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C++数论库:NTL

平嘉熙
2023-12-01

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) x ∈ { 0 , 1 , ⋯ n − 1 } x \in \{0,1,\cdots n-1\} x{0,1,n1},如果 n ≤ 0 n \le 0 n0 那么 x = 0 x=0 x=0

RandomBits(ZZ& x, long l):随机生成 l l 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 Z_p Zp

ZZ_p a(2):初始化为 2 m o d    p 2 \mod p 2modp,这里参数类型是long

random(a):随机生成 Z p Z_p Zp中元素

ZZ_pX m Z p [ x ] Z_p[x] Zp[x]中的多项式,记录为向量 Z p n Z_p^n Zpn

SetCoeff(m, 5):将 x 5 x^5 x5系数置为 1

m[0]=1:将 x 0 x^0 x0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 ) ) Z_p[x]/(m(x)) Zp[x]/(m(x)),若 p p p是素数且 m ( x ) m(x) m(x)是d次不可约多项式,那么它同构于有限域 G F ( p d ) GF(p^d) GF(pd)

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

random(f, 5):随机生成5次多项式

h = sqr(g) % f:计算 h ≡ g 2 m o d    f h \equiv g^2 \mod f hg2modf

  • G F ( p d ) [ x ] / ( x n − 1 ) GF(p^d)[x]/(x^n-1) GF(pd)[x]/(xn1)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_p
    ZZ_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 - 1
	SetCoeff(f, 8); //将 x^8 系数置为 1
    SetCoeff(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 f
	h = 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;
}
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