Modular Arithmetic （ Arithmetic and Algebra） CGAL 4.13 -User Manual

1 Introduction

Modular arithmetic is a fundamental tool in modern algebra systems. In conjunction with the Chinese remainder theorem it serves as the workhorse in several algorithms computing the gcd, resultant etc. Moreover, it can serve as a very efficient filter, since it is often possible to exclude that some value is zero by computing its modular correspondent with respect to one prime only.

模运算是现代代数系统的基本工具。与中国余数定理（Chinese remainder theorem）结合，是若干计算GCD和合矢量（resultant ）算法的核心。另外，它能够作为一个十分高效的工具，原因在于它常常可能排除计算一个素数（ since it is often possible to exclude that some value is zero by computing its modular correspondent with respect to one prime only.）

2 Residue and Modularizable

First of all, this package introduces a type Residue. It represents Z/pZ for some prime p. The prime number p is stored in a static member variable. The class provides static member functions to change this value.

Changing the prime invalidates already existing objects of this type. However, already existing objects do not lose their value with respect to the old prime and can be reused after restoring the old prime. Since the type is based on double arithmetic the prime is restricted to values less than 226. The initial value of p is 67108859.

Moreover, the package introduces the concept Modularizable. An algebraic structure T is considered as Modularizable if there is a mapping from T into an algebraic structure that is based on the type Residue. For scalar types, e.g. Integers, this mapping is just the canonical homomorphism into Z/pZ represented by Residue. For compound types, e.g. Polynomials, the mapping is applied to the coefficients of the compound type. The mapping is provided by the class Modular_traits<T>. The class Modular_traits<T> is designed such that the concept Modularizable can be considered as optional, i.e., Modular_traits<T> provides a tag that can be used for dispatching.

首先，本包引入了Residue类型。它表示某素数p的乘法群（Z/pZ ）；素数（prime）p是保存在一个静态成员变量。一个类提供的静态函数用来改变这个值。（数学上，同余（英语：congruence modulo，符号：≡）是数论中的一种等价关系。当两个整数除以同一个正整数，若得相同余数，则二整数同余。同余是抽象代数中的同余关系的原型。在同余理论中，模 n 的互质同余类组成一个乘法群，称为整数模 n 乘法群，也称为模 n 既约剩余类。在环理论中，一个抽象代数的分支，也称这个群为整数模 n 的环的单位群（单位是指乘法可逆元）。这个群是数论的基石，在密码学、整数分解和素性测试均有运用。例如，关于这个群的阶（即群的“大小”），我们可以确定如果 n 是质数当且仅当阶数为 n-1。）

改变素数的值使现存的这个类的对象非法。但现存的对象不会丢失该值，通过恢复旧的素数就可以重用。因该类型基于双精度算术运算，素数被限制在小于226，初始p的值是67108859。

另外，本包引入了“可余的”或可模的 Modularizable概念。如果一个代数结构T与一个基于Residue的结构存在一个映射（mapping ），则T被认为是Modularizable。对于标量，即整数，映射仅仅是Z/pZ（乘法群）中Residue表示的典型同态（canonical homomorphism）；对于复合类型，即多项式，映射运用于复合类型的系数。映射由类 Modular_traits<T>提供。类 Modular_traits<T>的设计可以将概念Modularizable 作为可选的，即 Modular_traits<T> 提供了一个标签用于调度。

2.1 Example

In the following example modular arithmetic is used as a filter on order to avoid unnecessary gcd computations of polynomials. A gcd computation can be very costly due to coefficient growth within the Euclidean algorithm.

The general idea is that firstly the gcd is computed with respect to one prime only. If this modular gcd is constant we can (in most cases) conclude that the actual gcd is constant as well.

For this purpose the example introduces the function may_have_common_factor(). Note that there are two versions of this function, namely for the case that the coefficient type is Modularizable and that it is not. If the type is not Modularizable the filter is just not applied and the function returns true.

Further note that the implementation of class Residue requires a mantissa precision according to the IEEE Standard for Floating-Point Arithmetic (IEEE 754). However, on some processors the traditional FPU uses an extended precision. Hence, it is indispensable that the proper mantissa length is enforced before performing any arithmetic operations. Moreover, it is required that numbers are rounded to the next nearest value. This can be ensured using Protect_FPU_rounding with CGAL_FE_TONEAREST, which also enforces the required precision as a side effect.

下面的例子中，求余的运算用于过滤，避免多项式中不必要的GCD计算。GCD计算十分耗时，由于欧几里德算法（Euclidean algorithm）中的系数的增长。

通用的思路是，首先GCD计算只涉及一个素数。如果这个模GCD（modular gcd ）是一个常数，我们可以推测大多数情况下实际的GCD也是一个常数。（不懂！！！！）

出于这个目的，例子中引入了一个函数may_have_common_factor()。这个函数有两个版本，分别用于系数类型是Modularizable和非Modularizable时。如果是后者，则过滤器则不会被使用，函数返回true.

另外，根据IEEE浮点运算标准（IEEE754）Residue类的实现要求尾数的精度。但在有些处理器中传统的FPU使用一个扩展的精度，所以必须在进行任何算法运算前将尾数精度进行适当设置。同是，要求将数舍入到最近的值，这个要求可通过带CGAL_FE_TONEAREST参数的Protect_FPU_rounding 来完成，它也同时将尾数精度进行了适当设置。

File Modular_arithmetic/modular_filter.cpp

 

#include <CGAL/basic.h>

 

#ifdef CGAL_USE_GMP

 

#include <CGAL/Gmpz.h>

#include <CGAL/Polynomial.h>

 

// Function in case Polynomial is Modularizable

template< typename Polynomial >

bool may_have_common_factor(

const Polynomial& p1, const Polynomial& p2, CGAL::Tag_true)

{

std::cout<< "The type is modularizable" << std::endl;

 

// Enforce IEEE double precision and rounding mode to nearest

// before useing modular arithmetic

CGAL::Protect_FPU_rounding<true> pfr(CGAL_FE_TONEAREST);

 

// Use Modular_traits to convert to polynomials with modular coefficients

typedef CGAL::Modular_traits<Polynomial> MT;

typedef typename MT::Residue_type MPolynomial;

typedef typename MT::Modular_image Modular_image;

MPolynomial mp1 = Modular_image()(p1);

MPolynomial mp2 = Modular_image()(p2);

 

// check for unlucky primes, the polynomials should not lose a degree

typename CGAL::Polynomial_traits_d<Polynomial>::Degree degree;

typename CGAL::Polynomial_traits_d<MPolynomial>::Degree mdegree;

if ( degree(p1) != mdegree(mp1)) return true;

if ( degree(p2) != mdegree(mp2)) return true;

 

// compute gcd for modular images

MPolynomial mg = CGAL::gcd(mp1,mp2);

 

// if the modular gcd is not trivial: return true

if ( mdegree(mg) > 0 ){

std::cout << "The gcd may be non trivial" << std::endl;

return true;

}else{

std::cout << "The gcd is trivial" << std::endl;

return false;

}

}

 

// This function returns true, since the filter is not applicable

template< typename Polynomial >

bool may_have_common_factor(

const Polynomial&, const Polynomial&, CGAL::Tag_false){

std::cout<< "The type is not modularizable" << std::endl;

return true;

}

 

template< typename Polynomial >

Polynomial modular_filtered_gcd(const Polynomial& p1, const Polynomial& p2){

typedef CGAL::Modular_traits<Polynomial> MT;

typedef typename MT::Is_modularizable Is_modularizable;

 

// Try to avoid actual gcd computation

if (may_have_common_factor(p1,p2, Is_modularizable())){

// Compute gcd, since the filter indicates a common factor

return CGAL::gcd(p1,p2);

}else{

typename CGAL::Polynomial_traits_d<Polynomial>::Univariate_content content;

typename CGAL::Polynomial_traits_d<Polynomial>::Construct_polynomial construct;

return construct(CGAL::gcd(content(p1),content(p2))); // return trivial gcd

}

}

 

int main(){

CGAL::set_pretty_mode(std::cout);

 

typedef CGAL::Gmpz NT;

typedef CGAL::Polynomial<NT> Poly;

CGAL::Polynomial_traits_d<Poly>::Construct_polynomial construct;

 

Poly f1=construct(NT(2), NT(6), NT(4));

Poly f2=construct(NT(12), NT(4), NT(8));

Poly f3=construct(NT(3), NT(4));

 

std::cout << "f1 : " << f1 << std::endl;

std::cout << "f2 : " << f2 << std::endl;

 

std::cout << "compute modular filtered gcd(f1,f2): " << std::endl;

Poly g1 = modular_filtered_gcd(f1,f2);

std::cout << "gcd(f1,f2): " << g1 << std::endl;

 

std::cout << std::endl;

Poly p1 = f1*f3;

Poly p2 = f2*f3;

 

std::cout << "f3 : " << f3 << std::endl;

std::cout << "p1=f1*f3 : " << p1 << std::endl;

std::cout << "p2=f2*f3 : " << p2 << std::endl;

 

std::cout << "compute modular filtered gcd(p1,p2): " << std::endl;

Poly g2 = modular_filtered_gcd(p1,p2);

std::cout << "gcd(p1,p2): " << g2 << std::endl;

}

 

#else

 

int main (){

std::cout << " This examples needs GMP! " << std::endl;

}

 

#endif

3 Design and Implementation History

The class Residue is based on the C-code of Sylvain Pion et. al. as it was presented in [2].

The remaining part of the package is the result of the integration process of the NumeriX library of Exacus [1] into CGAL.