C++数论库：NTL

NTL

官网：https://libntl.org/doc/tour.html

NTL is a high-performance, portable C++ library providing data structures and algorithms for arbitrary length integers; for vectors, matrices, and polynomials over the integers and over finite fields; and for arbitrary precision floating point arithmetic.

NTL provides high quality implementations of state-of-the-art algorithms for:

arbitrary length integer arithmetic and arbitrary precision floating point arithmetic;
polynomial arithmetic over the integers and finite fields including basic arithmetic, polynomial factorization, irreducibility testing, computation of minimal polynomials, traces, norms, and more;
lattice basis reduction, including very robust and fast implementations of Schnorr-Euchner, block Korkin-Zolotarev reduction, and the new Schnorr-Horner pruning heuristic for block Korkin-Zolotarev;
basic linear algebra over the integers, finite fields, and arbitrary precision floating point numbers.

类型介绍

The basic ring classes are:

ZZ: big integers
ZZ_p: big integers modulo p
zz_p: integers mod “single precision” p
GF2: integers mod 2
ZZX: univariate polynomials over ZZ
ZZ_pX: univariate polynomials over ZZ_p
zz_pX: univariate polynomials over zz_p
GF2X: polynomials over GF2
ZZ_pE: ring/field extension over ZZ_p
zz_pE: ring/field extension over zz_p
GF2E: ring/field extension over GF2
ZZ_pEX: univariate polynomials over ZZ_pE
zz_pEX: univariate polynomials over zz_pE
GF2EX: univariate polynomials over GF2E

使用

常用函数

SetSeed(const ZZ& s)：设置PRF种子

RandomBnd(ZZ& x, const ZZ& n)： $\in \{0,1,\cdots n-1\}$ ，如果 $\le 0$ 那么 $x = 0$

RandomBits(ZZ& x, long l)：随机生成 $l$ 比特的整数

ZZ p(17)：初始化整数为17，这里参数类型是long

p = to_ZZ("123")：读入字符串，可输入大整数

GenPrime(p, 8)：随机生成8比特素数

ZZ_p::init(p)：初始化环 $Z_p$

ZZ_p a(2)：初始化为 $\mod p$ ，这里参数类型是long

random(a)：随机生成 $Z_p$ 中元素

ZZ_pX m： $Z_p[x]$ 中的多项式，记录为向量 $Z_p^n$

SetCoeff(m, 5)：将 $x^5$ 系数置为 1

m[0]=1：将 $x^0$ 系数置为 1

BuildIrred(m, 3)：随机生成3次不可约多项式

ZZ_pE::init(m)：初始化环 $Z_p[x]/(m(x))$ ，若 $p$ 是素数且 $m (x)$ 是d次不可约多项式，那么它同构于有限域 $GF(p^d)$

ZZ_pEX f, g, h： $GF(p^d)[x]$ 上的多项式，记录为向量 $GF(p^d)^n$

random(f, 5)：随机生成5次多项式

h = sqr(g) % f：计算 $\equiv g^2 \mod f$

环 $GF(p^d)[x]/(x^n-1)$ 上多项式运算：

#include <iostream>#include <NTL/ZZ_p.h> // integers mod p
#include <NTL/ZZ_pX.h> // polynomials over ZZ_p
#include <NTL/ZZ_pE.h> // ring/field extension of ZZ_p
#include <NTL/ZZ_pEX.h> // polynomials over ZZ_pE
#include <NTL/ZZ_pXFactoring.h>
#include <NTL/ZZ_pEXFactoring.h>using namespace std;
using namespace NTL;#pragma comment(lib, "NTL")int main()
{ZZ p(17); //初始化为17//群Z_pZZ_p::init(p); //随机生成Z_p[x]中的d次不可约多项式int d = 4;ZZ_pX m;BuildIrred(m, d); //域GF(p^d) = Z_p[x]/m(x)ZZ_pE::init(m); //GF(p^d)[x]中的多项式ZZ_pEX f, g, h; // f(x) = x^8 - 1SetCoeff(f, 8); //将 x^8 系数置为 1SetCoeff(f, 0, -1); //将 x^0 系数置为 -1//随机生成5次多项式random(g, 5);// 环上多项式的运算：h = g^2 mod fh = sqr(g) % f; cout << "p = " << p << endl;cout << "d = " << d << endl;cout << "m(x) = " << m << endl;cout << "f = " << f << endl;cout << "g = " << g << endl;cout << "h = " << h << endl;return 0;
}