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  <title>Axiom Documentation</title>
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 <body>
  <div align="center"><img align="middle" src="doctitle.png"/></div>
  <hr/>
<center>
 <h2>RCM3720 Cryptography, Network and Computer Security</h2>
 <h3>Laboratory Class 3: Number Theory</h3>
</center>
<hr/>

<ul>

 <li> Check out the commands <tt>gcd</tt> and <tt>factor</tt>, and test them
  on different numbers, small and large.
 </li>  
 <li> Axiom provides a few useful commands for taking apart the factors of an
  object:
  <ul>
   <li> <span class="cmd">n:=5040</span></li>
   <li> <span class="cmd">f:=factor(n)</span></li>
   <li> <span class="cmd">numf:=numberOfFactors(f)</span></li>
   <li> <span class="cmd">fs:=[nthFactor(f,i) for i in 1..numf]</span></li>
   <li> <span class="cmd">es:=[nthExponent(f,i) for i in 1..numf]</span></li>
   <li> <span class="cmd">reduce(*,[fs.i^es.i for i in 1..numf])</span></li>
  </ul>
 </li>
 <li> The last command simply multiplies all the factors to their powers.</li>
 
 <li> Check out the commands <tt>prime?</tt>, <tt>nextPrime</tt> and
      <tt>prevPrime</tt>.
 </li>
 <li> To compute the <tt>i</tt>-th prime, we can construct a <i>stream</i>
      (an infinite list) in Axiom:
  <ul>
   <li> 
    <span class="cmd">
     primes:Stream Integer:=[i for i in 2.. | prime? i]
    </span>
   </li>
  </ul>
 </li>
 <li> Now we can find, for example, the 100-th prime, and the 2500-th prime:
  <ul>
   <li> <span class="cmd">primes.100</span></li>
   <li> <span class="cmd">primes.2500</span></li>
  </ul>
 </li>
 <li> Create random 10 digit primes:
  <ul>
   <li> <span class="cmd">p := nextPrime(random(10^10))</span></li>
   <li> <span class="cmd">q := nextPrime(random(10^10))</span></li>
  </ul>
 </li>
 <li> Now multiply them and factor the product.  How long did it take?</li>

 <li> Try the same thing with 12 digit primes and 15 digit primes.</li>
  
 <li> The extended Euclidean algorithm is implemented by the command
  <tt>extendedEuclidean</tt>.  Here's how to use it:
  <ul>
   <li> <span class="cmd">a:=1149</span></li>
   <li> <span class="cmd">b:=3137</span></li>
   <li> <span class="cmd">g:=extendedEuclidean(a,b)</span></li>
   <li> <span class="cmd">s:=g.coef1</span></li>
   <li> <span class="cmd">t:=g.coef2</span></li>
  </ul>
 </li>
 <li> and now test them:
  <ul>
   <li> <span class="cmd">s*a+t*b</span></li>
  </ul>
 </li>
 <li> Try this on a few other numbers.</li>
  
 <li> Axiom uses the command <tt>positiveRemainder</tt> instead of
      <tt>mod</tt> command, so let's define <tt>mod</tt> to be a renaming 
      of the <tt>positiveRemainder</tt> function:
  <ul>
   <li> <span class="cmd">mod ==> positiveRemainder</span></li>
  </ul>
 </li>
 <li> Now the commands <tt>addmod</tt>, <tt>submod</tt>, <tt>mulmod</tt>, and
      <tt>invmod</tt> can be used to perform modular arithmetic.  Here's a few
      examples; first a simple modulus calculation:
  <ul>
   <li> <span class="cmd">-10 mod 3</span></li>
  </ul>
 </li>
 <li> Addition, subtraction and multiplication mod 14:
  <ul>
   <li> <span class="cmd">addmod(10,13,14)</span></li>
   <li> <span class="cmd">submod(17,23,14)</span></li>
   <li> <span class="cmd">mulmod(13,27,14)</span></li>
  </ul>
 </li>
 <li> Powers and inverses:
  <ul>
   <li> <span class="cmd">powmod(19,237,14)</span></li>
   <li> <span class="cmd">invmod(11,14)</span></li>
  </ul>
 </li>
 <li> Find out what happens if you try to take an inverse of a number not
      relatively prime to the modulus:
  <ul>
   <li> <span class="cmd">invmod(12,14)</span></li>
  </ul>
 </li>
 <li> Try these command with a few other numbers, and test out the examples in
  the notes.
 </li>
 <li> The second method, which can be more powerful, is to treat all numbers
  as elements of the residue values 0 to <tt>n-1</tt>.  This can be done with 
  the <tt>IntegerMod</tt> construction, or its abbreviation <tt>ZMOD</tt>.  
  Here's a few examples:
  <ul>
   <li> <span class="cmd">a:=11::ZMOD 14</span></li>
  </ul>
 </li>
 <li> This declares the variable <tt>a</tt> to be a member of the residue 
      class modulo 14.  Now all arithmetic including <tt>a</tt> will be 
      reduced to this same class of values:
  <ul>
   <li> <span class="cmd">a+25</span></li>
   <li> <span class="cmd">a*39</span></li>
   <li> <span class="cmd">a^537</span></li>
  </ul>
 </li>
 <li> Inversion can be done with the <tt>recip</tt> command:
  <ul>
   <li> <span class="cmd">recip(a)</span></li>
  </ul>
 </li>  
 <li> We don't have to define a variable first.  All the above commands could
      be equivalently written as:
  <ul>
   <li> <span class="cmd">(11::ZMOD 14)+25</span></li>
   <li> <span class="cmd">11::ZMOD 14*39</span></li>
   <li> <span class="cmd">11::ZMOD 14^537</span></li>
   <li> <span class="cmd">recip(11::ZMOD 14)</span></li>
  </ul>
 </li>  
 <li> If the modulus is a prime, then division (by non-zero values) is also
      possible.  Axiom provides the alternative construction 
      <tt>PrimeField</tt> or more simply <tt>PF</tt>.  For example:
  <ul>
   <li> <span class="cmd">a:=7::PF 11</span></li>
  </ul>
 </li>
 <li> All the above arithmetic operations of addition, subtraction, 
      multiplication and powers work, but now we also have inversion:
  <ul>
   <li> <span class="cmd">1/a</span></li>
  </ul>
 </li>
 <li> Using any of the methods you like, test out Fermat's theorem for a large
      prime <tt>p</tt> and an integer <tt>a</tt>.
 </li>  
 <li> Euler's totient function is implemented with <tt>eulerPhi</tt>.  Choose
      a large integer <tt>n</tt>, a random <tt>a</tt> with 
      <tt>gcd(a,n)=1</tt> , and test Euler's theorem
 </li>
</ul>
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