/src/cryptopp/algebra.h

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// algebra.h - originally written and placed in the public domain by Wei Dai

/// \file algebra.h
/// \brief Classes for performing mathematics over different fields

#ifndef CRYPTOPP_ALGEBRA_H
#define CRYPTOPP_ALGEBRA_H

#include "config.h"
#include "integer.h"
#include "misc.h"

NAMESPACE_BEGIN(CryptoPP)

class Integer;

/// \brief Abstract group
/// \tparam T element class or type
/// \details <tt>const Element&</tt> returned by member functions are references
///   to internal data members. Since each object may have only
///   one such data member for holding results, the following code
///   will produce incorrect results:
///   <pre>    abcd = group.Add(group.Add(a,b), group.Add(c,d));</pre>
///   But this should be fine:
///   <pre>    abcd = group.Add(a, group.Add(b, group.Add(c,d));</pre>
template <class T> class CRYPTOPP_NO_VTABLE AbstractGroup
{
public:
  typedef T Element;

  virtual ~AbstractGroup() {}

  /// \brief Compare two elements for equality
  /// \param a first element
  /// \param b second element
  /// \return true if the elements are equal, false otherwise
  /// \details Equal() tests the elements for equality using <tt>a==b</tt>
  virtual bool Equal(const Element &a, const Element &b) const =0;

  /// \brief Provides the Identity element
  /// \return the Identity element
  virtual const Element& Identity() const =0;

  /// \brief Adds elements in the group
  /// \param a first element
  /// \param b second element
  /// \return the sum of <tt>a</tt> and <tt>b</tt>
  virtual const Element& Add(const Element &a, const Element &b) const =0;

  /// \brief Inverts the element in the group
  /// \param a first element
  /// \return the inverse of the element
  virtual const Element& Inverse(const Element &a) const =0;

  /// \brief Determine if inversion is fast
  /// \return true if inversion is fast, false otherwise
  virtual bool InversionIsFast() const {return false;}

  /// \brief Doubles an element in the group
  /// \param a the element
  /// \return the element doubled
  virtual const Element& Double(const Element &a) const;

  /// \brief Subtracts elements in the group
  /// \param a first element
  /// \param b second element
  /// \return the difference of <tt>a</tt> and <tt>b</tt>. The element <tt>a</tt> must provide a Subtract member function.
  virtual const Element& Subtract(const Element &a, const Element &b) const;

  /// \brief TODO
  /// \param a first element
  /// \param b second element
  /// \return TODO
  virtual Element& Accumulate(Element &a, const Element &b) const;

  /// \brief Reduces an element in the congruence class
  /// \param a element to reduce
  /// \param b the congruence class
  /// \return the reduced element
  virtual Element& Reduce(Element &a, const Element &b) const;

  /// \brief Performs a scalar multiplication
  /// \param a multiplicand
  /// \param e multiplier
  /// \return the product
  virtual Element ScalarMultiply(const Element &a, const Integer &e) const;

  /// \brief TODO
  /// \param x first multiplicand
  /// \param e1 the first multiplier
  /// \param y second multiplicand
  /// \param e2 the second multiplier
  /// \return TODO
  virtual Element CascadeScalarMultiply(const Element &x, const Integer &e1, const Element &y, const Integer &e2) const;

  /// \brief Multiplies a base to multiple exponents in a group
  /// \param results an array of Elements
  /// \param base the base to raise to the exponents
  /// \param exponents an array of exponents
  /// \param exponentsCount the number of exponents in the array
  /// \details SimultaneousMultiply() multiplies the base to each exponent in the exponents array and stores the
  ///   result at the respective position in the results array.
  /// \details SimultaneousMultiply() must be implemented in a derived class.
  /// \pre <tt>COUNTOF(results) == exponentsCount</tt>
  /// \pre <tt>COUNTOF(exponents) == exponentsCount</tt>
  virtual void SimultaneousMultiply(Element *results, const Element &base, const Integer *exponents, unsigned int exponentsCount) const;
};

/// \brief Abstract ring
/// \tparam T element class or type
/// \details <tt>const Element&</tt> returned by member functions are references
///   to internal data members. Since each object may have only
///   one such data member for holding results, the following code
///   will produce incorrect results:
///   <pre>    abcd = group.Add(group.Add(a,b), group.Add(c,d));</pre>
///   But this should be fine:
///   <pre>    abcd = group.Add(a, group.Add(b, group.Add(c,d));</pre>
template <class T> class CRYPTOPP_NO_VTABLE AbstractRing : public AbstractGroup<T>
{
public:
  typedef T Element;

  /// \brief Construct an AbstractRing
  AbstractRing() {m_mg.m_pRing = this;}

  /// \brief Copy construct an AbstractRing
  /// \param source other AbstractRing
  AbstractRing(const AbstractRing &source)
    {CRYPTOPP_UNUSED(source); m_mg.m_pRing = this;}

  /// \brief Assign an AbstractRing
  /// \param source other AbstractRing
  AbstractRing& operator=(const AbstractRing &source)
    {CRYPTOPP_UNUSED(source); return *this;}

  /// \brief Determines whether an element is a unit in the group
  /// \param a the element
  /// \return true if the element is a unit after reduction, false otherwise.
  virtual bool IsUnit(const Element &a) const =0;

  /// \brief Retrieves the multiplicative identity
  /// \return the multiplicative identity
  virtual const Element& MultiplicativeIdentity() const =0;

  /// \brief Multiplies elements in the group
  /// \param a the multiplicand
  /// \param b the multiplier
  /// \return the product of a and b
  virtual const Element& Multiply(const Element &a, const Element &b) const =0;

  /// \brief Calculate the multiplicative inverse of an element in the group
  /// \param a the element
  virtual const Element& MultiplicativeInverse(const Element &a) const =0;

  /// \brief Square an element in the group
  /// \param a the element
  /// \return the element squared
  virtual const Element& Square(const Element &a) const;

  /// \brief Divides elements in the group
  /// \param a the dividend
  /// \param b the divisor
  /// \return the quotient
  virtual const Element& Divide(const Element &a, const Element &b) const;

  /// \brief Raises a base to an exponent in the group
  /// \param a the base
  /// \param e the exponent
  /// \return the exponentiation
  virtual Element Exponentiate(const Element &a, const Integer &e) const;

  /// \brief TODO
  /// \param x first element
  /// \param e1 first exponent
  /// \param y second element
  /// \param e2 second exponent
  /// \return TODO
  virtual Element CascadeExponentiate(const Element &x, const Integer &e1, const Element &y, const Integer &e2) const;

  /// \brief Exponentiates a base to multiple exponents in the Ring
  /// \param results an array of Elements
  /// \param base the base to raise to the exponents
  /// \param exponents an array of exponents
  /// \param exponentsCount the number of exponents in the array
  /// \details SimultaneousExponentiate() raises the base to each exponent in the exponents array and stores the
  ///   result at the respective position in the results array.
  /// \details SimultaneousExponentiate() must be implemented in a derived class.
  /// \pre <tt>COUNTOF(results) == exponentsCount</tt>
  /// \pre <tt>COUNTOF(exponents) == exponentsCount</tt>
  virtual void SimultaneousExponentiate(Element *results, const Element &base, const Integer *exponents, unsigned int exponentsCount) const;

  /// \brief Retrieves the multiplicative group
  /// \return the multiplicative group
  virtual const AbstractGroup<T>& MultiplicativeGroup() const
    {return m_mg;}

private:
  class MultiplicativeGroupT : public AbstractGroup<T>
  {
  public:
    const AbstractRing<T>& GetRing() const
      {return *m_pRing;}

    bool Equal(const Element &a, const Element &b) const
      {return GetRing().Equal(a, b);}

    const Element& Identity() const
      {return GetRing().MultiplicativeIdentity();}

    const Element& Add(const Element &a, const Element &b) const
      {return GetRing().Multiply(a, b);}

    Element& Accumulate(Element &a, const Element &b) const
      {return a = GetRing().Multiply(a, b);}

    const Element& Inverse(const Element &a) const
      {return GetRing().MultiplicativeInverse(a);}

    const Element& Subtract(const Element &a, const Element &b) const
      {return GetRing().Divide(a, b);}

    Element& Reduce(Element &a, const Element &b) const
      {return a = GetRing().Divide(a, b);}

    const Element& Double(const Element &a) const
      {return GetRing().Square(a);}

    Element ScalarMultiply(const Element &a, const Integer &e) const
      {return GetRing().Exponentiate(a, e);}

    Element CascadeScalarMultiply(const Element &x, const Integer &e1, const Element &y, const Integer &e2) const
      {return GetRing().CascadeExponentiate(x, e1, y, e2);}

    void SimultaneousMultiply(Element *results, const Element &base, const Integer *exponents, unsigned int exponentsCount) const
      {GetRing().SimultaneousExponentiate(results, base, exponents, exponentsCount);}

    const AbstractRing<T> *m_pRing;
  };

  MultiplicativeGroupT m_mg;
};

// ********************************************************

/// \brief Base and exponent
/// \tparam T base class or type
/// \tparam E exponent class or type
template <class T, class E = Integer>
struct BaseAndExponent
{
public:
  BaseAndExponent() {}
  BaseAndExponent(const T &base, const E &exponent) : base(base), exponent(exponent) {}
  bool operator<(const BaseAndExponent<T, E> &rhs) const {return exponent < rhs.exponent;}
  T base;
  E exponent;
};

// VC60 workaround: incomplete member template support
template <class Element, class Iterator>
  Element GeneralCascadeMultiplication(const AbstractGroup<Element> &group, Iterator begin, Iterator end);
template <class Element, class Iterator>
  Element GeneralCascadeExponentiation(const AbstractRing<Element> &ring, Iterator begin, Iterator end);

// ********************************************************

/// \brief Abstract Euclidean domain
/// \tparam T element class or type
/// \details <tt>const Element&</tt> returned by member functions are references
///   to internal data members. Since each object may have only
///   one such data member for holding results, the following code
///   will produce incorrect results:
///   <pre>    abcd = group.Add(group.Add(a,b), group.Add(c,d));</pre>
///   But this should be fine:
///   <pre>    abcd = group.Add(a, group.Add(b, group.Add(c,d));</pre>
template <class T> class CRYPTOPP_NO_VTABLE AbstractEuclideanDomain : public AbstractRing<T>
{
public:
  typedef T Element;

  /// \brief Performs the division algorithm on two elements in the ring
  /// \param r the remainder
  /// \param q the quotient
  /// \param a the dividend
  /// \param d the divisor
  virtual void DivisionAlgorithm(Element &r, Element &q, const Element &a, const Element &d) const =0;

  /// \brief Performs a modular reduction in the ring
  /// \param a the element
  /// \param b the modulus
  /// \return the result of <tt>a%b</tt>.
  virtual const Element& Mod(const Element &a, const Element &b) const =0;

  /// \brief Calculates the greatest common denominator in the ring
  /// \param a the first element
  /// \param b the second element
  /// \return the greatest common denominator of a and b.
  virtual const Element& Gcd(const Element &a, const Element &b) const;

protected:
  mutable Element result;
};

// ********************************************************

/// \brief Euclidean domain
/// \tparam T element class or type
/// \details <tt>const Element&</tt> returned by member functions are references
///   to internal data members. Since each object may have only
///   one such data member for holding results, the following code
///   will produce incorrect results:
///   <pre>    abcd = group.Add(group.Add(a,b), group.Add(c,d));</pre>
///   But this should be fine:
///   <pre>    abcd = group.Add(a, group.Add(b, group.Add(c,d));</pre>
template <class T> class EuclideanDomainOf : public AbstractEuclideanDomain<T>
{
public:
  typedef T Element;

  EuclideanDomainOf() {}

  bool Equal(const Element &a, const Element &b) const
    {return a==b;}

  const Element& Identity() const
    {return Element::Zero();}

  const Element& Add(const Element &a, const Element &b) const
    {return result = a+b;}

  Element& Accumulate(Element &a, const Element &b) const
    {return a+=b;}

  const Element& Inverse(const Element &a) const
    {return result = -a;}

  const Element& Subtract(const Element &a, const Element &b) const
    {return result = a-b;}

  Element& Reduce(Element &a, const Element &b) const
    {return a-=b;}

  const Element& Double(const Element &a) const
    {return result = a.Doubled();}

  const Element& MultiplicativeIdentity() const
    {return Element::One();}

  const Element& Multiply(const Element &a, const Element &b) const
    {return result = a*b;}

  const Element& Square(const Element &a) const
    {return result = a.Squared();}

  bool IsUnit(const Element &a) const
    {return a.IsUnit();}

  const Element& MultiplicativeInverse(const Element &a) const
    {return result = a.MultiplicativeInverse();}

  const Element& Divide(const Element &a, const Element &b) const
    {return result = a/b;}

  const Element& Mod(const Element &a, const Element &b) const
    {return result = a%b;}

  void DivisionAlgorithm(Element &r, Element &q, const Element &a, const Element &d) const
    {Element::Divide(r, q, a, d);}

  bool operator==(const EuclideanDomainOf<T> &rhs) const
    {CRYPTOPP_UNUSED(rhs); return true;}

private:
  mutable Element result;
};

/// \brief Quotient ring
/// \tparam T element class or type
/// \details <tt>const Element&</tt> returned by member functions are references
///   to internal data members. Since each object may have only
///   one such data member for holding results, the following code
///   will produce incorrect results:
///   <pre>    abcd = group.Add(group.Add(a,b), group.Add(c,d));</pre>
///   But this should be fine:
///   <pre>    abcd = group.Add(a, group.Add(b, group.Add(c,d));</pre>
template <class T> class QuotientRing : public AbstractRing<typename T::Element>
{
public:
  typedef T EuclideanDomain;
  typedef typename T::Element Element;

  QuotientRing(const EuclideanDomain &domain, const Element &modulus)
    : m_domain(domain), m_modulus(modulus) {}

  const EuclideanDomain & GetDomain() const
    {return m_domain;}

  const Element& GetModulus() const
    {return m_modulus;}

  bool Equal(const Element &a, const Element &b) const
    {return m_domain.Equal(m_domain.Mod(m_domain.Subtract(a, b), m_modulus), m_domain.Identity());}

  const Element& Identity() const
    {return m_domain.Identity();}

  const Element& Add(const Element &a, const Element &b) const
    {return m_domain.Add(a, b);}

  Element& Accumulate(Element &a, const Element &b) const
    {return m_domain.Accumulate(a, b);}

  const Element& Inverse(const Element &a) const
    {return m_domain.Inverse(a);}

  const Element& Subtract(const Element &a, const Element &b) const
    {return m_domain.Subtract(a, b);}

  Element& Reduce(Element &a, const Element &b) const
    {return m_domain.Reduce(a, b);}

  const Element& Double(const Element &a) const
    {return m_domain.Double(a);}

  bool IsUnit(const Element &a) const
    {return m_domain.IsUnit(m_domain.Gcd(a, m_modulus));}

  const Element& MultiplicativeIdentity() const
    {return m_domain.MultiplicativeIdentity();}

  const Element& Multiply(const Element &a, const Element &b) const
    {return m_domain.Mod(m_domain.Multiply(a, b), m_modulus);}

  const Element& Square(const Element &a) const
    {return m_domain.Mod(m_domain.Square(a), m_modulus);}

  const Element& MultiplicativeInverse(const Element &a) const;

  bool operator==(const QuotientRing<T> &rhs) const
    {return m_domain == rhs.m_domain && m_modulus == rhs.m_modulus;}

protected:
  EuclideanDomain m_domain;
  Element m_modulus;
};

NAMESPACE_END

#ifdef CRYPTOPP_MANUALLY_INSTANTIATE_TEMPLATES
#include "algebra.cpp"
#endif

#endif

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Created: 2024-11-21 07:03