A Tour of NTL: Examples: Modular Arithmetic

NTL also supports modular integer arithmetic.
The class `ZZ_p`
represents the integers mod `p`.
Despite the notation, `p` need not in general be prime,
except in situations where this is mathematically required.
The classes `vec_ZZ_p`, `mat_ZZ_p`,
and `ZZ_pX` represent vectors, matrices, and polynomials
mod `p`, and work much the same way as the corresponding
classes for `ZZ`.

Here is a program that reads a prime number `p`,
and a polynomial `f` modulo `p`, and factors it.

#include <NTL/ZZ_pXFactoring.h> int main() { ZZ p; cin >> p; ZZ_p::init(p); ZZ_pX f; cin >> f; vec_pair_ZZ_pX_long factors; CanZass(factors, f); cout << factors << "\n"; // calls "Cantor/Zassenhaus" algorithm }

As a program is running, NTL keeps track of a "current modulus"
for the class `ZZ_p`, which can be initialized or changed
using `ZZ_p::init`.
This must be done before any variables are declared or
computations are done that depend on this modulus.

Please note that for efficiency reasons, NTL does not make any attempt to ensure that variables declared under one modulus are not used under a different one. If that happens, the behavior of a program in this case is completely unpredictable.

Here are two more examples that illustrate the `ZZ_p`-related
classes.
The first is a vector addition routine (already supplied by NTL):

#include <NTL/vec_ZZ_p.h> void add(vec_ZZ_p& x, const vec_ZZ_p& a, const vec_ZZ_p& b) { long n = a.length(); if (b.length() != n) Error("vector add: dimension mismatch"); x.SetLength(n); long i; for (i = 0; i < n; i++) add(x[i], a[i], b[i]); }

The second example is an inner product routine (also supplied by NTL):

#include <NTL/vec_ZZ_p.h> void InnerProduct(ZZ_p& x, const vec_ZZ_p& a, const vec_ZZ_p& b) { long n = min(a.length(), b.length()); long i; ZZ accum, t; accum = 0; for (i = 0; i < n; i++) { mul(t, rep(a[i]), rep(b[i])); add(accum, accum, t); } conv(x, accum); }

This second example illustrates two things.
First, it illustrates the use of function `rep` which
returns a read-only reference to the representation of a `ZZ_p`
as a `ZZ` between `0` and `p-1`.
Second, it illustrates a useful algorithmic technique,
whereby one computes over `ZZ`, reducing mod `p`
only when necessary.
This reduces the number of divisions that need to be performed significantly,
leading to much faster execution.

The class `ZZ_p` supports all the basic arithmetic
operations in both operator and procedural form.
All of the basic operations support a "promotion logic",
promoting `long` to `ZZ_p`.

Note that the class `ZZ_p` is mainly useful only
when you want to work with vectors, matrices, or polynomials
mod `p`.
If you just want to do some simple modular arithemtic,
it is probably easier to just work with `ZZ`s directly.
This is especially true if you want to work with many different
moduli: modulus switching is supported, but it is a bit awkward.

The class `ZZ_pX` supports all the basic arithmetic
operations in both operator and procedural form.
All of the basic operations support a "promotion logic",
promoting both `long` and `ZZ_p` to `ZZ_pX`.

See `ZZ_p.txt` for details on `ZZ_p`;
see `ZZ_pX.txt` for details on `ZZ_pX`;
see `ZZ_pXFactoring.txt` for details on
the routines for factoring polynomials over `ZZ_p`;
see `vec_ZZ_p.txt` for details on `vec_ZZ_p`;
see `mat_ZZ_p.txt` for details on `mat_ZZ_p`.

There is a mechanism for saving and restoring a modulus, which the following example illustrates. This routine takes as input an integer polynomial and a prime, and tests if the polynomial is irreducible modulo the prime.

#include <NTL/ZZX.h> #include <NTL/ZZ_pXFactoring.h> long IrredTestMod(const ZZX& f, const ZZ& p) { ZZ_pBak bak; // save current modulus in bak bak.save(); ZZ_p::init(p); // set the current modulus to p return DetIrredTest(to_ZZ_pX(f)); // old modulus is restored automatically when bak is destroyed // upon return }

The modulus switching mechanism is actually quite a bit more general and flexible than this example illustrates.

The function `to_ZZ_pX` is yet another of NTL's many
conversion functions.
We could also have used the equivalent procedural form:

ZZ_pX f1; conv(f1, f); return DetIrredTest(f1);

Suppose in the above example that `p` is known in advance
to be a small, single-precision prime.
In this case, NTL provides a class `zz_p`, that
acts just like `ZZ_p`,
along with corresponding classes `vec_zz_p`,
`mat_zz_p`, and `zz_pX`.
The interfaces to all of the routines are generally identical
to those for `ZZ_p`.
However, the routines are much more efficient, in both time and space.

For small primes, the routine in the previous example could be coded as follows.

#include <NTL/ZZX.h> #include <NTL/lzz_pXFactoring.h> long IrredTestMod(const ZZX& f, long p) { zz_pBak bak; bak.save(); zz_p::init(p); return DetIrredTest(to_zz_pX(f)); }

The following is a routine (essentially the same as implemented in NTL) for computing the GCD of polynomials with integer coefficients. It uses a "modular" approach: the GCDs are computed modulo small primes, and the results are combined using the Chinese Remainder Theorem (CRT). The small primes are specially chosen "FFT primes", which are of a special form that allows for particular fast polynomial arithmetic.

#include <NTL/ZZX.h> void GCD(ZZX& d, const ZZX& a, const ZZX& b) { if (a == 0) { d = b; if (LeadCoeff(d) < 0) negate(d, d); return; } if (b == 0) { d = a; if (LeadCoeff(d) < 0) negate(d, d); return; } ZZ c1, c2, c; ZZX f1, f2; content(c1, a); divide(f1, a, c1); content(c2, b); divide(f2, b, c2); GCD(c, c1, c2); ZZ ld; GCD(ld, LeadCoeff(f1), LeadCoeff(f2)); ZZX g, res; ZZ prod; zz_pBak bak; bak.save(); long FirstTime = 1; long i; for (i = 0; ;i++) { zz_p::FFTInit(i); long p = zz_p::modulus(); if (divide(LeadCoeff(f1), p) || divide(LeadCoeff(f2), p)) continue; zz_pX G, F1, F2; zz_p LD; conv(F1, f1); conv(F2, f2); conv(LD, ld); GCD(G, F1, F2); mul(G, G, LD); if (deg(G) == 0) { res = 1; break; } if (FirstTime || deg(G) < deg(g)) { prod = 1; g = 0; FirstTime = 0; } else if (deg(G) > deg(g)) { continue; } if (!CRT(g, prod, G)) { PrimitivePart(res, g); if (divide(f1, res) && divide(f2, res)) break; } } mul(d, res, c); if (LeadCoeff(d) < 0) negate(d, d); }

See `lzz_p.txt` for details on `zz_p`;
see `lzz_pX.txt` for details on `zz_pX`;
see `lzz_pXFactoring.txt` for details on
the routines for factoring polynomials over `zz_p`;
see `vec_lzz_p.txt` for details on `vec_zz_p`;
see `mat_lzz_p.txt` for details on `mat_zz_p`.

Arithmetic mod 2 is such an important special case that NTL
provides a class `GF2`, that
acts just like `ZZ_p` when `p == 2`,
along with corresponding classes `vec_GF2`,
`mat_GF2`, and `GF2X`.
The interfaces to all of the routines are generally identical
to those for `ZZ_p`.
However, the routines are much more efficient, in both time and space.

This example illustrates the `GF2X` and `mat_GF2`
classes with a simple routine to test if a polynomial over GF(2)
is irreducible using linear algebra.
NTL's built-in irreducibility test is to be preferred, however.

#include <NTL/GF2X.h> #include <NTL/mat_GF2.h> long MatIrredTest(const GF2X& f) { long n = deg(f); if (n <= 0) return 0; if (n == 1) return 1; if (GCD(f, diff(f)) != 1) return 0; mat_GF2 M; M.SetDims(n, n); GF2X x_squared = GF2X(2, 1); GF2X g; g = 1; for (long i = 0; i < n; i++) { VectorCopy(M[i], g, n); M[i][i] += 1; g = (g * x_squared) % f; } long rank = gauss(M); if (rank == n-1) return 1; else return 0; }

Note that the statement

g = (g * x_squared) % f;could be replace d by the more efficient code sequence

MulByXMod(g, g, f); MulByXMod(g, g, f);but this would not significantly impact the overall running time, since it is the Gaussian elimination that dominates the running time.

See `GF2.txt` for details on `GF2`;
see `GF2X.txt` for details on `GF2X`;
see `GF2XFactoring.txt` for details on
the routines for factoring polynomials over `GF2`;
see `vec_GF2.txt` for details on `vec_GF2`;
see `mat_GF2.txt` for details on `mat_GF2`.