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<!DOCTYPE html>
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<meta http-equiv="Content-Type" content="text/html; charset=UTF-8"/>
<title>highlight.mac</title>
<meta name="generator" content="KF5::SyntaxHighlighting - Definition (Maxima) - Theme (Breeze Light)"/>
</head><body style="background-color:#ffffff;color:#1f1c1b"><pre>
<span style="color:#898887">/*</span>
<span style="color:#898887">------------------------------------------------------------------------</span>
<span style="color:#898887">Efficient Galois Fields in Maxima</span>

<span style="color:#898887">by Alasdair McAndrew</span>
<span style="color:#898887">and later extended by Fabrizio Caruso and Jacopo Daurizio</span>

<span style="color:#898887">it is distribuited together with gf_roots by Jacopo Daurizio</span>

<span style="color:#898887">Authors:</span>

<span style="color:#898887">Fabrizio Caruso   (optimizations, testing)</span>
<span style="color:#898887">Jacopo D'Aurizio   (optimizations, modular roots)</span>
<span style="color:#898887">Alasdair McAndrew (original version of the package, pohlig-helman log, etc... )</span>
<span style="color:#898887">------------------------------------------------------------------------*/</span>

<span style="color:#898887">/* Released under terms of the GNU General Public License, version 2,</span>
<span style="color:#898887"> * by permission of the authors to Robert Dodier circa 2007-12-02.</span>
<span style="color:#898887"> */</span>

<span style="color:#898887">/* Defines a flag for dealing with large fields.  If it is set to "false",</span>
<span style="color:#898887">then lookup tables are used for exponentiation and logarithms.  Otherwise</span>
<span style="color:#898887">other algorithms are used */</span>

<span style="color:#644a9b">define_variable</span>(<span style="color:#aa5500">largefield</span>,<span style="font-weight:bold">true</span>,<span style="color:#aa5500">bool</span>)$
<span style="color:#644a9b">define_variable</span>(<span style="color:#aa5500">gf_char</span>,<span style="color:#b08000">0</span>,<span style="color:#aa5500">integer</span>)$
<span style="color:#644a9b">define_variable</span>(<span style="color:#aa5500">gf_exp</span>,<span style="color:#b08000">0</span>,<span style="color:#aa5500">integer</span>)$
<span style="color:#644a9b">define_variable</span>(<span style="color:#aa5500">gf_order</span>,<span style="color:#b08000">0</span>,<span style="color:#aa5500">integer</span>)$
<span style="color:#644a9b">define_variable</span> (<span style="color:#aa5500">gf_one</span>, <span style="color:#607880;font-weight:bold">'</span><span style="color:#aa5500">gf_one</span>, <span style="color:#aa5500">any_check</span>)$
<span style="color:#644a9b">define_variable</span> (<span style="color:#aa5500">gf_prim</span>, <span style="color:#607880;font-weight:bold">'</span><span style="color:#aa5500">gf_prim</span>, <span style="color:#aa5500">any_check</span>)$
<span style="color:#644a9b">define_variable</span> (<span style="color:#aa5500">gf_irr</span>, <span style="color:#607880;font-weight:bold">'</span><span style="color:#aa5500">gf_irr</span>, <span style="color:#aa5500">any_check</span>)$
<span style="color:#644a9b">define_variable</span> (<span style="color:#aa5500">gf_list</span>, <span style="color:#607880;font-weight:bold">'</span><span style="color:#aa5500">gf_list</span>, <span style="color:#aa5500">any_check</span>)$
<span style="color:#644a9b">define_variable</span> (<span style="color:#aa5500">gen_powers</span>, <span style="color:#607880;font-weight:bold">'</span><span style="color:#aa5500">gf_list</span>, <span style="color:#aa5500">any_check</span>)$
<span style="color:#644a9b">remvalue</span>(<span style="color:#aa5500">x</span>,<span style="color:#aa5500">z</span>,<span style="color:#aa5500">gf_char</span>,<span style="color:#aa5500">gf_exp</span>,<span style="color:#aa5500">gf_irr</span>,<span style="color:#aa5500">pg</span>,<span style="color:#aa5500">gp</span>,<span style="color:#aa5500">lg</span>,<span style="color:#aa5500">gf_prim</span>,<span style="color:#aa5500">gf_one</span>,<span style="color:#aa5500">gf_order</span>,<span style="color:#aa5500">gf_list</span>,<span style="color:#aa5500">gen_powers</span>)$


<span style="color:#898887">/* --------------------------------------------------------------------------------------------- */</span>
<span style="color:#898887">/* Settings */</span>

<span style="color:#aa5500">GF_VERBOSE</span>:<span style="font-weight:bold">false</span>; <span style="color:#898887">/* Verbosity */</span>
<span style="color:#aa5500">GF_WARNING</span>: <span style="font-weight:bold">true</span>; <span style="color:#898887">/* Warnings */</span>
<span style="color:#aa5500">GF_IRREDUCIBILITY_CHECK</span>:<span style="font-weight:bold">false</span>;   <span style="color:#898887">/* Irreducibility test for the minimal polynomial of the extension */</span>

<span style="color:#898887">/*</span>
<span style="color:#898887">------------------------------------------------------------------------------------------------ */</span>


<span style="color:#898887">/* It defines a new current field with gf_char=b, min. pol.= p of deg= e*/</span>
<span style="color:#aa5500">gf_set</span>([<span style="color:#aa5500">ars</span>]):=<span style="color:#644a9b">block</span>([<span style="color:#aa5500">gj</span>,<span style="color:#aa5500">m</span>,<span style="color:#aa5500">i</span>,<span style="color:#aa5500">j</span>,<span style="color:#aa5500">dg</span>],
  <span style="font-weight:bold">if</span> <span style="color:#644a9b">length</span>(<span style="color:#aa5500">ars</span>)=<span style="color:#b08000">1</span> <span style="font-weight:bold">then</span>
    (
    <span style="color:#aa5500">gf_setp</span>(<span style="color:#aa5500">ars</span>[<span style="color:#b08000">1</span>]),
    <span style="color:#644a9b">return</span>(<span style="font-weight:bold">true</span>)
    )
  <span style="font-weight:bold">else</span>
    (
    <span style="font-weight:bold">if</span> <span style="color:#644a9b">length</span>(<span style="color:#aa5500">ars</span>)=<span style="color:#b08000">2</span> <span style="font-weight:bold">then</span>
       (
       <span style="font-weight:bold">if</span> <span style="color:#644a9b">numberp</span>(<span style="color:#aa5500">ars</span>[<span style="color:#b08000">2</span>]) <span style="font-weight:bold">then</span>
         (
         <span style="font-weight:bold">if</span> <span style="color:#aa5500">ars</span>[<span style="color:#b08000">2</span>]=<span style="color:#b08000">0</span> <span style="font-weight:bold">and</span> <span style="color:#aa5500">GF_WARNING</span> <span style="font-weight:bold">then</span>
           (
           <span style="color:#644a9b">print</span>(<span style="color:#bf0303">"WARNING: the irreducible is zero! We assume GF("</span>,<span style="color:#aa5500">ars</span>[<span style="color:#b08000">1</span>],<span style="color:#bf0303">")"</span>),
           <span style="color:#aa5500">gf_setp</span>(<span style="color:#aa5500">ars</span>[<span style="color:#b08000">1</span>]),
           <span style="color:#644a9b">return</span>(<span style="font-weight:bold">true</span>)
           )
         <span style="font-weight:bold">else</span>
           (
           <span style="color:#644a9b">error</span>(<span style="color:#bf0303">"ERROR: you tried to extend with a non-zero constant!"</span>),
           <span style="color:#644a9b">return</span>(<span style="font-weight:bold">false</span>)
           )
         )
       <span style="font-weight:bold">else</span>
         (
         <span style="color:#aa5500">dg</span>:<span style="color:#644a9b">hipow</span>(<span style="color:#aa5500">ars</span>[<span style="color:#b08000">2</span>],<span style="color:#aa5500">x</span>),

         <span style="font-weight:bold">if</span> <span style="color:#aa5500">dg</span>=<span style="color:#b08000">1</span> <span style="font-weight:bold">then</span>
           <span style="color:#aa5500">gf_setp</span>(<span style="color:#aa5500">ars</span>[<span style="color:#b08000">1</span>]),
         <span style="color:#aa5500">gf_irr</span>:<span style="color:#aa5500">ars</span>[<span style="color:#b08000">2</span>],
         <span style="color:#aa5500">gf_exp</span>:<span style="color:#aa5500">dg</span>,
         <span style="color:#644a9b">return</span>(<span style="font-weight:bold">true</span>)
         )
       )
    <span style="font-weight:bold">else</span>
       (
       <span style="color:#aa5500">gf_exp</span>:<span style="color:#aa5500">ars</span>[<span style="color:#b08000">2</span>],
       <span style="font-weight:bold">if</span> <span style="color:#aa5500">gf_exp</span>=<span style="color:#b08000">1</span> <span style="font-weight:bold">then</span>
          (
          <span style="color:#aa5500">gf_setp</span>(<span style="color:#aa5500">ars</span>[<span style="color:#b08000">1</span>]),
          <span style="color:#aa5500">gf_irr</span>:<span style="color:#644a9b">rat</span>(<span style="color:#aa5500">ars</span>[<span style="color:#b08000">3</span>]),
          <span style="color:#644a9b">return</span>(<span style="font-weight:bold">true</span>)
          ),
       <span style="color:#aa5500">gf_irr</span>:<span style="color:#644a9b">rat</span>(<span style="color:#aa5500">ars</span>[<span style="color:#b08000">3</span>])
       )
    ),

  <span style="color:#aa5500">gf_char</span>:<span style="color:#aa5500">ars</span>[<span style="color:#b08000">1</span>],
  <span style="color:#aa5500">gf_one</span>:<span style="color:#644a9b">rat</span>(<span style="color:#b08000">1</span>,<span style="color:#aa5500">x</span>),
  <span style="color:#aa5500">gf_order</span>:<span style="color:#aa5500">gf_char</span>^<span style="color:#aa5500">gf_exp</span><span style="color:#b08000">-1</span>,

  <span style="color:#aa5500">m</span>:<span style="color:#644a9b">makelist</span>(<span style="color:#644a9b">coeff</span>(<span style="color:#aa5500">gf_irr</span>,<span style="color:#aa5500">x</span>,<span style="color:#aa5500">i</span>),<span style="color:#aa5500">i</span>,<span style="color:#b08000">0</span>,<span style="color:#aa5500">gf_exp</span>),
  <span style="color:#aa5500">gf_list</span>:[[<span style="color:#644a9b">first</span>(<span style="color:#aa5500">m</span>),<span style="color:#b08000">0</span>]],<span style="color:#aa5500">j</span>:<span style="color:#b08000">1</span>,
  <span style="font-weight:bold">for</span> <span style="color:#aa5500">i</span>:<span style="color:#b08000">2</span> <span style="font-weight:bold">thru</span> <span style="color:#aa5500">gf_exp</span><span style="color:#b08000">+1</span> <span style="font-weight:bold">do</span> <span style="font-weight:bold">if</span> <span style="color:#aa5500">m</span>[<span style="color:#aa5500">i</span>]=<span style="color:#b08000">0</span> <span style="font-weight:bold">then</span> <span style="color:#aa5500">j</span>:<span style="color:#aa5500">j</span><span style="color:#b08000">+1</span> <span style="font-weight:bold">else</span> ( <span style="color:#aa5500">gf_list</span>:<span style="color:#644a9b">endcons</span>([<span style="color:#aa5500">m</span>[<span style="color:#aa5500">i</span>],<span style="color:#aa5500">j</span>],<span style="color:#aa5500">gf_list</span>), <span style="color:#aa5500">j</span>:<span style="color:#b08000">1</span> ),

  <span style="font-weight:bold">if</span> <span style="font-weight:bold">not</span>(<span style="color:#644a9b">primep</span>(<span style="color:#aa5500">gf_char</span>)) <span style="font-weight:bold">then</span> <span style="color:#644a9b">error</span>(<span style="color:#bf0303">"ERROR: Gf_Char must be a prime number."</span>)
    <span style="font-weight:bold">else</span>
      <span style="color:#0057ae;font-style:italic">modulus</span>:<span style="color:#aa5500">gf_char</span>,
  <span style="font-weight:bold">if</span> <span style="color:#aa5500">GF_IRREDUCIBILITY_CHECK</span> <span style="font-weight:bold">and</span>
       <span style="color:#644a9b">hipow</span>(<span style="color:#644a9b">args</span>(<span style="color:#644a9b">factor</span>(<span style="color:#aa5500">ars</span>[<span style="color:#b08000">3</span>]))[<span style="color:#b08000">1</span>],<span style="color:#aa5500">x</span>)#<span style="color:#644a9b">hipow</span>(<span style="color:#644a9b">rat</span>(<span style="color:#aa5500">ars</span>[<span style="color:#b08000">3</span>]),<span style="color:#aa5500">x</span>) <span style="font-weight:bold">then</span>
      <span style="color:#644a9b">error</span>(<span style="color:#bf0303">"ERROR: Polynomial is not irreducible"</span>),

  <span style="font-weight:bold">if</span> <span style="font-weight:bold">not</span>(<span style="color:#aa5500">largefield</span>) <span style="font-weight:bold">then</span>
     (
     <span style="color:#aa5500">pg</span>:<span style="color:#aa5500">mkpowers</span>(),
     <span style="color:#aa5500">lg</span>:<span style="color:#aa5500">mklogs</span>()
     )
  <span style="font-weight:bold">else</span>
     (
     <span style="font-weight:bold">if</span> <span style="color:#aa5500">GF_VERBOSE</span> <span style="font-weight:bold">then</span>
       <span style="color:#644a9b">print</span>(<span style="color:#bf0303">"finding a primitive element..."</span>),

     <span style="color:#aa5500">gf_prim</span>:<span style="color:#644a9b">rat</span>(<span style="color:#aa5500">gf_findprim</span>(),<span style="color:#aa5500">x</span>),
     <span style="font-weight:bold">if</span> <span style="color:#aa5500">GF_VERBOSE</span> <span style="font-weight:bold">then</span>
     <span style="color:#644a9b">print</span>(<span style="color:#bf0303">"...primitive element found."</span>)

     ),
  <span style="color:#0057ae;font-style:italic">modulus</span>:<span style="font-weight:bold">false</span>, <span style="color:#898887">/* it resets the modulus */</span>
  <span style="color:#644a9b">return</span>(<span style="font-weight:bold">true</span>)

  )$


<span style="color:#898887">/* Prints out information about the field */</span>
<span style="color:#aa5500">gf_info</span>():=<span style="color:#644a9b">block</span>(
  <span style="color:#644a9b">print</span>(<span style="color:#bf0303">"Prime gf_char value: "</span>,<span style="color:#aa5500">gf_char</span>),
  <span style="color:#644a9b">print</span>(<span style="color:#bf0303">"Exponent: "</span>, <span style="color:#aa5500">gf_exp</span>),
  <span style="color:#644a9b">print</span>(<span style="color:#bf0303">"Multiplicative order: "</span>,<span style="color:#aa5500">gf_order</span>),
  <span style="color:#644a9b">print</span>(<span style="color:#bf0303">"Irreducible polynomial: "</span>,<span style="color:#aa5500">gf_irr</span>),
  <span style="color:#644a9b">print</span>(<span style="color:#bf0303">"Primitive element: "</span>,<span style="color:#aa5500">gf_prim</span>),
  <span style="font-weight:bold">if</span> (<span style="color:#aa5500">largefield</span>) <span style="font-weight:bold">then</span>
    <span style="color:#644a9b">print</span>(<span style="color:#bf0303">"Largefield flag is true; powers and logarithms not computed."</span>)
    <span style="font-weight:bold">else</span>
    <span style="color:#644a9b">print</span>(<span style="color:#bf0303">"Largefield flag is false; powers and logarithms computed."</span>),
  <span style="color:#644a9b">disp</span>()
)$
</pre></body></html>