>log, alog, a, b </em>) to divide a and b. Make
sure you use the <em>alog</em> table returned by <b>gf_w16_get_mult_alog_table()</b> for multiplication and the one returned by
<b>gf_w16_get_div_alog_table()</b> for division. Here are some timings: </p> <br><br>


UNIX> gf_time 4 M 0 10240 10240 - <br>
Seed: 0 <br>
Multiply: 0.228860 s Mops: 100.000 436.949 Mega-ops/s <br>
UNIX> gf_inline_time 4 0 10240 10240 <br>
Seed: 0 <br>
Inline mult: 0.096859 s Mops: 100.000 1032.424 Mega-ops/s <br>
UNIX> gf_time 8 M 0 10240 10240 - <br>
Seed: 0 <br>
Multiply: 0.228931 s Mops: 100.000 436.812 Mega-ops/s <br>
UNIX> gf_inline_time 8 0 10240 10240 <br>
Seed: 0 <br>
Inline mult: 0.114300 s Mops: 100.000 874.889 Mega-ops/s <br>
UNIX> gf_time 16 M 0 10240 10240 - <br>
Seed: 0 <br>
Multiply: 0.193626 s Mops: 50.000 258.229 Mega-ops/s <br>
UNIX> gf_inline_time 16 0 10240 10240 <br>
Seed: 0 <br>
Inline mult: 0.310229 s Mops: 100.000 322.342 Mega-ops/s <br>
UNIX> <br> <br>

<h3>
7.2 &nbsp &nbsp Using different techniques for single and region multiplication </h3>


You may want to "mix and match" the techniques. For example, suppose you'd like to use "-m SPLIT 8 8" for
<b>multiply()</b> in <em>GF(2<sup>32</sup>),</em> because it's fast, and you don't mind consuming all of that space for tables. However, for
<b>multiply_region(),</b> you'd like to use "-m SPLIT 32 4 -r ALTMAP," because that's the fastest way to implement
<b>multiply_region().</b> Unfortunately, There is no way to create a <b>gf_t</b> that does this combination. In this case, you should
simply create two <b>gf_t's,</b> and use one for <b>multiply()</b> and the other for <b>multiply_region().</b> All of the implementations
may be used interchangably with the following exceptions:

<ul>
<li>
<b>"COMPOSITE"</b> implements a different Galois Field. </li><br>

<li>If you change a field's polynomial, then the resulting Galois Field will be different. </li>

</ul>








<br/>


6 &nbsp &nbsp  <em>  FURTHER INFORMATION ON OPTIONS AND ALGORITHMS     </em>   <span id="index_number">28  </span> <br><br><br>

<ul>
<li>

If you are using <b>"ALTMAP"</b> to multiply regions, then the contents of the resulting regions of memory will
depend on the multiplication technique, the size of the region and its alignment. Please see section 7.9 for a
detailed explanation of this. </li>

<li>If you are using <b>"CAUCHY"</b> to multiply regions, then like <b>"ALTMAP,"</b> the contents of the result regions of
memory the multiplication technique and the size of the region. You don't have to worry about alignment. </li>

<h3>7.3 &nbsp &nbsp General <em>w </em>  </h3>
The library supports Galois Field arithmetic with 2 < <em>w </em> &#8804 32. Values of <em>w </em> which are not whole number powers of
2 are handled by the functions in <b>gf_wgen.c</b> . For these values of <em>w </em>, the available multiplication types are <b>"SHIFT,"
"BYT<em>w </em>O p," "BYT<em>w </em>O b," "GROUP," "TABLE"</b> and <b>"LOG." "LOG" </b> is only valid for <em>w </em> < 28 and <b>"TABLE"</b>

is only valid for <em>w </em> < 15. The defaults for these values of <em>w </em> are <b>"TABLE"</b> for <em>w </em> < 8, <b>"LOG"</b> for <em>w </em> < 16, and
<b>"BYT<em>w </em>O p"</b> for <em>w </em> < 32.<br><br>

<h3>7.4 Arguments to "SPLIT" </h3>

The "SPLIT" technique is based on the distributive property of multiplication and addition: <br><br>
<center>
a * (b + c) = (a * b) + (a * c). </center>
<br>
This property allo<em>w </em>s us to, for example, split an eight bit <em>w </em>ord into t<em>w </em>o four-bit components and calculate the product
by performing t<em>w </em>o table lookups in 16-element tables on each of the compoents, and adding the result. There is much
more information on <b>"SPLIT"</b> in The Paper. Here <em>w </em>e describe the version of <b>"SPLIT"</b> implemented in GF-Complete.

<p>
<b>"SPLIT"</b> takes t<em>w </em>o arguments, <em>w </em>hich are the number of bits in each component of a, <em>w </em>hich <em>w </em>e call <em>w </em><sub>a</sub>, and the
number of bits in each component of b, <em>w </em>hich <em>w </em>e call <em>w </em><sub>b.</sub> If the t<em>w </em>o differ, it does not matter <em>w </em>hich is bigger - the
library recognizes this and performs the correct implementation. The legal values of <em>w </em><sub>a</sub> and <em>w </em><sub>b</sub> fall into five categories:
</p><br>


<ol>
<li>
 <em>w </em><sub>a</sub> is equal to <em>w </em> and <em>w </em><sub>b</sub> is equal to four. In this case, b is broken up into <em>w </em>/4
four-bit <em>w </em>ords <em>w </em>hich are used
in 16-element lookup tables. The tables are created on demand in <b>multiply_region()</b> and the SSSE3 instruction

<b>mm_shuffle_epi8()</b> is leveraged to perform 16 lookups in parallel. Thus, these are very fast implementations.
<em>w </em>hen <em>w </em> &#8805 16, you should combine this <em>w </em>ith <b>"ALTMAP"</b> to get the best performance (see The Paper
or [PGM13b] for explanation). If you do this please see section 7.9 for information about <b>"ALTMAP"</b> and
alignment.<br><br>


If you don't use <b>"ALTMAP,"</b> the implementations for <em>w </em> &#8712 {16, 32, 64} convert the standard representation into
<b>"ALTMAP,"</b> perform the multiplication <em>w </em>ith <b>"ALTMAP"</b> and then convert back to the standard representation.
The performance difference using <b>"ALTMAP"</b> can be significant: <br><br><br>

<div id="number_spacing">
<center>
<div id="table_page28">
<table cellpadding="6" cellspacing="0" style="text-align:center;font-size:19px">
<tr>
<td> gf_time 16 G 0 1048576 100 -m SPLIT 16 4 -</td> <td>Speed = 8,389 MB/s </td> 
</tr>
<tr>
<td>gf_time 16 G 0 1048576 100 -m SPLIT 16 4 -r ALTMAP - </td> <td>Speed = 8,389 MB/s </td> 
</tr>

<tr>
<td>gf_time 32 G 0 1048576 100 -m SPLIT 32 4 -</td> <td> Speed = 5,304 MB/s</td> 
</tr>
<tr>
<td>gf_time 32 G 0 1048576 100 -m SPLIT 32 4 -r ALTMAP -</td> <td> Speed = 7,146 MB/s</td> 
</tr>


<tr>
<td>gf_time 64 G 0 1048576 100 -m SPLIT 64 4 - </td> <td>Speed = 2,595 MB/s </td> 
</tr>

<tr>
<td>gf_time 64 G 0 1048576 100 -m SPLIT 64 4 -r ALTMAP - </td> <td>Speed = 3,436 MB/s </td> 
</tr>
</div>



</table>
</div>









<br/>


6 &nbsp &nbsp  <em>  FURTHER INFORMATION ON OPTIONS AND ALGORITHMS     </em>   <span id="index_number">29  </span> <br><br><br>

<ol style="list-style-type:none">


<li>2. &nbsp w<sub>a</sub> is equal to <em>w </em> and w<sub>b</sub> is equal to eight. Now, b is broken into bytes, each of these is used in its own 256-element
lookup table. This is typically the best w<sub>a</sub>y to perform <b>multiply_region()</b> without SSE.</li> 
Because this is a region optimization, when you specify these options, you get a default <b>multiply()</b>  see
Table 1 for a listing of the defaults. See section 7.2 for using a different <b>multiply()</b> than the defaults.<br><br>


<li>
3. &nbsp w<sub>a</sub> is equal to <em>w </em> and <em>w </em><sub>b</sub> is equal to 16. This is only valid for <em>w </em> = 32 and <em>w </em> = 64. No<em>w </em>, b is broken into shorts,
each of these is used in its own 64K-element lookup table. This is typically slower than when <em>w </em><sub>b</suB> equals 8, and
requires more amortization (larger buffer sizes) to be effective. </li><br>


<li>4. &nbsp <em>w </em><sub>a</sub> and <em>w </em><sub>b</sub> are both equal to eight. Now  both <em>a</em> and <em>b</em> are broken into bytes, 
and the products of the various bytes
are looked up in multiple 256 &#215 256 tables. In <em>GF(2<sup>16</sup>),</em> there are three of these tables. In <em>GF(232),</em> there are
seven, and in <em>GF(2<sup>64</sup>)</em> there are fifteen. Thus, this implementation can be a space hog. How ever, for <em>w </em> = 32,
this is the fastest way to perform <b>multiply()</b> on some machines.
when this option is employed, <b>multiply_region()</b> is implemented in an identical fashion to when <em>w </em><sub>a</sub> = <em>w </em>
and <em>w </em><sub>b</sub> = 8. </li><br>

<li>5.&nbsp w<sub>a</sub> = 32 and w<sub>b</sub> = 2. (<em>w</em> = 32 only). I was playing with a different way to use <b>mm_shuffle_epi8().</b> It works,
but it's slower than when w<sub>b</sub> = 4.
</li>

</ul>



<h2>7.5 &nbsp&nbsp Arguments to "GROUP" </h3>

The <b>"GROUP"</b> multiplication option takes t<em>w </em>o arguments, g<sub>s</sub> and g<sub>r</sub>. It implements multiplication in the same manner
as <b>"SHIFT,"</b> except it uses a table of size 2<sup>gs</sup> to perform g<sup>s</sup> shifts at a time, and a table of size 2<sup>gr</sup> to perform g<sup>r</sup>
reductions at at time. The program <b>gf_methods</b> only prints the options 4 4 and 4 8 as arguments for <b>"GROUP."</b>
However, other values of g<sub>s</sub> and g<sub>r</sub> are legal and sometimes desirable: <br><br>

<ol>
<li>
 For <em>w </em> &#8804 32 and <em>w </em> = 64, any values of g<sub>s</sub> and g<sub>r</sub> may be used, so long as they are less than or equal to <em>w </em> and so
long as the tables fit into memory. There are four exceptions to this, listed belo<em>w </em>. </li><br>
<li> For <em>w </em> = 4, <b>"GROUP"</b> is not supported. </li><br>
<li> For <em>w </em> = 8, <b>"GROUP"</b> is not supported. </li><br>
<li> For <em>w </em> = 16, <b>"GROUP"</b> is only supported for gs = gr = 4. </li><br>
<li> For <em>w </em> = 128 <b>"GROUP"</b> only supports <em>g<sub>s</sub></em> = 4 and <em> g<sub>r</b> </em> &#8712 {4, 8, 16}.</li><br>
</ol>
<p>
The way that gs and gr impact performance is as follows. The <b>"SHIFT"</b> implementation works by performing a
carry-free multiplication in <em>w </em> steps, and then performing reduction in <em>w </em> steps. In "GROUP," the carry-free multiplication
is reduced to  <em>w /</em>g<sub>s</sub>steps, and the reduction is reduced to <em>w /</em>g<sub>r</sub>

. Both require tables. The table for the carry-free
multiplication must be created at the beginning of each <b>multiply()</b> or <b>multiply_region(),</b> while the table for reduction
is created when the <b>gf_t</b> is initialized. For that reason, it makes sense for g<sub>r</sub> to be bigger than g<sub>s.</sub></p>

<p>
To give a flavor for the impact of these arguments, Figure 3 show </em>s the performance of varying g<sub>s</sub> and g<sub>r</sub> for
single multiplication and region multiplication respectively, in <em> GF(2<sup>32</sup>)</em> and <em>GF(2<sup>64</sup>).</em> As the graphs demonstrate,
<b>multiply()</b> performs better <em>w </em>ith smaller values of gs, <em>w </em>hile multiply region() amortizes the creation of the shifting
table, and can tolerate larger values of g<sub>s.</sub> <em>w </em>hen g<sub>s</sub> equals g<sub>r,</sub> there are some optimizations that we hand-encode.
These can be seen clearly in the <b>multiply_region()</b> graphs.
</p>








<br/>
7 &nbsp &nbsp  <em>   FURTHER INFORMATION ON OPTIONS AND ALGORITHMS     </em>   <span id="index_number">30 </span> 


<div id="box_1"> 
 
<div class="image-cell_3"> </div>

<div class="image-cell_4"> </div>
</div>
Figure 3: The performance of <b>multiply()</b> and <b>multiply_region()</b> using <b>"GROUP,"</b> and varying the arguments <br> g<sub>s</sub>
and g<sub>r.</sub> All graphs are heat maps with black equaling zero. The region size is 100KB.

<h3>7.6 &nbspConsiderations with "COMPOSITE" </h3>


As mentioned above, using <b>"ALTMAP"</b> with <b>"COMPOSITE"</b> allows <b>multiply_region()</b> to recursively call <b>multiply_
region(),</b> rather than simply calling <b>multiply()</b> on every word in the region. The difference can be pronounced:<br><br>

<div id="table_page28"><center>

<table cellpadding="6" cellspacing="0" style="text-align:center;font-size:19px"><tr>
<td>
gf_time 32 G 0 10240 10240 -m COMPOSITE 2 - -
Speed = 322 MB/s </td> </tr>
<tr>
<td>gf_time 32 G 0 10240 10240 -m COMPOSITE 2 - -r ALTMAP -
Speed = 3,368 MB/s </td> </tr>

<tr>
<td>
gf_time 32 G 0 10240 10240 -m COMPOSITE 2 -m SPLIT 16 4 -r ALTMAP - -r ALTMAP -
Speed = 3,925 MB/s </td> </tr>
</center>
</table>
</div>


<br><br>
<p>
There is support for performing <b>multiply()</b> inline for the <b>"TABLE"</b> implementations for w &#8712 {4, 8} and for the
"LOG" implementation for <em>w</em> = 16 (see section 7.1). These are leveraged by <b>multiply()</b> in <b>"COMPOSITE,"</b> and
by <b>multiply_region()</b> if you are not using <b>"ALTMAP."</b> To demonstrate this, in the table below, you can see that the
performance of <b>multiply()</b> with <b>"SPLIT 8 4"</b> is 88 percent as fast than the default in <em>w</em> = 8 (which is <b>"TABLE"</b>).
When you use each as a base field for <b>"COMPOSITE"</b> with <em>w</em> = 16, the one with <b>"SPLIT 8 4"</b> is now just 37 percent
as fast. The difference is the inlining of multiplication in the base field when <b>"TABLE"</b> is employed:</p><br><br>

<div id="table_page28" border="0"><center>

    <table cellpadding="6" cellspacing="0" style="text-align:center;font-size:19px">

      <tr><td>gf_time 8 M 0 1048576 100 - Speed = 501 Mega-ops/s</td> </tr>
      <tr><td>gf_time 8 M 0 1048576 100 -m SPLIT 8 4 - Speed = 439 Mega-ops/s </td> </tr>
      <tr><td>gf_time 8 M 0 1048576 100 -m COMPOSITE 2 - - Speed = 207 Mega-ops/s </td> </tr>
      <tr><td>gf_time 8 M 0 1048576 100 -m COMPOSITE 2 -m SPLIT 8 4 - - Speed = 77 Mega-ops/s </td> </tr>

    </table> 
    </center>
<br><br>
</div>

You can keep making recursive definitions of composites field if you want. For example, this one's not too slow for
region operations (641 MB/s):








<br/>
<br/>


6 &nbsp &nbsp  <em>  FURTHER INFORMATION ON OPTIONS AND ALGORITHMS     </em>   <span id="index_number">31  </span> <br><br><br>

<div id="number_spacing">
<center>
gf_time 128 G 0 1048576 100 -m COMPOSITE 2 <span style="color:red">-m COMPOSITE 2 </span> <span style="color:blue">-m COMPOSITE 2 </span> <br>
<span style="color:rgb(250, 149, 167)">-m SPLIT 16 4 -r ALTMAP -</span> <span style="color:blue">-r ALTMAP -</span> <span style="color:red"> -r ALTMAP -</span> -r ALTMAP -
</center>
</div><br>

<p>Please see section 7.8.1 for a discussion of polynomials in composite fields.</p>

<h2>7.7 &nbsp &nbsp &nbsp "CARRY_FREE" and the Primitive Polynomial </h2>


If your machine supports the PCLMUL instruction, then we leverage that in <b>"CARRY_FREE."</b> This implementation
first performs a carry free multiplication of two <em>w</em>-bit numbers, which yields a 2<em>w</em>-bit number. It does this with
one PCLMUL instruction. To reduce the 2<em>w</em>-bit number back to a <em>w</em>-bit number requires some manipulation of the
polynomial. As it turns out, if the polynomial has a lot of contiguous zeroes following its leftmost one, the number of
reduction steps may be minimized. For example, with <em>w </em> = 32, we employ the polynomial 0x100400007, because that
is what other libraries employ. This only has 9 contiguous zeros following the one, which means that the reduction
takes four steps. If we instead use 0x1000000c5, which has 24 contiguous zeros, the reduction takes just two steps.
You can see the difference in performance:
<br><br>
<center>
<div id="table_page28">

<table cellpadding="6" cellspacing="0" style="text-align:center;font-size:19px">
<tr>

<td>gf_time 32 M 0 1048576 100 -m CARRY_FREE - </td> <td> Speed = 48 Mega-ops/s</td> </tr>

<tr><td>gf_time 32 M 0 1048576 100 -m CARRY_FREE -p 0xc5 -</td> <td> Speed = 81 Mega-ops/s </td> </tr>

</table></center>
</div>
<br><br>

<p>
This is relevant for <em>w </em> = 16 and <em>w </em> = 32, where the "standard" polynomials are sub-optimal with respect to
<b>"CARRY_FREE."</b> For <em>w </em> = 16, the polynomial 0x1002d has the desired property. It’s less important, of course,
with <em>w </em> = 16, because <b>"LOG"</b> is so much faster than <b>CARRY_FREE.</b> </p>

<h2>7.8 &nbsp  More on Primitive Polynomials </h3>

<h3>7.8.1 &nbsp Primitive Polynomials that are not Primitive </h4>

The library is willing to work with most polynomials, even if they are not primitive or irreducible. For example, the
polynomial x<sup>4</sup> + x<sup>3</sup> +x<sup>2</sup> +x+1 is irreducible, and therefore generates a valid Galois Field for <em>GF(2<sup>4</sup>).</em> However, it
is not primitive, because 2<sup>5</sup> = 1. For that reason, if you use this polynomial, you cannot use the <b>"LOG"</b> method. The
other methods will work fine: <br><br>

<div id="number_spacing">

UNIX> gf_mult 2 2 4 -p 0xf -  <br>
4 <br>
UNIX> gf_mult 4 2 4 -p 0xf - <br>
8 <br>
UNIX> gf_mult 8 2 4 -p 0xf - <br>
15 <br>
UNIX> gf_mult 15  2 4 -p 0xf - <br>
1 <br>
UNIX> gf_div 1 15 4 -p 0xf - <br>
2 <br>
UNIX> gf_div 1 15 4 -p 0xf -m LOG - <br>
usage: gf_div a b w [method] - does division of a and b in GF(2&#710;w) <br>
Bad Method Specification: Cannot use Log tables because the polynomial is not primitive. <br>
UNIX>  <br>
</div>
<p>
If a polynomial is reducible, then it does not define a Galois Field, but instead a ring. GF-Complete attempts to
work here where it can; however certain parts of the library will not work:
</p>






<br/>


6 &nbsp &nbsp  <em>  FURTHER INFORMATION ON OPTIONS AND ALGORITHMS     </em>   <span id="index_number">32  </span> <br><br><br>
<ol>
<li>
Division is a best effort service. The problemis that often quotients are not unique. If <b>divide()</b> returns a non-zero
number, then that number will be a valid quotient, but it may be one of many. If the multiplication technique is
<b>"TABLE,"</b> then if a quotient exists, one is returned. Otherwise, zero is returned. Here are some examples - the
polynomial x<sup>4</sup> + 1 is reducible, and therefore produces a ring. Below, we see that with this polynomal, 1*6 = 6
and 14*6 = 6. Therefore, 6/6 has two valid quotients: 1 and 14. GF-Complete returns 14 as the quotient:</li><br>

<div id="number_spacing">
UNIX> gf_mult 1 6 4 -p 0x1 -<br>
6 <br>
UNIX> gf_mult 14 6 4 -p 0x1 - <br>
6 <br>
UNIX> gf_div 6 6 4 -p 0x1 - <br>
14 <br>
UNIX> <br><br>
</div>


<li>When <b>"EUCLID"</b> is employed for division, it uses the extended Euclidean algorithm for GCD to find a number's
inverse, and then it multiplies by the inverse. The problem is that not all numbers in a ring have inverses. For
example, in the above ring, there is no number <em>a</em> such that 6a = 1. Thus, 6 has no inverse. This means that even
though 6/6 has quotients in this ring, <b>"EUCLID"</b> will fail on it because it is unable to find the inverse of 6. It will
return 0:
</li><br>
<div id="number_spacing">
UNIX> gf_div 6 6 4 -p 0x1 -m TABLE -d EUCLID -<br>
0<br>
UNIX><br>
</div><br>

<li> Inverses only work if a number has an inverse. Inverses may not be unique. </li><br>

<li> <b>"LOG"</b> will not work. In cases where the default would be <b>"LOG,"</b> <b>"SHIFT"</b> is used instead. </li>
</ol>

<p>
Due to problems with division, <b>gf_unit</b> may fail on a reducible polynomial. If you are determined to use such a
polynomial, don't let this error discourage you.
</p>

<h3>7.8.2 Default Polynomials for Composite Fields </h3>

GF-Complete will successfully select a default polynomial in the following composite fields:
<ul>
<li> <em>w </em> = 8 and the default polynomial (0x13) is employed for <em>GF(2<sup>4</sup>)</em></li><br>
<li> w = 16 and the default polynomial (0x11d) is employed for <em>GF(2<sup>8</sup>)</em></li><br>
<li> <em>w </em> = 32 and the default polynomial (0x1100b) is employed for <em>GF(2<sup>16</sup>) </em></li><br>
<li> <em>w </em> = 32 and 0x1002d is employed for <em>GF(2<sup>16</sup>) </em></li><br>
<li> <em>w </em> = 32 and the base field for <em>GF(w<em>16</em>) </em> is a composite field that uses a default polynomial</li><br>
<li> <em>w </em> = 64 and the default polynomial (0x100400007) is employed for <em>GF(2<sup>32</sup>)</em></li><br>
<li> <em>w </em> = 64 and 0x1000000c5 is employed for <em>GF(2<sup>32</sup>) </em></li><br>
<li> <em>w </em> = 64 and the base field for <em>GF(w<sup>32</sup>) </em> is a composite field that uses a default polynomial</li><br>
<li> <em>w </em> = 128 and the default polynomial (0x1b) is employed for <em>GF(2<sup>64</sup>) </em></li><br>
<li> <em>w </em> = 128 and the base field for <em> GF(w<sup>64 </sup>) </em> is a composite field that uses a default polynomial</li><br>
</ul>








<br/>


6 &nbsp &nbsp  <em>  FURTHER INFORMATION ON OPTIONS AND ALGORITHMS     </em>   <span id="index_number">33  </span> <br><br><br>


<h3>7.8.3 The Program gf_poly for Verifying Irreducibility of Polynomials </h3>

The program <b>gf_poly</b> uses the Ben-Or algorithm[GP97] to determine whether a polynomial with coefficients in <em> GF(2<sup>w </sup>) </em>
is reducible. Its syntax is:<br><br>
<div id="number_spacing">
gf_poly w method power:coef power:coef ... 
</div>

<br>
<p>You can use it to test for irreducible polynomials with binary coefficients by specifying w = 1. For example, from
the discussion above, we know that x<sup>4</sup> +x+1 and x<sup>4</sup> +x<sup>3</sup> +x<sup>2</sup> +x+1 are both irreducible, but x<sup>4</sup> +1 is reducible.
<b>gf_poly</b> confirms:<p><br>

<div id="number_spacing">
UNIX> gf_poly 1 - 4:1 1:1 0:1 <br>
Poly: x&#710;4 + x + 1 <br>
Irreducible. <br>
UNIX> gf_poly 1 - 4:1 3:1 2:1 1:1 0:1 <rb>
Poly: x&#710;4 + x&#710;3 + x&#710;2 + x + 1 <br>
Irreducible. <br>
UNIX> gf_poly 1 - 4:1 0:1 r<br>
Poly: x&#710;4 + 1 <br>
Reducible. <br>
UNIX> <br>

</div>


<p>
For composite fields <em>GF((2<sup>l</sup>)<sup>2</sup>),</em> we are looking for a value s such that x<sup>2</sup> + sx + 1 is irreducible. That value
depends on the base field. For example, for the default field <em>GF(2<sup>32</sup>),</em> a value of <em>s</em> = 2 makes the polynomial
irreducible. However, if the polynomial 0xc5 is used (so that PCLMUL is fast - see section 7.7), then <em>s</em> = 2 yields a
reducible polynomial, but <em>s</em> = 3 yields an irreducible one. You can use <b>gf_poly</b> to help verify these things, and to help
define s if you need to stray from the defaults:</p> <br>

<div id="number_spacing">
UNIX> gf_poly 32 - 2:1 1:2 0:1<br>
Poly: x&#710;2 + (0x2)x + 1 <br>
Irreducible. <br>
UNIX> gf_poly 32 -p 0xc5 - 2:1 1:2 0:1 <br>
Poly: x&#710;2 + (0x2)x + 1 <br>
Reducible. <br>
UNIX> gf_poly 32 -p 0xc5 - 2:1 1:3 0:1 <br>
Poly: x&#710;2 + (0x3)x + 1 <br>
Irreducible. <br>
UNIX> <br>
</div>

<p>
<b>gf_unit</b> does random sampling to test for problems. In particular, it chooses a random a and a random b, multiplies
them, and then tests the result by dividing it by a and b. When w is large, this sampling does not come close to
providing complete coverage to check for problems. In particular, if the polynomial is reducible, there is a good
chance that <b>gf_unit</b> won't discover any problems. For example, the following <b>gf_unit</b> call does not flag any problems,
even though the polynomial is reducible.</p>
<br>
<div id="number_spacing">
UNIX> gf_unit 64 A 0 -m COMPOSITE 2 -p 0xc5 - -p 2 -<br>
UNIX>
</div>

<p>
How can we demonstrate that this particular field has a problem? Well, when the polynomial is 0xc5, we can factor
x<sup>2</sup> + 2x + 1 as (x + 0x7f6f95f9)(x + 0x7f6f95fb). Thus, in the composite field, when we multiply 0x17f6f95f9 by
0x17f6f95fb, we get zero. That's the problem:
</p>








<br/>


6 &nbsp &nbsp  <em>  FURTHER INFORMATION ON OPTIONS AND ALGORITHMS     </em>   <span id="index_number">34  </span> <br><br><br>

<div id="number_spacing">

UNIX> gf_mult 7f6f95f9 7f6f95fb 32h -p 0xc5 - <br>
1 <br>
UNIX> gf_mult 17f6f95f9 17f6f95fb 64h -m COMPOSITE 2 -p 0xc5 - -p 2 - <br>
0 <br>
UNIX> <br>

</div>

<h2>7.9 "ALTMAP" considerations and extract_word() </h2>

There are two times when you may employ alternate memory mappings:
<ol>
<li> When using <b>"SPLIT"</b> and w<sub>b</sub> = 4. </li>
<li> When using <b>"COMPOSITE."</b> </li>
</ol>

Additionally, by default, the <b>"CAUCHY"</b> region option also employs an alternate memory mapping.

<p>When you use alternate memory mappings, the exact mapping of words in <em> GF(2<sup>w </sup>) </em> to memory depends on the
situation, the size of the region, and the alignment of the pointers. To help you figure things out, we have included the
procedures <b>extract_word.wxx()</b> as part of the <b>gf_t</b> struct. This procedure takes four parameters: </p>
<ul>
<li>A pointer to the <b>gf_t.</b> </li>
<li> The beginning of the memory region. </li>
<li>The number of bytes in the memory region. </li>
<li>The desired word number: <em>n.</em> </li>
</ul>

<p>
It then returns the <em>n</em>-th word in memory. When the standard mapping is employed, this simply returns the <em>n</em>-
th contiguous word in memory. With alternate mappings, each word may be split over several memory regions, so
<b>extract_word()</b> grabs the relevant parts of each memory region to extract the word. Below, we go over each of the
above situations in detail. Please refer to Figure 2 in Section 5 for reference. </p>


<h3>7.9.1 Alternate mappings with "SPLIT" </h3>

The alternate mapping with <b>"SPLIT"</b> is employed so that we can best leverage <b>mm_shuffle_epi8().</b> Please read [PGM13b]
for details as to why. Consider an example when <em>w</em> = 16. In the main region of memory (the middle region in Figure
2), multiplication proceeds in units of 32 bytes, which are each broken into two 16-byte regions. The first region
holds the high bytes of each word in <em>GF(2<sup>16</sup>),</em> and the second region holds the low bytes.
Let's look at a very detailed example, from <b>gf_example_5.c.</b> This program makes the following call, where <b>gf</b> has

been initialized for <em>w</em> = 16, using <b>"SPLIT"</b> and <b>"ALTMAP:"</b><br><br>
<div id="number_spacing">
gf.multiply_region.w32(&gf, a, b, 0x1234, 30*2, 0);
</div><br>


<p>In other words, it is multiplying a region a of 60 bytes (30 words) by the constant 0x1234 in <em> GF(2<sup>16</sup>),</em> and placing
the result into <em>b.</em> The pointers <em>a</em> and <em>b</em> have been set up so that they are not multiples of 16. The first line of output
prints <em>a</em> and <em>b:</em></p><br>

a: 0x10010008c b: 0x10010015c <br><br>

As described in Section 5, the regions of memory are split into three parts:








<br/>


6 &nbsp &nbsp  <em>  FURTHER INFORMATION ON OPTIONS AND ALGORITHMS     </em>   <span id="index_number">35  </span> <br><br><br>


<ol>
<li> 4 bytes starting at 0x1001008c / 0x10010015c. </li>
<li> 32 bytes starting at 0x10010090 / 0x100100160. </li>
<li> 24 bytes starting at 0x100100b0 / 0x100100180. </li>

</ol>


<p>In the first and third parts, the bytes are laid out according to the standard mapping. However, the second part is
split into two 16-byte regions- one that holds the high bytes of each word and one that holds the low bytes. To help
illustrate, the remainder of the output prints the 30 words of <em>a</em> and <em>b</em> as they appear in memory, and then the 30 return
values of <b>extract_word.w32():</b> </p><br>

<div id="number_spacing">
<table cellspacing="6" style="text-align:right">

<tr>
<td></td> <td> 1</td> <td> 2 </td> <td> 3 </td> <td> 4</td> <td> 5 </td> <td> 6 </td> <td> 7</td> <td> 8 </td> <td> 9</td> </tr>
<tr>
<td>a:</td><td> 640b</td> <td> 07e5</td> <td> 2fba </td> <td> ce5d </td> <td> f1f9</td> <td> 3ab8</td> <td> c518 </td> <td> 1d97</td> <td> 45a7</td>
 <td> 0160</td> </tr>
 
<tr><td>b:</td> <td>1ba3</td><td> 644e</td> <td> 84f8</td> <td> be3c</td> <td> 4318</td> <td> 4905</td> <td> b2fb </td> <td> 46eb </td> <td> ef01 </td>
 <td>a503</td> 
</tr>
</table> 
 <br><br>
<table cellspacing="6" style="text-align:right">

<tr>
<td> 10</td> <td> 11 </td> <td> 12</td> <td> 13</td> <td> 14 </td> <td> 15 </td> <td> 16</td> <td> 17</td> <td>18</td> <td> 19 </td></tr>
<tr>
<td>a:</td><td> 3759</td> <td> b107</td> <td> 9660 </td> <td> 3fde </td> <td> b3ea</td> <td> 8a53</td> <td> 75ff </td> <td> 46dc</td> <td> c504</td>
 <td> 72c2</td> </tr>
 
<tr><td>b:</td> <td>da27</td><td> e166</td> <td> a0d2</td> <td> b3a2</td> <td> 1699</td> <td> 3a3e</td> <td> 47fb </td> <td> 39af </td> <td> 1314 </td>
 <td>8e76</td> 
</tr>
</table> 

<table cellspacing="6" style="text-align:right">
<br><br>
<tr>
<td> 20</td> <td> 21 </td> <td> 22</td> <td> 23</td> <td> 24 </td> <td> 25 </td> <td> 26</td> <td> 27</td> <td>28</td> <td> 29 </td></tr>
<tr>
<td>a:</td><td> b469</td> <td> 1b97</td> <td> e91d </td> <td> 1dbc </td> <td> 131e</td> <td> 47e0</td> <td> c11a </td> <td> 7f07</td> <td> 76e0</td>
 <td> fe86</td> </tr>
 
<tr><td>b:</td> <td>937c</td><td> a5db</td> <td> 01b7</td> <td> 7f5f</td> <td> 8974</td> <td> 05e1</td> <td> cff3 </td> <td> a09c </td> <td> de3c </td>
 <td>4ac0</td> 
</tr>
</table> 
<br><br>
<table cellspacing="6">


<tr><td>Word</td><td> 0:</td> <td>0x640b </td><td>*</td> <td>0x1234</td> <td>=</td> <td>0x1ba3 Word 15:</td> <td>0x4575 </td><td>*</td> <td>0x1234</td> <td>=</td> <td>0xef47</td></tr>     
<tr><td>Word</td> <td> 1:</td> <td>0x07e5 </td><td>*</td> <td>0x1234</td> <td>=</td> <td>0x644e Word 16:</td> <td>0x60dc </td><td>*</td> <td>0x1234</td> <td>=</td> <td>0x03af</td></tr>
<tr><td>Word</td> <td> 2:</td> <td>0xba59 </td><td>*</td> <td>0x1234</td> <td>=</td> <td>0xf827 Word 17:</td> <td>0x0146 </td><td>*</td> <td>0x1234</td> <td>=</td> <td>0xa539 </td> </tr>
<tr><td>Word</td> <td>3:</td> <td>0x2f37 </td><td>*</td> <td>0x1234</td> <td>=</td> <td>0x84da Word 18:</td> <td>0xc504 </td><td>*</td> <td>0x1234</td> <td>=</td> <td>0x1314 </td> </tr>
<tr><td>Word</td> <td>4:</td> <td>0x5d07 </td><td>*</td> <td>0x1234</td> <td>=</td> <td>0x3c66 Word 19:</td> <td>0x72c2 </td><td>*</td> <td>0x1234</td> <td>=</td> <td>0x8e76 </td> </tr>
<tr><td>Word</td> <td>5:</td> <td>0xceb1 </td><td>*</td> <td>0x1234</td> <td>=</td> <td>0xbee1 Word 20:</td> <td>0xb469 </td><td>*</td> <td>0x1234</td> <td>=</td> <td>0x937c </td> </tr>
<tr><td>Word</td> <td>6:</td> <td>0xf960 </td><td>*</td> <td>0x1234</td> <td>=</td> <td>0x18d2 Word 21:</td> <td>0x1b97 </td><td>*</td> <td>0x1234</td> <td>=</td> <td>0xa5db </td> </tr>
<tr><td>Word</td> <td>7:</td> <td>0xf196 </td><td>*</td> <td>0x1234</td> <td>=</td> <td>0x43a0 Word 22:</td> <td>0xe91d </td><td>*</td> <td>0x1234</td> <td>=</td> <td>0x01b7 </td> </tr>
<tr><td>Word</td> <td>8:</td> <td>0xb8de </td><td>*</td> <td>0x1234</td> <td>=</td> <td>0x05a2 Word 23:</td> <td>0x1dbc </td><td>*</td> <td>0x1234</td> <td>=</td> <td>0x7f5f </td> </tr>
<tr><td>Word</td> <td>9:</td> <td>0x3a3f </td><td>*</td> <td>0x1234</td> <td>=</td> <td>0x49b3 Word 24:</td> <td>0x131e </td><td>*</td> <td>0x1234</td> <td>=</td> <td>0x8974 </td> </tr>
<tr><td>Word</td> <td>10:</td> <td>0x18ea </td><td>*</td> <td>0x1234</td> <td>=</td> <td>0xfb99 Word 25:</td> <td>0x47e0 </td><td>*</td> <td>0x1234</td> <td>=</td> <td>0x05e1 </td> </tr>
<tr><td>Word</td> <td>11:</td> <td>0xc5b3 </td><td>*</td> <td>0x1234</td> <td>=</td> <td>0xb216 Word 26:</td> <td>0xc11a </td><td>*</td> <td>0x1234</td> <td>=</td> <td>0xcff3  </td> </tr>
<tr><td>Word</td> <td>12:</td> <td>0x9753 </td><td>*</td> <td>0x1234</td> <td>=</td> <td>0xeb3e Word 27:</td> <td>0x7f07 </td><td>*</td> <td>0x1234</td> <td>=</td> <td>0xa09c  </td> </tr>
<tr><td>Word</td> <td>13:</td> <td>0x1d8a </td><td>*</td> <td>0x1234</td> <td>=</td> <td>0x463a Word 28:</td> <td>0x76e0 </td><td>*</td> <td>0x1234</td> <td>=</td> <td>0xde3c  </td> </tr>
<tr><td>Word</td> <td>14:</td> <td>0xa7ff </td><td>*</td> <td>0x1234</td> <td>=</td> <td>0x01fb Word 29:</td> <td>0xfe86 <td>*</td> <td>0x1234</td> <td>=</td> <td>0x4ac0 </td> </tr>

</table>
</div>
<br>
In the first region are words 0 and 1, which are identical to how they appear in memory: 0x640b and 0x07e5. In
the second region are words 2 through 17. These words are split among the two sixteen-byte regions. For example,
word 2, which <b>extract_word()</b> reports is 0xba59, is constructed from the low byte in word 2 (0xba) and the low byte
in word 10 (0x59). Since 0xba59 * 0x1234 = 0xf827, we see that the low byte in word 2 of <em> b </em> is 0xf8, and the low byte
in word 10 is 0x27.
<p>When we reach word 22, we are in the third region of memory, and words are once again identical to how they
appear in memory.</p>

<p>While this is confusing, we stress that that so long as you call <b>multiply_region()</b> with pointers of the same alignment
and regions of the same size, your results with <b>ALTMAP</b> will be consistent. If you call it with pointers of </p>






<br/>


7 &nbsp &nbsp  <em>  FURTHER INFORMATION ON OPTIONS AND ALGORITHMS     </em>   <span id="index_number">36  </span> <br><br><br>

different alignments, or with different region sizes, then the results will not be consistent. To reiterate, if you don't use
<b>ALTMAP,</b> you don't have to worry about any of this - words will always be laid out contiguously in memory.
<p>
When <em>w</em> = 32, the middle region is a multiple of 64, and each word in the middle region is broken into bytes, each
of which is in a different 16-byte region. When <em>w</em> = 64, the middle region is a multiple of 128, and each word is
stored in eight 16-byte regions. And finally, when<em>w</em> = 128, the middle region is a multiple of 128, and each word is
stored in 16 16-byte regions.</p><br>

<h3>7.9.2 &nbsp Alternate mappings with "COMPOSITE" </h3>

With <b>"COMPOSITE,"</b> the alternate mapping divides the middle region in half. The lower half of each word is stored
in the first half of the middle region, and the higher half is stored in the second half. To illustrate, gf_example_6
performs the same example as gf_example_5, except it is using <b>"COMPOSITE"</b> in GF((2<sup>16</sup>)<sup>2</sup>), and it is multiplying
a region of 120 bytes rather than 60. As before, the pointers are not aligned on 16-bit quantities, so the region is broken
into three regions of 4 bytes, 96 bytes, and 20 bytes. In the first and third region, each consecutive four byte word is a
word in <em>GF(2<sup>32</sup>).</em> For example, word 0 is 0x562c640b, and word 25 is 0x46bc47e0. In the middle region, the low two
bytes of each word come from the first half, and the high two bytes come from the second half. For example, word 1
as reported by <b>extract_word()</b> is composed of the lower two bytes of word 1 of memory (0x07e5), and the lower two
bytes of word 13 (0x3fde). The product of 0x3fde07e5 and 0x12345678 is 0x211c880d, which is stored in the lower
two bytes of words 1 and 13 of <em>b.</em><br><br>

a: 0x10010011c b: 0x1001001ec

<br><br>

<div id="number_spacing">
<table cellspacing="6" style="text-align:right">

<tr>
<td></td> <td> 1</td> <td> 2 </td> <td> 3 </td> <td> 4</td> <td> 5 </td> <td> 6 </td> <td> 7</td> <td> 8 </td> <td> 9</td> </tr>
<tr>
<td>a:</td><td> 562c640b</td> <td> 959407e5</td> <td> 56592fba </td> <td> cbadce5d </td> <td> 1d1cf1f9</td> <td> 35d73ab8</td> <td> 6493c518 </td> <td> b37c1d97</td> 
<td> 8e4545a7</td>
 <td> c0d80160</td> </tr>
 
<tr><td>b:</td> <td>f589f36c</td><td> f146880d</td> <td> 74f7b349</td> <td> 7ea7c5c6</td> <td> 34827c1a</td> <td> 93cc3746</td> <td> bfd9288b </td>
 <td> 763941d1 </td> 
<td> bcd33a5d </td>
 <td>da695e64</td> 
</tr>
</table> 


<br><br>
<table cellspacing="6" style="text-align:right">

<tr>
<td> 10</td> <td> 11 </td> <td> 12</td> <td> 13</td> <td> 14 </td> <td> 15 </td> <td> 16</td> <td> 17</td> <td>18</td> <td> 19 </td></tr>
<tr>
<td>a:</td><td> 965b3759</td> <td> cb3eb107</td> <td> 1b129660 </td> <td> 95a33fde </td> <td> 95a7b3ea</td> <td> d16c8a53</td> <td> 153375ff </td> 
<td> f74646dc</td> <td> 35aac504</td>
 <td> 98f972c2</td> </tr>
 
<tr><td>b:</td> <td>fd70f125</td><td> 3274fa8f</td> <td> d9dd34ee</td> <td> c01a211c</td> <td> d4402403</td> <td> 8b55c08b</td> <td> da45f0ad </td> 
<td> 90992e18 </td> <td> b65e0902 </td>
 <td>d91069b5</td> 
</tr>
</table> 


<table cellspacing="6" style="text-align:right">
<br><br>
<tr>
<td> 20</td> <td> 21 </td> <td> 22</td> <td> 23</td> <td> 24 </td> <td> 25 </td> <td> 26</td> <td> 27</td> <td>28</td> <td> 29 </td></tr>
<tr>
<td>a:</td><td> 5509b469</td> <td> 7f8a1b97</td> <td> 3472e91d </td> <td> 9ee71dbc </td> <td> de4e131e</td> <td> 46bc47e0</td> <td> 5bc9c11a </td>
 <td> 931d7f07</td> <td> c85cfe86</td>
 <td> fe86</td> </tr>
 
<tr><td>b:</td> <td>fc92b8f5</td><td> edd59668</td> <td> b4bc0d90</td> <td> a679e4ce</td> <td> 1a98f7d0</td> <td> 6038765f</td> <td> b2ff333f </td> <td> e7937e49 </td> 
<td> fa5a5867 </td>
 <td>79c00ea2</td> 
</tr>
</table> 
<br><br>


<table cellspacing="6" style="text-align:right">


<tr><td>Word</td><td> 0:</td> <td>0x562c640b </td><td>*</td> <td>0x12345678</td> <td>=</td> <td>0xf589f36c Word 15:</td> <td>0xb46945a7 </td><td>*</td> <td>0x12345678</td> <td>=</td> <td>0xb8f53a5d</td></tr>     
<tr><td>Word</td> <td> 1:</td> <td>0x3fde07e5 </td><td>*</td> <td>0x12345678</td> <td>=</td> <td>0x211c880d Word 16:</td> <td>0x55098e45 </td><td>*</td> <td>0x12345678</td> <td>=</td> <td>0xfc92bcd3</td></tr>
<tr><td>Word</td> <td> 2:</td> <td>0x95a39594 </td><td>*</td> <td>0x12345678</td> <td>=</td> <td>0xc01af146 Word 17:</td> <td>0x1b970160 </td><td>*</td> <td>0x12345678</td> <td>=</td> <td>0x96685e64 </td> </tr>
<tr><td>Word</td> <td>3:</td> <td>0xb3ea2fba </td><td>*</td> <td>0x12345678</td> <td>=</td> <td>0x2403b349 Word 18:</td> <td>0x7f8ac0d8 </td><td>*</td> <td>0x12345678</td> <td>=</td> <td>0xedd5da69 </td> </tr>
<tr><td>Word</td> <td>4:</td> <td>0x95a75659 </td><td>*</td> <td>0x12345678</td> <td>=</td> <td>0xd44074f7 Word 19:</td> <td>0xe91d3759 </td><td>*</td> <td>0x12345678</td> <td>=</td> <td>0x0d90f125 </td> </tr>
<tr><td>Word</td> <td>5:</td> <td>0x8a53ce5d </td><td>*</td> <td>0x12345678</td> <td>=</td> <td>0xc08bc5c6 Word 20:</td> <td>0x3472965b </td><td>*</td> <td>0x12345678</td> <td>=</td> <td>0xb4bcfd70 </td> </tr>
<tr><td>Word</td> <td>6:</td> <td>0xd16ccbad </td><td>*</td> <td>0x12345678</td> <td>=</td> <td>0x8b557ea7 Word 21:</td> <td>0x1dbcb107 </td><td>*</td> <td>0x12345678</td> <td>=</td> <td>0xe4cefa8f </td> </tr>
<tr><td>Word</td> <td>7:</td> <td>0x75fff1f9 </td><td>*</td> <td>0x12345678</td> <td>=</td> <td>0xf0ad7c1a Word 22:</td> <td>0x9ee7cb3e </td><td>*</td> <td>0x12345678</td> <td>=</td> <td>0xa6793274 </td> </tr>
<tr><td>Word</td> <td>8:</td> <td>0x15331d1c </td><td>*</td> <td>0x12345678</td> <td>=</td> <td>0xda453482 Word 23:</td> <td>0x131e9660 </td><td>*</td> <td>0x12345678</td> <td>=</td> <td>0xf7d034ee </td> </tr>
<tr><td>Word</td> <td>9:</td> <td>0x46dc3ab8 </td><td>*</td> <td>0x12345678</td> <td>=</td> <td>0x2e183746 Word 24:</td> <td>0xde4e1b12 </td><td>*</td> <td>0x12345678</td> <td>=</td> <td>0x1a98d9dd </td> </tr>
<tr><td>Word</td> <td>10:</td> <td>0xf74635d7 </td><td>*</td> <td>0x12345678</td> <td>=</td> <td>0x909993cc Word 25:</td> <td>0x46bc47e0 </td><td>*</td> <td>0x12345678</td> <td>=</td> <td>0x6038765f </td> </tr>
<tr><td>Word</td> <td>11:</td> <td>0xc504c518 </td><td>*</td> <td>0x12345678</td> <td>=</td> <td>0x0902288b Word 26:</td> <td>0x5bc9c11a </td><td>*</td> <td>0x12345678</td> <td>=</td> <td>0xb2ff333f  </td> </tr>
<tr><td>Word</td> <td>12:</td> <td>0x35aa6493 </td><td>*</td> <td>0x12345678</td> <td>=</td> <td>0xb65ebfd9 Word 27:</td> <td>0x931d7f07 </td><td>*</td> <td>0x12345678</td> <td>=</td> <td>0xe7937e49  </td> </tr>

</table>
</div>








<br/>


8 &nbsp &nbsp  <em>  THREAD SAFETY     </em>   <span id="index_number">37  </span> <br><br><br>
<div id="number_spacing">
<table cellpadding="6" cellspacing="0">
<tr>
<td>Word 13:</td> <td> 0x72c21d97</td> <td> *</td> <td> 0x12345678</td> <td> =</td> <td> 0x69b541d1</td> <td> Word 28:</tD>

<td> 0xd40676e0 </td> <td> * </td> <td> 0x12345678 </td> <td> = </td> <td> 0xfa5a5867 </td> </tr>

<tr><td>Word 14:</td> <td> 0x98f9b37c</td> <td> * </td> <td> 0x12345678 </td> <td> = </td> <td> 0xd9107639</td> <td> Word 29:</td>
<td> 0xc85cfe86</td> <td>*</td> <td> 0x12345678</td> <td> =</td> <td> 0x79c00ea2</td></tr>

</table>
</div><br>


<p>
As with <b>"SPLIT,"</b> using <b>multiply_region()</b> with <b>"COMPOSITE"</b> and <b>"ALTMAP"</b> will be consistent only if the
alignment of pointers and region sizes are identical. </p>


<h3>7.9.3 The mapping of "CAUCHY" </h3>

With <b>"CAUCHY,"</b> the region is partitioned into <em>w</em> subregions, and each word in the region is broken into <em>w</em> bits,
each of which is stored in a different subregion. To illustrate, <b>gf_example_7</b> multiplies a region of three bytes by 5
in <em>GF(2<sup>3</sup>)</em> using <b>"CAUCHY:"</b><br><br>

<div id="number_spacing">

UNIX> gf_example_7 <br>
a: 0x100100190 b: 0x1001001a0 <br><br>
a: 0x0b 0xe5 0xba <br>
b: 0xee 0xba 0x0b <br><br>
a bits: 00001011 11100101 10111010 <br>
b bits: 11101110 10111010 00001011<br><br>
Word 0: 3 * 5 = 4 <br>
Word 1: 5 * 5 = 7 <br>
Word 2: 2 * 5 = 1 <br>
Word 3: 5 * 5 = 7 <br>  
Word 4: 4 * 5 = 2 <br>
Word 5: 6 * 5 = 3 <br>
Word 6: 2 * 5 = 1 <br>
Word 7: 6 * 5 = 3 <br>
UNIX><br><br> </div>
<p>

The program prints the three bytes of a and b in hexadecimal and in binary. To see how words are broken up,
consider word 0, which is the lowest bit of each of the three bytes of a (and b). These are the bits 1, 1 and 0 in a, and
0, 0, and 1 in b. Accordingly, the word is 3 in a, and 3*5 = 4 in b. Similarly, word 7 is the high bit in each byte: 0, 1, 1
(6) in a, and 1, 1, 0 (3) in b.</p>
<p>With <b>"CAUCHY," multiply_region()</b>may be implemented exclusively with XOR operations. Please see [BKK<sup>+</sup>95]
for more information on the motivation behind <b>"CAUCHY."</b> </p>

<h2>8 &nbsp Thread Safety </h2>

Once you initialize a <b>gf_t,</b> you may use it wontonly in multiple threads for all operations except for the ones below.
With the implementations listed below, the scratch space in the <b>gf_t</b> is used for temporary tables, and therefore you
cannot call <b>region_multiply,</b> and in some cases <b>multiply</b> from multiple threads because they will overwrite each
others' tables. In these cases, if you want to call the procedures from multiple threads, you should allocate a separate
gf_t for each thread:
<ul>
<li>
 All "GROUP" implementations are not thread safe for either <b>region_multiply()</b> or <b> multiply().</b> Other than
<b>"GROUP," multiply() </b> is always thread-safe.

</li>
</ul>









<br/>


9 &nbsp &nbsp  <em>  LISTING OF PROCEDURES     </em>   <span id="index_number">38  </span> <br><br><br>
<ul>
<li>

For <em>w </em> = 4, <b>region_multiply.w32()</b> is unsafe in in "-m TABLE -r QUAD -r LAZY." </li><br>
<li> For <em>w </em> = 8, <b> region_multiply.w32()</b> is unsafe in in "-m TABLE -r DOUBLE -r LAZY."</li><br>
<li> For <em>w </em> = 16, <b>region_multiply.w32() </b> is unsafe in in "-m TABLE."</li><br>
<li> For <em>w </em> &#8712 {32, 64, 128}, all <b>"SPLIT"</b> implementations are unsafe for <b>region_multiply().</b> This means that if the
default uses <b>"SPLIT"</b> (see Table 1 for when that occurs), then <b>region_multiply()</b> is not thread safe.</li><br>
<li> The <b>"COMPOSITE"</b> operations are only safe if the implementations of the underlying fields are safe.</li>
</ul>

<h2>9 &nbspListing of Procedures </h2>

The following is an alphabetical listing of the procedures, data types and global variables for users to employ in
GF-complete.<br>

<ul>
<li> <b>GF_W16_INLINE_DIV()</b> in <b>gf_complete.h:</b> This is a macro for inline division when <em>w </em> = 16. See section 7.1.</li><br>
<li> <b>GF_W16_INLINE_MULT()</b> in <b>gf_complete.h:</b> This is a macro for inline multiplication when <em>w </em> = 16. See
section 7.1.</li><br>
<li> <b>GF_W4_INLINE_MULTDIV()</b> in <b>gf_complete.h:</b> This is a macro for inline multiplication/division when <em>w </em> =
4. See section 7.1.</li><br>

<li> <b>GF_W8_INLINE_MULTDIV()</b> in <b>gf_complete.h:</b> This is a macro for inline multiplication/division when <em>w </em> =
8. See section 7.1.</li><br>
<li> <b>MOA_Fill_Random_Region()</b> in <b>gf_rand.h:</b> Fills a region with random numbers.</li><br>
<li> <b>MOA_Random_128()</b> in <b>gf_rand.h:</b> Creates a random 128-bit number.</li><br>
<li> <b>MOA_Random_32()</b> in <b>gf_rand.h:</b> Creates a random 32-bit number. </li><br>
<li> <b>MOA_Random_64()</b> in <b>gf_rand.h:</b> Creates a random 64-bit number. </li><br>
<li> <b>MOA_Random_W()</b> in <b>gf_rand.h:</b> Creates a random w-bit number, where <em>w </em> &#8804 32. </li><br>
<li> <b>MOA_Seed()</b> in <b>gf_rand.h:</b> Sets the seed for the random number generator. </li><br>
<li> <b>gf_errno</b> in <b>gf_complete.h:</b> This is to help figure out why an initialization call failed. See section 6.1.</li><br>
<li> <b>gf_create_gf_from_argv()</b> in <b>gf_method.h:</b> Creates a gf_t using C style argc/argv. See section 6.1.1. </li><br>
<li> <b>gf_division_type_t</b> in <b>gf_complete.h:</b> the different ways to specify division when using <b>gf_init_hard().</b> See 
section 6.4. </li><br>
<li> <b>gf_error()</b> in <b>gf_complete.h:</b> This prints out why an initialization call failed. See section 6.1. </li><br>

<li> <b>gf_extract</b> in <b>gf_complete.h:</b> This is the data type of <b>extract_word()</b> in a gf_t. See section 7.9 for an example
of how to use extract word().</li>
</ul>





<br/>


9 &nbsp &nbsp  <em>  LISTING OF PROCEDURES     </em>   <span id="index_number">39  </span> <br><br><br>
<ul>
<li>
<b>gf_free()</b> in <b>gf_complete.h:</b> If <b>gf_init easy(), gf_init hard()</b> or <b>create_gf_from_argv()</b> allocated memory, this
frees it. See section 6.4. </li>

<li> <b>gf_func_a_b</b> in <b>gf_complete.h:</b> This is the data type of <b>multiply()</b> and <b>divide()</b> in a gf_t. See section 4.2 for
examples of how to use <b>multiply()</b> and <b>divide()</b></li><br>

<li> <b>gf_func_a_b</b> in <b>gf_complete.h:</b> This is the data type of <b>multiply()</b> and <b>divide()</b> in a <b>gf_t.</b> See section 4.2 for
examples of how to use <b>multiply()</b> and <b>divide()</b></li><br>

<li> <b>gf_func_a</b> in <b>gf_complete.h:</b> This is the data type of <b>inverse()</b> in a <b>gf_t</b></li><br>

<li> <b>gf_general_add()</b> in <b>gf_general.h:</b> This adds two <b>gf_general_t's </b></li><br>

<li> <b>gf_general_divide()</b> in <b>gf_general.h:</b> This divides two <b>gf_general t's </b></li><br>

<li> <b>gf_general_do_region_check() </b> in <b>gf_general.h:</b> This checks a region multiply of <b>gf_general_t's </b></li><br>

<li> <b>gf_general_do_region_multiply() </b> in <b>gf_general.h:</b> This does a region multiply of <b>gf_general_t's </b></li><br>

<li> <b>gf_general_do_single_timing_test()</b> in <b>gf_general.h:</b> Used in <b>gf_time.c </b></li><br>

<li> <b>gf_general_inverse() </b> in <b>gf_general.h:</b> This takes the inverse of a <b>gf_general_t </b></li><br>

<li> <b>gf_general_is_one() </b> in <b>gf_general.h:</b> This tests whether a <b>gf_general_t </b> is one</li><br>

<li> <b>gf_general_is_two() </b> in <b>gf_general.h:</b> This tests whether a <b>gf_general_t  </b>is two</li><br>

<li> <b>gf_general_is_zero() </b> in <b>gf_general.h:</b> This tests whether a <b>gf_general_t </b> is zero</li><br>

<li> <b>gf_general_multiply() </b> in <b>gf_general.h:</b> This multiplies two <b>gf_general_t's.</b> See the implementation of gf_mult.c

for an example</li><br>
<li> <b>gf_general_s_to_val() </b> in <b>gf_general.h:</b> This converts a string to a <b>gf_general t.</b> See the implementation of
gf_mult.c for an example</li><br>
<li> <b>gf_general_set_one() </b> in <b>gf_general.h:</b> This sets a <b>gf_general_t</b> to one</li><br>
<li> <b>gf_general_set_random()</b> in <b>gf_general.h:</b> This sets a <b>gf_general_t </b> to a random number</li><br>
<li> <b>gf_general_set_two() in </b><b>gf_general.h:</b> This sets a <b>gf_general_t </b> to two</li><br>
<li> <b>gf_general_set_up_single_timing_test() </b> in <b>gf_general.h:</b> Used in <b>gf_time.c</b></li><br>
<li> <b>gf_general_set_zero() in </b><b>gf_general.h:</b> This sets a <b>gf_general_t_to_zero</b></li><br>
<li> <b>gf_general_t_in .</b><b>gf_general.h:</b> This is a general data type for all values of w. See the implementation of gf_mult.c
for examples of using these</li><br>
<li> <b>gf_general_val_to_s()</b> in<b>gf_general.h:</b> This converts a <b>gf_general_t </b> to a string. See the implementation of
<b>gf_mult.c</b> for an example</li><br>

<li> <b>gf_init_easy()</b> in <b>gf_complete.h:</b> This is how you initialize a default <b>gf_t.</b> See 4.2 through 4.5 for examples of
calling <b>gf_init_easy()</b></li><br>
</ul>







<br/>


9 &nbsp &nbsp  <em>  LISTING OF PROCEDURES     </em>   <span id="index_number">40  </span> <br><br><br>

<ul>

<li><b>gf_init hard()</b> in <b>gf_complete.h: </b> This allows you to initialize a <b>gf_t</b> without using the defaults. See 6.4. We
recommend calling create <b>gf_from argv()</b> when you can, instead of <b>gf_ init_hard()</b></li><br>

<li> <b>gf_ mult_type_t </b> in <b>gf_complete.h: </b> the different ways to specify multiplication when using <b>gf_init hard()</b>. See
section 6.4</li><br>

<li> <b>gf_region_type_t</b> in <b>gf_complete.h: </b> the different ways to specify region multiplication when using <b>gf_init_hard()</b>.
See section 6.4</li><br>

<li> <b>gf_region_in</b> <b>gf_complete.h: </b> This is the data type of <b>multiply_region()</b> in a <b>gf_t.</b> See section 4.3 for an example
of how to use <b>multiply_region()</b></li><br>

<li> <b>gf_scratch_size()</b> in <b>gf_complete.h: </b> This is how you calculate how much memory a <b>gf_t</b> needs. See section 6.4.</li><br>

<li> <b>gf_size()</b> in <b>gf_complete.h: </b> Returns the memory consumption of a <b>gf_t.</b> See section 6.5.</li><br>

<li> <b>gf_ val_128_t</b> in <b>gf_complete.h: </b> This is how you store a value where <em>w </em> &#8804 128. It is a pointer to two 64-bit
unsigned integers. See section 4.4</li><br>


<li> <b>gf_val_32_t</b> in <b>gf_ complete.h: </b> This is how you store a value where <em>w </em> &#8804 32. It is equivalent to a 32-bit unsigned
integer. See section 4.2</li><br>

<li> <b>gf_ val_64_t</b> in <b>gf_complete.h: </b> This is how you store a value where <em>w </em> &#8804 64. It is equivalent to a 64-bit unsigned
integer. See section 4.5</li><br>

<li> <b>gf_w16_get_div_alog_table()</b> in <b>gf_ complete.h: </b> This returns a pointer to an inverse logarithm table that can be
used for inlining division when <em>w </em> = 16. See section 7.1</li><br>


<li> <b>gf_w16_get_log_table()</b> in <b>gf_complete.h: </b> This returns a pointer to a logarithm table that can be used for inlining
when <em>w </em> = 16. See section 7.1</li><br>


<li> <b>gf_w16_get_mult_alog_table()</b> in <b>gf_complete.h: </b> This returns a pointer to an inverse logarithm table that can be
used for inlining multiplication when <em>w </em> = 16. See section 7.1</li><br>


<li> <b>gf_ w4 get div table()</b> in <b>gf_complete.h: </b> This returns a pointer to a division table that can be used for inlining
when <em>w </em> = 4. See section 7.1</li><br>


<li> <b>gf_w4_get_mult_table()</b> in <b>gf_complete.h: </b> This returns a pointer to a multiplication table that can be used for
inlining when <em>w </em> = 4. See section 7.1</li><br>

<li> <b>gf_w8_get_div_table()</b> in <b>gf_complete.h: </b> This returns a pointer to a division table that can be used for inlining
when <em>w </em> = 8. See section 7.1</li><br>

<li> <b>gf_w8_get_mult_table()</b> in <b>gf_complete.h: </b> This returns a pointer to a multiplication table that can be used for
inlining when <em>w </em> = 8. See section 7.1</li><br>

</ul>









<br/>
10 &nbsp &nbsp  <em>TROUBLESHOOTING </em>   <span id="index_number">41  </span> <br><br><br>

<ul>
<li><b> SSE support.</b> Leveraging SSE instructions requires processor support as well as compiler support. For example,
the Mac OS 10.8.4 (and possibly earlier versions) default compile environment fails to properly compile
PCLMUL instructions. This issue can be fixed by installing an alternative compiler; see Section 3 for details</li><br>

<li> <b>Initialization segfaults.</b> You have to already have allocated your <b>gf_t</b> before you pass a pointer to it in
<b>bgf_init_easy()</b>, <b>create_gf_ from_argv()</b>, or <b>bgf_ini_hard()</b></li><br>


<li> <b>GF-Complete is slower than it should be.</b> Perhaps your machine has SSE, but you haven't specified the SSE
compilation flags. See section 3 for how to compile using the proper flags</li><br>


<li> <b>Bad alignment.</b> If you get alignment errors, see Section 5</li><br>

<li> <b>Mutually exclusive region types.</b> Some combinations of region types are invalid. All valid and implemented
combinations are printed by <b>bgf_methods.c </b></li><br>

<li><b>Incompatible division types.</b> Some choices of multiplication type constrain choice of divide type. For example,
<b>"COMPOSITE"</b> methods only allow the default division type, which divides by finding inverses (i.e.,
neither <b>"EUCLID"</b> nor <b>"MATRIX"</b> are allowed). For each multiplication method printed by <b>gf_methods.c,</b> the
corresponding valid division types are also printed</li><br>


<li><b> Arbitrary "GROUP" arguments.</b> The legal arguments to <b>"GROUP"</b> are specified in section 7.5</li><br>

<li> <b> Arbitrary "SPLIt" arguments.</b> The legal arguments to <b>"SPLIt"</b> are specified in section 7.4</li><br>

<li> <b>Threading problems.</b> For threading questions, see Section 8</li><br>

<li> <b>No default polynomial.</b> If you change the polynomial in a base field using <b>"COMPOSITE,"</b> then unless it is
a special case for which GF-Complete finds a default polynomial, you'll need to specify the polynomial of the
composite field too. See 7.8.2 for the fields where GF-Complete will support default polynomials</li><br>
<li> Encoding/decoding with different fields. Certain fields are not compatible. Please see section 7.2 for an
explanation</li><br>


<li> <b>"ALTMAP" is confusing.</b> We agree. Please see section 7.9 for more explanation.</li><br>

<li> <b>I used "ALTMAP" and it doesn't appear to be functioning correctly.</b> With 7.9, the size of the region and
its alignment both matter in terms of how <b>"ALTMAP"</b> performs <b>multiply_region()</b>. Please see section 7.9 for
detailed explanation</li><br>

<li><b>Where are the erasure codes?.</b> This library only implements Galois Field arithmetic, which is an underlying
component for erasure coding. Jerasure will eventually be ported to this library, so that you can have fast erasure
coding</li><br>
</ul>
<h2>11 &nbsp &nbsp Timings </h2>

We don't want to get too detailed with timing, because it is quite machine specific. However, here are the timings on
an Intel Core i7-3770 CPU running at 3.40 GHz, with 4 &#215 256 KB L2 caches and an 8MB L3 cache. All timings are
obtained with <b>gf_time</b> or <b>gf_inline_time,</b> in user mode with the machine dedicated solely to running these jobs.









<br/>
10 &nbsp &nbsp  <em>TROUBLESHOOTING </em>   <span id="index_number">41  </span> <br><br><br>

<div class="image-cell_5"> </div>
<center>Figure 4: Speed of doing single multiplications for w &#8712 {4, 8, 16}. </center>
<h2>11.1 &nbsp Multiply() </h2>

The performance of <b>multiply()</b> is displayed in Figures 4 for w &#8712 {4, 8, 16} and 5 for w &#8712 {32, 64, 128}. These
numbers were obtained by calling <b>gf_time</b> with the size and iterations both set to 10240. We plot the speed in megaops
per second.

<p>As would be anticipated, the inlined operations (see section 7.1) outperform the others. Additionally, in all
cases with the exception of <em>w</em> = 32, the defaults are the fastest performing implementations. With w = 32,
"CARRY_FREE" is the fastest with an alternate polynomial (see section 7.7). Because we require the defaults to
use a "standard" polynomial, we cannot use this implementation as the default. </p>

<h2>11.2 &nbsp Divide() </h2>

For the  <b>"TABLE"</b> and <b>"LOG"</b> implementations, the performance of division is the same as multiplication. This means
that for w &#8712 {4, 8, 16}, it is very fast indeed. For the other implementations, division is implemented with Euclid's
method, and is several factors slower than multiplication.
In Figure 6, we plot the speed of a few implementations of the larger word sizes. Compared to the <b>"TABLE"</b> and
<b>"LOG"</b> implemenations for the smaller word sizes, where the speeds are in the hundreds of mega-ops per second,
these are very slow. Of note is the <b>"COMPOSITE"</b> implementation for <em>w</em> = 32, which is much faster than the others








<br/>
10 &nbsp &nbsp  <em>TROUBLESHOOTING </em>   <span id="index_number">43  </span> <br><br><br>

<div class="image-cell_6"> </div>

<center>Figure 5: Speed of doing single multiplications for w &#8712 {32, 64, 128}. </center><br>

because it uses a special application of Euclid's method, which relies on division in <em>GF(2<sup>16</sup>),</em> which is very fast.<br><br>

<h3>11.3 &nbsp Multiply_Region() </h2>

Tables 3 through 8 show the performance of the various region operations. It should be noted that for <em>GF(2<sup>16 </sup>) </em>
through <em>GF(2<sup>128</sup>),</em> the default is not the fastest implementation of <b>multiply_region().</b> The reasons for this are outlined
in section 6
<p>
For these tables, we performed 1GB worth of <b>multiply_region()</b> calls for all regions of size 2i bytes for 10 &#8804 i &#8804
30. In the table, we plot the fastest speed obtained.</p>
<p>We note that the performance of <b>"CAUCHY"</b> can be improved with techniques from [LSXP13] and [PSR12].</p>









<br/>
<em>REFERENCES </em>   <span id="index_number">44  </span> <br><br><br>

<div class="image-cell_7"> </div>

<center>Figure 6: Speed of doing single divisions for w &#8712 {32, 64, 128}. </center><br>

<center>
<div id="data2">
<table cellpadding="2" cellspacing="0" style="text-align:center;font-size:19px">

<tr><th>Method</td> <th>Speed (MB/s)</td> </tr>

<tr><td>-m TABLE (Default) -</td> <td>11879.909</td> </tr>
<tr><td>-m TABLE -r CAUCHY -</td> <td>9079.712</td> </tr>
<tr><td>-m BYTWO_b -</td> <td>5242.400</td> </tr>
<tr><td>-m BYTWO_p -</td> <td>4078.431</td> </tr>
<tr><td>-m BYTWO_b -r NOSSE -</td> <td>3799.699</td> </tr>
<tr><td>-m TABLE -r QUAD -</td> <td>3014.315</td> </tr>

<tr><td>-m TABLE -r DOUBLE -</td> <td>2253.627</td> </tr>
<tr><td>-m TABLE -r NOSSE -</td> <td>2021.237</td> </tr>
<tr><td>-m TABLE -r NOSSE -</td> <td>1061.497</td> </tr>
<tr><td>-m LOG -</td> <td>503.310</td> </tr>


<tr><td>m SHIFT -</td> <td>157.749</td> </tr>
<tr><td>-m CARRY_FREE -</td> <td>86.202</td> </tr>
</div>
</table> <br><br>
</div> </center>
<center>Table 3: Speed of various calls to <b>multiply_region()</b> for <em>w</em> = 4. </center>

<h3>References </h3>

[Anv09] H. P. Anvin. The mathematics of RAID-6.<a href=""> http://kernel.org/pub/linux/kernel/people/hpa/
raid6.pdf,</a> 2009.<br><br>

[BKK<sup>+</sup>95] J. Blomer, M. Kalfane, M. Karpinski, R. Karp, M. Luby, and D. Zuckerman. An XOR-based erasureresilient
coding scheme. Technical Report TR-95-048, International Computer Science Institute, August
1995. <br><br>

[GMS08] K. Greenan, E. Miller, and T. J. Schwartz. Optimizing Galois Field arithmetic for diverse processor
architectures and applications. In MASCOTS 2008: <em>16th IEEE Symposium on Modeling, Analysis and
Simulation of Computer and Telecommunication Systems,</em> Baltimore, MD, September 2008.<br><br>


[GP97] S. Gao and D. Panario. Tests and constructions of irreducible polynomials over finite fields. In <em> Foundations
of Computational Mathematics,</em> pages 346–361. Springer Verlag, 1997.
















<br/>
<em>REFERENCES </em>   <span id="index_number">45  </span> <br><br><br>


<center>
<div id="data2">
<table cellpadding="2" cellspacing="0" style="text-align:center;font-size:19px">

<tr><th>Method</td> <th>Speed (MB/s)</td> </tr>
<tr><td>-m SPLIT 8 4 (Default)</td> <td>13279.146</td> </tr>
<tr><td>-m COMPOSITE 2 - -r ALTMAP -</td> <td>5516.588</td> </tr>
<tr><td>-m TABLE -r CAUCHY -</td> <td>4968.721</td> </tr>
<tr><td>-m BYTWO_b -</td> <td>2656.463</td> </tr>
<tr><td>-m TABLE -r DOUBLE -</td> <td>2561.225</td> </tr>
<tr><td>-m TABLE -</td> <td>1408.577</td> </tr>

<tr><td>-m BYTWO_b -r NOSSE -</td> <td>1382.409</td> </tr>
<tr><td>-m BYTWO_p -</td> <td>1376.661</td> </tr>
<tr><td>-m LOG_ZERO_EXT -</td> <td>1175.739</td> </tr>
<tr><td>-m LOG_ZERO -</td> <td>1174.694</td> </tr>


<tr><td>-m LOG -</td> <td>997.838</td> </tr>
<tr><td>-m SPLIT 8 4 -r NOSSE -</td> <td>885.897</td> </tr>


<tr><td>-m BYTWO_p -r NOSSE -</td> <td>589.520</td> </tr>
<tr><td>-m COMPOSITE 2 - -</td> <td>327.039</td> </tr>


<tr><td>-m SHIFT -</td> <td>106.115</td> </tr>

<tr><td>-m CARRY_FREE -</td> <td>104.299</td> </tr>


</div>
</table> <br><br>
</div> </center>
<center>Table 4: Speed of various calls to multiply region() for <em>w</em> = 4. </center><br><br>

[LBOX12] J. Luo, K. D. Bowers, A. Oprea, and L. Xu. Efficient software implementations of large finite fields
<em>GF(2<sup>n</sup>) </em> for secure storage applications.<em> ACM Transactions on Storage, 8(2),</em> February 2012.<br><br>

[LD00] J. Lopez and R. Dahab. High-speed software multiplication in f<sub>2<sup>m</sup></sub>. In <em>Annual International Conference
on Cryptology in India,</em> 2000.<br><br>

[LHy08] H. Li and Q. Huan-yan. Parallelized network coding with SIMD instruction sets. In <em>International Symposium
on Computer Science and Computational Technology,</em> pages 364-369. IEEE, December 2008.<br><br>

[LSXP13] J. Luo, M. Shrestha, L. Xu, and J. S. Plank. Efficient encoding schedules for XOR-based erasure codes.
<em>IEEE Transactions on Computing,</em>May 2013.<br><br>

[Mar94] G. Marsaglia. The mother of all random generators.<a href=""> ftp://ftp.taygeta.com/pub/c/mother.
c,</a> October 1994.<br>

[PGM13a] J. S. Plank, K. M. Greenan, and E. L. Miller. A complete treatment of software implementations of
finite field arithmetic for erasure coding applications. Technical Report UT-CS-13-717, University of
Tennessee, September 2013.<br><br>

[PGM13b] J. S. Plank, K. M. Greenan, and E. L. Miller. Screaming fast Galois Field arithmetic using Intel SIMD
instructions. In FAST-2013: <em>11th Usenix Conference on File and Storage Technologies,</em> San Jose, February
2013.<br><br>

[Pla97] J. S. Plank. A tutorial on Reed-Solomon coding for fault-tolerance in RAID-like systems.<em> Software -
Practice & Experience,</em> 27(9):995-1012, September 1997.












<br/>
<em>REFERENCES </em>   <span id="index_number">46  </span> <br><br><br>


<center>
<div id="data2">
<table cellpadding="2" cellspacing="0" style="text-align:center;font-size:19px">

<tr><th>Method</td> <th>Speed (MB/s)</td> </tr>
<tr><td>-m SPLIT 16 4 -r ALTMAP -</td> <td>10460.834</td> </tr>
<tr><td>-m SPLIT 16 4 -r SSE (Default) - </td> <td>8473.793</td> </tr>
<tr><td>-m COMPOSITE 2 - -r ALTMAP -</td> <td>5215.073</td> </tr>
<tr><td>-m LOG -r CAUCHY -</td> <td>2428.824</td> </tr>
<tr><td>-m TABLE -</td> <td>2319.129</td> </tr>
<tr><td>-m SPLIT 16 8 -</td> <td>2164.111</td> </tr>

<tr><td>-m SPLIT 8 8 -</td> <td>2163.993</td> </tr>
<tr><td>-m SPLIT 16 4 -r NOSSE -</td> <td>1148.810</td> </tr>
<tr><td>-m LOG -</td> <td>1019.896</td> </tr>
<tr><td>-m LOG_ZERO -</td> <td>1016.814</td> </tr>
<tr><td>-m BYTWO_b -</td> <td>738.879</td> </tr>
<tr><td>-m COMPOSITE 2 - -</td> <td>596.819</td> </tr>
<tr><td>-m BYTWO_p -</td> <td>560.972</td> </tr>
<tr><td>-m GROUP 4 4 -</td> <td>450.815</td> </tr>
<tr><td>-m BYTWO_b -r NOSSE -</td> <td>332.967</td> </tr>
<tr><td>-m BYTWO_p -r NOSSE -</td> <td>249.849</td> </tr>
<tr><td>-m CARRY_FREE -</td> <td>111.582</td> </tr>
<tr><td>-m SHIFT -</td> <td>95.813</td> </tr>


</div>
</table> <br><br>
</div> </center>
<center>Table 5: Speed of various calls to multiply region()  for <em>w</em> = 4. </center><br><br>

[PMG<sup>+</sup>13] J. S. Plank, E. L. Miller, K. M. Greenan, B. A. Arnold, J. A. Burnum, A. W. Disney, and A. C. McBride.
GF-Complete: A comprehensive open source library for Galois Field arithmetic. version 1.0. Technical
Report UT-CS-13-716, University of Tennessee, September 2013.<br><br>

[PSR12] J. S. Plank, C. D. Schuman, and B. D. Robison. Heuristics for optimizing matrix-based erasure codes for
fault-tolerant storage systems. In DSN-2012:<em> The International Conference on Dependable Systems and
Networks,</em> Boston, MA, June 2012. IEEE.<br><br>

[Rab89] M. O. Rabin. Efficient dispersal of information for security, load balancing, and fault tolerance. <em>Journal
of the Association for Computing Machinery,</em> 36(2):335-348, April 1989.









<br/>
<em>REFERENCES </em>   <span id="index_number">47  </span> <br><br><br>
<center>
<div id="data2">
<table cellpadding="2" cellspacing="0" style="text-align:center;font-size:19px">
<tr><th>Method</td> <th>Speed (MB/s)</td> </tr>
<tr>
 
<td>

-m SPLIT 32 4 -r SSE -r ALTMAP - <br>
-m SPLIT 32 4 (Default)  <br>
-m COMPOSITE 2 -m SPLIT 16 4 -r ALTMAP - -r ALTMAP - <br>
-m COMPOSITE 2 - -r ALTMAP -  <br>
-m SPLIT 8 8 - <br> 
-m SPLIT 32 8 - <br> 
-m SPLIT 32 16 - <br> 
-m SPLIT 8 8 -r CAUCHY <br> 
-m SPLIT 32 4 -r NOSSE <br> 
-m CARRY_FREE -p 0xc5 <br> 
-m COMPOSITE 2 - <br> 
-m BYTWO_b - <br> 
-m BYTWO_p - <br> 
-m GROUP 4 8 - <br> 
-m GROUP 4 4 - <br> 
-m CARRY_FREE - <br> 
-m BYTWO_b -r NOSSE - <br> 
-m BYTWO_p -r NOSSE - <br>
-m SHIFT - <br> 

</td>

<td>
7185.440 <br>
5063.966 <br>
 4176.440 <br>
3360.860 <br>
1345.678 <br>
1340.656 <br>
1262.676 <br>
1143.263  <br>
 480.859 <br>
393.185 <br>
332.964 <br>
309.971 <br>
258.623 <br>
242.076 <br>
227.399 <br>
226.785 <br>
143.403 <br>
111.956 <br>
52.295 <br>
</td>


</tr>

</div>
</table> <br><br>
</div> </center>
<center>Table 6: Speed of various calls to multiply region() <em>w</em> = 4. </center><br><br>

<center>
<div id="data2">
<table cellpadding="2" cellspacing="0" style="text-align:center;font-size:19px">
<tr><th>Method</td> <th>Speed (MB/s)</td> </tr>
<tr>
 
<td>
-m SPLIT 64 4 -r ALTMAP - <br>
-m SPLIT 64 4 -r SSE (Default) - <br>
-m COMPOSITE 2 -m SPLIT 32 4 -r ALTMAP - -r ALTMAP - <br>
-m COMPOSITE 2 - -r ALTMAP -  <br>
-m SPLIT 64 16 - <br>
-m SPLIT 64 8 -  <br>
-m CARRY_FREE -  <br>
-m SPLIT 64 4 -r NOSSE - <br>
-m GROUP 4 4 -  <br>
-m GROUP 4 8 -  <br>
-m BYTWO_b -  <br>
-m BYTWO_p -  <br>
-m SPLIT 8 8 - <br>
-m BYTWO_p -r NOSSE - <br>
-m COMPOSITE 2 - - <br>
-m BYTWO_b -r NOSSE - <br>
-m SHIFT - <br>

</td>

<td>3522.798 <br>
 2647.862 <br>
2461.572 <br>
1860.921 <br>
1066.490 <br>
998.461 <br>
975.290 <br>
545.479 <br>
230.137 <br>
153.947 <br>
144.052 <br>
124.538 <br>
98.892 <br>
77.912 <br>
77.522 <br>
36.391 <br>
25.282 <br>
</td>


</tr>

</div>
</table> <br><br>
</div> </center>
<center>Table 7: Speed of various calls to multiply region() for  <em>w</em> = 4. </center><br><br>













<br/>
<em>REFERENCES </em>   <span id="index_number">48  </span> <br><br><br>

<center>
<div id="data2">
<table cellpadding="2" cellspacing="0" style="text-align:center;font-size:19px">
<tr><th>Method</td> <th>Speed (MB/s)</td> </tr>
<tr>
 
<td>

-m SPLIT 128 4 -r ALTMAP - <br>
-m COMPOSITE 2 -m SPLIT 64 4 -r ALTMAP - -r ALTMAP - <br> 
-m COMPOSITE 2 - -r ALTMAP - <br> 
-m SPLIT 128 8 (Default) - <br>
-m CARRY_FREE -<br> 
-m SPLIT 128 4 -<br> 
-m COMPOSITE 2 - <br>
-m GROUP 4 8 -<br> 
-m GROUP 4 4 -<br> 
-m BYTWO_p -<br> 
-m BYTWO_b -<br> 
-m SHIFT -<br> 
</td>

<td>
1727.683 <br>
1385.693 <br>
1041.456 <br>
872.619 <br>
814.030 <br>
500.133  <br>
289.207 <br>
133.583 <br>
116.187 <br>
25.162 <br>
25.157 <br>
14.183 <br>
</td>


</tr>

</div>
</table> <br><br>
</div> </center>
<center>Table 8: Speed of various calls to multiply region() for <em>w</em> = 4. </center><br><br>
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