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<html><head><title>A Tour of NTL: Examples: Big Integers </title></head><body bgcolor="#fff9e6"><center><img src="arrow1.gif" alt="[Previous]" align=bottom> <a href="tour-examples.html"><img src="arrow2.gif" alt="[Up]" align=bottom></a> <a href="tour-ex2.html"> <img src="arrow3.gif" alt="[Next]" align=bottom></a></center><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>,whichrepresents "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 <NTL/ZZ.h>int main(){ ZZ a, b, c; cin >> a; cin >> b; c = (a+1)*(b+1); cout << c << "\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 andassignment operators can be used in a direct and natural way.The <tt>C++</tt> compiler and the NTL library routines automatically take careof all the bookkeeping involvedwith memory management and temporary objects.<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 <NTL/ZZ.h>int main(){ ZZ acc, val; acc = 0; while (SkipWhiteSpace(cin)) { cin >> val; acc += val*val; } cout << acc << "\n";}</pre><p>The function <tt>SkipWhiteSpace</tt> is defined by NTL.It skips over white space, and returns 1 if there is somethingfollowing it.This function is useful, becauseNTL's input operators raise an error if an inputis 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 leastan exception, is a bad idea, since the caller of the I/Oroutine must constantly check the status of the inputstream.<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& a, const ZZ& e, const ZZ& n){ if (e == 0) return to_ZZ(1); long k = NumBits(e); ZZ res; res = 1; for (long i = k-1; i >= 0; i--) { res = (res*res) % n; if (bit(e, i) == 1) res = (res*a) % n; } if (e < 0) return InvMod(res, n); else return res;}</pre><p>Note that as an alternative, we could implement the inner loopas 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 thatstores its result in its first argument.The reason for using the procedural variant is efficieny:on every iteration through the above loop, the functional formof <tt>SqrMod</tt> will cause a temporary <tt>ZZ</tt> object tobe created and destroyed, whereas the procedural version will not create any temporaries.Where performance is critical, the procedural versionis to be preferred.Although it is usually silly to get too worked up about performance,it may be reasonable to argue that modular exponentiationis an important enough routine that it should be as fast as possible.<p>Note that when the functional version of a functioncan be naturally named with an operator, this is done.So for example, NTL provides a 3-argument <tt>mul</tt> routinefor <tt>ZZ</tt> multiplication, and a functional versionwhose name is <tt>operator *</tt>, and not <tt>mul</tt>.<p>While we are taking about temporaries, consider the first versionof 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 forcleaning 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 sophisticatedprime test.<p><pre>#include <NTL/ZZ.h>long witness(const ZZ& n, const ZZ& 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 < k && z != 1); return z != 1 || y != n-1;}long PrimeTest(const ZZ& n, long t){ if (n <= 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 < 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 << "n: "; cin >> n; if (PrimeTest(n, 10)) cout << n << " is probably prime\n"; else cout << n << " is composite\n";}</pre><p>Note that in NTL, there are typically a number of ways tocompute 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 >> 1; while (!IsOdd(m)) { k++; m >>= 1; }</pre>It turns out that this is actually not significantly more efficient than the original version, because the implementationoptimizes multiplication and division by 2.<p>The following is more efficient:<pre> k = 1; while (bit(n, k) == 0) k++; m = n >> 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 thatallows 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 destinationof 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 ina natural way.As was already mentioned, NTL does not support implicit type conversion;rather, for basic operations, it simply overloads the operatorsor 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 convertedto a <tt>ZZ</tt>.Moreover, wherever possible, the implementation does this as efficiently as possible, and usually avoids the creationof a temporary <tt>ZZ</tt>.<p>There are also procedural versions for all the basic arithmeticoperations:<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 primesand 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 modularsquare root</ul><p>Most of these functions also have pure <tt>long</tt> versions as well, and as usual, there are both functional and proceduralvariants.<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 basicservices provided by NTL.<p><center><img src="arrow1.gif" alt="[Previous]" align=bottom> <a href="tour-examples.html"><img src="arrow2.gif" alt="[Up]" align=bottom></a> <a href="tour-ex2.html"> <img src="arrow3.gif" alt="[Next]" align=bottom></a></center></body></html>⌨️ 快捷键说明
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