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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>` (a.k.a., `vec_ZZ_p`), `Mat<ZZ_p>` (a.k.a., `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> using namespace std; using namespace NTL; int main() { ZZ p; cin >> p; ZZ_p::init(p); ZZ_pX f; cin >> f; Vec< Pair< ZZ_pX, long > > factors; CanZass(factors, f); // calls "Cantor/Zassenhaus" algorithm cout << factors << "\n"; }

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 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/ZZ_p.h> using namespace std; using namespace NTL; 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/ZZ_p.h> using namespace std; using namespace NTL; 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 mathematical operations on `Vec<ZZ_p>`'s; see `mat_ZZ_p.txt` for details on mathematical operations on `Mat<ZZ_p>`'s.

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> using namespace std; using namespace NTL; long IrredTestMod(const ZZX& f, const ZZ& p) { ZZ_pPush push(p); // save current modulus and install p // as current modulus return DetIrredTest(conv<ZZ_pX>(f)); // old modulus is restored automatically when push is destroyed // upon return }

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

Note the use of the conversion function `conv<ZZ_pX>`. 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> using namespace std; using namespace NTL; long IrredTestMod(const ZZX& f, long p) { zz_pPush push(p); return DetIrredTest(conv<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> using namespace std; using namespace NTL; 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_pPush push; // save current modulus, restore upon return 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`.

NTL provides a number of "residue class" types with a dynamic modulus stored as a global variable: the types `ZZ_p` and `zz_p`, discussed above, as well as the types `ZZ_pE`, `zz_pE`, and `GF2E`, discussed later.

Some caution must be used so that a variable constructed under one modulus is not used "out of context", when a different modulus, or perhaps no modulus, is installed as the current modulus. While arithmetic operations should certainly be avoided, NTL does take care to allow for certain operations to be safely performed "out of context". These operations include default and copy constructors, as well as assignment.

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. Note that `Vec<GF2>` is an explicit specialization of the template class `Vec<T>`, with a special implementation that packs the coefficients into the bits of a machine word. You need to include the header file `<NTL/vec_GF2.h>` to use the class `Vec<GF2>`.

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> using namespace std; using namespace NTL; 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(INIT_MONO, 2); 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`.