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A Tour of NTL: Examples: Big Integers </title>
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<h1> 
<p align=center>
A Tour of NTL: Examples: Big Integers
</p>
</h1>

<p> <hr>  <p>

The first example makes use of the class
<tt>ZZ</tt>,
which
represents "big integers": signed, arbitrary length integers.
This program reads two big integers <tt>a</tt> and <tt>b</tt>,
and prints <tt>(a+1)*(b+1)</tt>.

<p>
<pre>
#include &lt;NTL/ZZ.h&gt;

NTL_CLIENT

int main()
{
   ZZ a, b, c; 

   cin &gt;&gt; a; 
   cin &gt;&gt; b; 
   c = (a+1)*(b+1);
   cout &lt;&lt; c &lt;&lt; "\n";
}
</pre>

<p>

This program declares three variables <tt>a</tt>, <tt>b</tt>,
and <tt>c</tt> of type <tt>ZZ</tt>.
The values <tt>a</tt> and <tt>b</tt> are read from standard input.
The value <tt>c</tt> is then computed as <tt>(a+1)*(b+1)</tt>.
Finally, the value of <tt>c</tt> is printed to the standard output.

<p>
Note that one can compute with <tt>ZZ</tt>s much as with ordinary
<tt>int</tt>s, in that most of the standard arithmetic and
assignment operators can be used in a direct and natural way.
The <tt>C++</tt> compiler and the NTL library routines 
automatically take care
of all the bookkeeping involved
with memory management and temporary objects.

<p>
Note the macro <tt>NTL_CLIENT</tt>.
When compiling NTL is ISO mode (the default), this
expands to 
<pre>
   using namespace std;
   using namespace NTL;
</pre>
When compiling NTL in traditional mode, this expands
to the empty string.

<p> <hr> <p>

Here's a program that reads a list of integers from standard 
input and prints the sum of their squares.

<p>
<pre>
#include &lt;NTL/ZZ.h&gt;

NTL_CLIENT

int main()
{
   ZZ acc, val;

   acc = 0;
   while (SkipWhiteSpace(cin)) {
      cin &gt;&gt; val;
      acc += val*val;
   }

   cout &lt;&lt; acc &lt;&lt; "\n";
}
</pre>

<p>

The function <tt>SkipWhiteSpace</tt> is defined by NTL.
It skips over white space, and returns 1 if there is something
following it.
This function is useful, because
NTL's input operators raise an error if an input
is missing or ill-formed. 
This is different from the standard I/O library,
which does not raise an error.
Personally, I find that not raising an error, or at least
an exception, is a bad idea, since the caller of the I/O
routine must constantly check the status of the input
stream.




<p>
<hr>
<p>

Here's a simple modular exponentiation routine for computing
<tt>a^e mod n</tt>.
NTL already provides a more sophisticated one, though.

<p>
<pre>
ZZ PowerMod(const ZZ&amp; a, const ZZ&amp; e, const ZZ&amp; n)
{
   if (e == 0) return to_ZZ(1);

   long k = NumBits(e);

   ZZ res;
   res = 1;

   for (long i = k-1; i &gt;= 0; i--) {
      res = (res*res) % n;
      if (bit(e, i) == 1) res = (res*a) % n;
   }

   if (e &lt; 0)
      return InvMod(res, n);
   else
      return res;
}
</pre>
<p>

Note that as an alternative, we could implement the inner loop
as follows:

<pre>
   res = SqrMod(res, n);
   if (bit(e, i) == 1) res = MulMod(res, a, n);
</pre>

We could also write this as:

<pre>
   SqrMod(res, res, n);
   if (bit(e, i) == 1) MulMod(res, res, a, n);
</pre>

This illustrates an important point about NTL's programming interface.
For every function in NTL, there is a procedural version that
stores its result in its first argument.
The reason for using the procedural variant is efficieny:
on every iteration through the above loop, the functional form
of <tt>SqrMod</tt> will cause a temporary <tt>ZZ</tt> object to
be created and destroyed, whereas the procedural version 
will not create any temporaries.
Where performance is critical, the procedural version
is to be preferred.
Although it is usually silly to get too worked up about performance,
it may be reasonable to argue that modular exponentiation
is an important enough routine that it should be as fast as possible.

<p>

Note that when the functional version of a function
can be naturally named with an operator, this is done.
So for example, NTL provides a 3-argument <tt>mul</tt> routine
for <tt>ZZ</tt> multiplication, and a functional version
whose name is <tt>operator *</tt>, and not <tt>mul</tt>.

<p>

While we are taking about temporaries, consider the first version
of the inner loop.
Execution of the statement
<pre>
   res = (res*res) % n;
</pre>
will result in the creation of two temporary objects,
one for the product, and one for the result of the mod operation,
whose value is copied into <tt>res</tt>.
Of course, the compiler automatically generates the code for
cleaning up temporaries and other local objects at the right time.
The programmer does not have to worry about this.


<p> <hr> <p>

This example is a bit more interesting.
The following program prompts the user for an input,
and applies a simple probabilistic primality test.
Note that NTL already provides a slightly more sophisticated
prime test.

<p>
<pre>
#include &lt;NTL/ZZ.h&gt;

NTL_CLIENT

long witness(const ZZ&amp; n, const ZZ&amp; x)
{
   ZZ m, y, z;
   long j, k;

   if (x == 0) return 0;

   // compute m, k such that n-1 = 2^k * m, m odd:

   k = 1;
   m = n/2;
   while (m % 2 == 0) {
      k++;
      m /= 2;
   }

   z = PowerMod(x, m, n); // z = x^m % n
   if (z == 1) return 0;

   j = 0;
   do {
      y = z;
      z = (y*y) % n; 
      j++;
   } while (j &lt; k &amp;&amp; z != 1);

   return z != 1 || y != n-1;
}


long PrimeTest(const ZZ&amp; n, long t)
{
   if (n &lt;= 1) return 0;

   // first, perform trial division by primes up to 2000

   PrimeSeq s;  // a class for quickly generating primes in sequence
   long p;

   p = s.next();  // first prime is always 2
   while (p && p < 2000) {
      if ((n % p) == 0) return (n == p);
      p = s.next();  
   }

   // second, perform t Miller-Rabin tests

   ZZ x;
   long i;

   for (i = 0; i &lt; t; i++) {
      x = RandomBnd(n); // random number between 0 and n-1

      if (witness(n, x)) 
         return 0;
   }

   return 1;
}

int main()
{
   ZZ n;

   cout &lt;&lt; "n: ";
   cin &gt;&gt; n;

   if (PrimeTest(n, 10))
      cout &lt;&lt; n &lt;&lt; " is probably prime\n";
   else
      cout &lt;&lt; n &lt;&lt; " is composite\n";
}
</pre>
<p>

Note that in NTL, there are typically a number of ways to
compute the same thing.
For example, consider the computation of <tt>m</tt> and <tt>k</tt>
in function <tt>witness</tt>.
We could have written it thusly:

<pre>
   k = 1;
   m = n &gt;&gt; 1;
   while (!IsOdd(m)) {
      k++;
      m &gt;&gt;= 1;
   }
</pre>

It turns out that this is actually not significantly more 
efficient than the original version, because the implementation
optimizes multiplication and division by 2.

<p>

The following is more efficient:

<pre>
   k = 1;
   while (bit(n, k) == 0) k++;
   m = n &gt;&gt; k;
</pre>

As it happens, there is a built-in NTL routine that does just what we want:

<pre>
   m = n-1;
   k = MakeOdd(m);
</pre>



<p> <hr> <p>

Having seen a number of examples involving <tt>ZZ</tt>s,
let's look at the <tt>ZZ</tt> interface in a bit more detail.

<p>

<b>
Constructors, assignment, and conversions
</b>

<p>

When you declare an object of type <tt>ZZ</tt>, 
the default constructor initializes to the value <tt>0</tt>.
As we have already seen, there is an assignment operator that
allows one to copy the value of one <tt>ZZ</tt> to another.
Note that these copies (like almost all copies in NTL) are "deep",
i.e., the actual data is copied, and not just a pointer.
Of course, if the amount of space allocated by the destination
of the assignment is insufficient to hold the value of the source,
space is automatically re-allocated.

<p>
One can also assign a value of type <tt>long</tt> to a <tt>ZZ</tt>:
<pre>
   ZZ x;
   x = 1;
</pre>

<p>
Note that one cannot write
<pre>
   ZZ x = 1;  // error
</pre>
to initialize a <tt>ZZ</tt>.
As a design principle, NTL avoids implicit conversions, and unfortunately,
the only way to allow initializations like this in <tt>C++</tt>
is to define an implicit conversion operator.
Instead, one could write
<pre>
   ZZ x = to_ZZ(1);
</pre>
This is an example of one of NTL's conversion routines.
For very large constants, one can write:
<pre>
   ZZ x = to_ZZ("99999999999999999999");
</pre>
These examples illustrate conversion rountines in their 
functional forms.
The corresponding procedural forms are all called <tt>conv</tt>, e.g.,
<pre>
   ZZ x;
   conv(x, 1);
   conv(x, "99999999999999999999");
</pre>

<p>
<b>
Functionality
</b>
<p>

All of the basic arithmetic operators are supported,
including comparison, arithmetic, shift, and bit-wise logical operations.
One can mix <tt>ZZ</tt>s and <tt>long</tt>s in any expresion in
a natural way.
As was already mentioned, NTL does not support implicit type conversion;
rather, for basic operations, it simply overloads the operators
or functions in a way to achieve a kind of "promotion logic":
if one input is a <tt>ZZ</tt> and the other is a <tt>long</tt>
(or something that implicitly converts to a <tt>long</tt>, like 
an <tt>int</tt>), the <tt>long</tt> input is effectively converted
to a <tt>ZZ</tt>.
Moreover, wherever possible, the implementation does this 
as efficiently as possible, and usually avoids the creation
of a temporary <tt>ZZ</tt>.

<p>
There are also procedural versions for all the basic arithmetic
operations:
<pre>
   add, sub, negate, mul, sqr, div, rem, DivRem, 
   LeftShift, RightShift,
   bit_and, bit_or, bit_xor
</pre>

<p>
There are many other routines.
Here is a brief summary:
<ul>
<li>
<tt>GCD</tt> -- computes greatest common divisor of two integers
<li>
<tt>XGCD</tt> -- extended Euclidean algorithm
<li>
<tt>AddMod</tt>, <tt>SubMod</tt>, <tt>NegateMod</tt>, 
<tt>MulMod</tt>, <tt>SqrMod</tt>, <tt>InvMod</tt>,
<tt>PowerMod</tt> -- routines for modular arithmetic,
including inversion and exponentiation
<li>
<tt>NumBits</tt> -- length of binary representation
<li>
<tt>bit</tt> -- extract a bit
<li>
<tt>ZZFromBytes</tt>, <tt>BytesFromZZ</tt> -- 
convert between octet strings and <tt>ZZ</tt>s
<li>
<tt>RandomBnd</tt>, <tt>RandomBits</tt>, <tt>RandomLen</tt> --
routines for generating pseudo-random numbers
<li>
<tt>GenPrime</tt>, <tt>ProbPrime</tt> -- routines for generating primes
and testing primality
<li>
<tt>power</tt> -- (non-modular) exponentiation
<li>
<tt>SqrRoot</tt> -- integer part of square root
<li>
<tt>Jacobi</tt>, <tt>SqrRootMod</tt> -- Jacobi symbol and modular
square root
</ul>

<p>
Most of these functions also have pure <tt>long</tt> versions as 
well, and as usual, there are both functional and procedural
variants.

<p>
There are other functions as well.
See <a href="ZZ.txt"><tt>ZZ.txt</tt></a> for complete details.
Also see <a href="tools.txt"><tt>tools.txt</tt></a> for some basic
services provided by NTL.


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