Chapter 6
Modular Arithmetic

Michael Hemmer and Sylvain Pion

6.1   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.

First of all, this package introduces a type CGAL::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. Note that 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 67111067.

Please note that the implementation of class CGAL::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 CGAL::Protect_FPU_rounding with CGAL_FE_TONEAREST, which also enforces the required precision as a side effect.

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 CGAL::Residue. For scalar types, e.g. Integers, this mapping is just the canonical homomorphism into ℤ_/pℤ represented by CGAL::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 CGAL::Modular_traits<T>. The class CGAL::Modular_traits<T> is designed such that the concept Modularizable can be considered as optional, i.e., CGAL::Modular_traits<T> provides a tag that can be used for dispatching.

6.1.1   Example

In the following example modular arithmetic is used as a filter.

File: examples/Modular_arithmetic/modular_filter.cpp

/* Modular arithmetic can be used as a filter, in this example modular 
   arithmetic is used 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. 
*/

#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 

6.2   Design and Implementation History

The class CGAL::Residue is based on the C-code of Sylvain Pion et. al. as it was presented in [BEPP99].

The remaining part of the package is the result of the integration process of the NumeriX library of Exacus [BEH⁺05] into Cgal.