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453 lines
16 KiB
453 lines
16 KiB
// algebra.h - originally written and placed in the public domain by Wei Dai |
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/// \file algebra.h |
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/// \brief Classes for performing mathematics over different fields |
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#ifndef CRYPTOPP_ALGEBRA_H |
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#define CRYPTOPP_ALGEBRA_H |
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#include "config.h" |
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#include "integer.h" |
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#include "misc.h" |
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NAMESPACE_BEGIN(CryptoPP) |
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class Integer; |
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/// \brief Abstract group |
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/// \tparam T element class or type |
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/// \details <tt>const Element&</tt> returned by member functions are references |
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/// to internal data members. Since each object may have only |
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/// one such data member for holding results, the following code |
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/// will produce incorrect results: |
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/// <pre> abcd = group.Add(group.Add(a,b), group.Add(c,d));</pre> |
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/// But this should be fine: |
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/// <pre> abcd = group.Add(a, group.Add(b, group.Add(c,d));</pre> |
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template <class T> class CRYPTOPP_NO_VTABLE AbstractGroup |
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{ |
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public: |
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typedef T Element; |
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virtual ~AbstractGroup() {} |
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/// \brief Compare two elements for equality |
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/// \param a first element |
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/// \param b second element |
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/// \returns true if the elements are equal, false otherwise |
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/// \details Equal() tests the elements for equality using <tt>a==b</tt> |
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virtual bool Equal(const Element &a, const Element &b) const =0; |
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/// \brief Provides the Identity element |
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/// \returns the Identity element |
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virtual const Element& Identity() const =0; |
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/// \brief Adds elements in the group |
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/// \param a first element |
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/// \param b second element |
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/// \returns the sum of <tt>a</tt> and <tt>b</tt> |
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virtual const Element& Add(const Element &a, const Element &b) const =0; |
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/// \brief Inverts the element in the group |
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/// \param a first element |
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/// \returns the inverse of the element |
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virtual const Element& Inverse(const Element &a) const =0; |
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/// \brief Determine if inversion is fast |
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/// \returns true if inversion is fast, false otherwise |
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virtual bool InversionIsFast() const {return false;} |
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/// \brief Doubles an element in the group |
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/// \param a the element |
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/// \returns the element doubled |
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virtual const Element& Double(const Element &a) const; |
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/// \brief Subtracts elements in the group |
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/// \param a first element |
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/// \param b second element |
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/// \returns the difference of <tt>a</tt> and <tt>b</tt>. The element <tt>a</tt> must provide a Subtract member function. |
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virtual const Element& Subtract(const Element &a, const Element &b) const; |
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/// \brief TODO |
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/// \param a first element |
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/// \param b second element |
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/// \returns TODO |
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virtual Element& Accumulate(Element &a, const Element &b) const; |
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/// \brief Reduces an element in the congruence class |
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/// \param a element to reduce |
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/// \param b the congruence class |
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/// \returns the reduced element |
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virtual Element& Reduce(Element &a, const Element &b) const; |
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/// \brief Performs a scalar multiplication |
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/// \param a multiplicand |
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/// \param e multiplier |
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/// \returns the product |
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virtual Element ScalarMultiply(const Element &a, const Integer &e) const; |
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/// \brief TODO |
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/// \param x first multiplicand |
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/// \param e1 the first multiplier |
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/// \param y second multiplicand |
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/// \param e2 the second multiplier |
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/// \returns TODO |
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virtual Element CascadeScalarMultiply(const Element &x, const Integer &e1, const Element &y, const Integer &e2) const; |
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/// \brief Multiplies a base to multiple exponents in a group |
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/// \param results an array of Elements |
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/// \param base the base to raise to the exponents |
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/// \param exponents an array of exponents |
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/// \param exponentsCount the number of exponents in the array |
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/// \details SimultaneousMultiply() multiplies the base to each exponent in the exponents array and stores the |
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/// result at the respective position in the results array. |
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/// \details SimultaneousMultiply() must be implemented in a derived class. |
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/// \pre <tt>COUNTOF(results) == exponentsCount</tt> |
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/// \pre <tt>COUNTOF(exponents) == exponentsCount</tt> |
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virtual void SimultaneousMultiply(Element *results, const Element &base, const Integer *exponents, unsigned int exponentsCount) const; |
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}; |
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/// \brief Abstract ring |
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/// \tparam T element class or type |
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/// \details <tt>const Element&</tt> returned by member functions are references |
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/// to internal data members. Since each object may have only |
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/// one such data member for holding results, the following code |
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/// will produce incorrect results: |
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/// <pre> abcd = group.Add(group.Add(a,b), group.Add(c,d));</pre> |
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/// But this should be fine: |
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/// <pre> abcd = group.Add(a, group.Add(b, group.Add(c,d));</pre> |
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template <class T> class CRYPTOPP_NO_VTABLE AbstractRing : public AbstractGroup<T> |
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{ |
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public: |
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typedef T Element; |
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/// \brief Construct an AbstractRing |
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AbstractRing() {m_mg.m_pRing = this;} |
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/// \brief Copy construct an AbstractRing |
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/// \param source other AbstractRing |
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AbstractRing(const AbstractRing &source) |
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{CRYPTOPP_UNUSED(source); m_mg.m_pRing = this;} |
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/// \brief Assign an AbstractRing |
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/// \param source other AbstractRing |
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AbstractRing& operator=(const AbstractRing &source) |
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{CRYPTOPP_UNUSED(source); return *this;} |
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/// \brief Determines whether an element is a unit in the group |
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/// \param a the element |
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/// \returns true if the element is a unit after reduction, false otherwise. |
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virtual bool IsUnit(const Element &a) const =0; |
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/// \brief Retrieves the multiplicative identity |
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/// \returns the multiplicative identity |
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virtual const Element& MultiplicativeIdentity() const =0; |
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/// \brief Multiplies elements in the group |
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/// \param a the multiplicand |
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/// \param b the multiplier |
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/// \returns the product of a and b |
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virtual const Element& Multiply(const Element &a, const Element &b) const =0; |
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/// \brief Calculate the multiplicative inverse of an element in the group |
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/// \param a the element |
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virtual const Element& MultiplicativeInverse(const Element &a) const =0; |
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/// \brief Square an element in the group |
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/// \param a the element |
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/// \returns the element squared |
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virtual const Element& Square(const Element &a) const; |
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/// \brief Divides elements in the group |
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/// \param a the dividend |
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/// \param b the divisor |
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/// \returns the quotient |
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virtual const Element& Divide(const Element &a, const Element &b) const; |
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/// \brief Raises a base to an exponent in the group |
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/// \param a the base |
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/// \param e the exponent |
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/// \returns the exponentiation |
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virtual Element Exponentiate(const Element &a, const Integer &e) const; |
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/// \brief TODO |
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/// \param x first element |
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/// \param e1 first exponent |
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/// \param y second element |
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/// \param e2 second exponent |
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/// \returns TODO |
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virtual Element CascadeExponentiate(const Element &x, const Integer &e1, const Element &y, const Integer &e2) const; |
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/// \brief Exponentiates a base to multiple exponents in the Ring |
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/// \param results an array of Elements |
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/// \param base the base to raise to the exponents |
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/// \param exponents an array of exponents |
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/// \param exponentsCount the number of exponents in the array |
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/// \details SimultaneousExponentiate() raises the base to each exponent in the exponents array and stores the |
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/// result at the respective position in the results array. |
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/// \details SimultaneousExponentiate() must be implemented in a derived class. |
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/// \pre <tt>COUNTOF(results) == exponentsCount</tt> |
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/// \pre <tt>COUNTOF(exponents) == exponentsCount</tt> |
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virtual void SimultaneousExponentiate(Element *results, const Element &base, const Integer *exponents, unsigned int exponentsCount) const; |
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/// \brief Retrieves the multiplicative group |
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/// \returns the multiplicative group |
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virtual const AbstractGroup<T>& MultiplicativeGroup() const |
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{return m_mg;} |
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private: |
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class MultiplicativeGroupT : public AbstractGroup<T> |
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{ |
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public: |
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const AbstractRing<T>& GetRing() const |
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{return *m_pRing;} |
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bool Equal(const Element &a, const Element &b) const |
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{return GetRing().Equal(a, b);} |
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const Element& Identity() const |
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{return GetRing().MultiplicativeIdentity();} |
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const Element& Add(const Element &a, const Element &b) const |
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{return GetRing().Multiply(a, b);} |
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Element& Accumulate(Element &a, const Element &b) const |
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{return a = GetRing().Multiply(a, b);} |
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const Element& Inverse(const Element &a) const |
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{return GetRing().MultiplicativeInverse(a);} |
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const Element& Subtract(const Element &a, const Element &b) const |
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{return GetRing().Divide(a, b);} |
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Element& Reduce(Element &a, const Element &b) const |
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{return a = GetRing().Divide(a, b);} |
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const Element& Double(const Element &a) const |
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{return GetRing().Square(a);} |
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Element ScalarMultiply(const Element &a, const Integer &e) const |
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{return GetRing().Exponentiate(a, e);} |
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Element CascadeScalarMultiply(const Element &x, const Integer &e1, const Element &y, const Integer &e2) const |
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{return GetRing().CascadeExponentiate(x, e1, y, e2);} |
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void SimultaneousMultiply(Element *results, const Element &base, const Integer *exponents, unsigned int exponentsCount) const |
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{GetRing().SimultaneousExponentiate(results, base, exponents, exponentsCount);} |
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const AbstractRing<T> *m_pRing; |
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}; |
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MultiplicativeGroupT m_mg; |
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}; |
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// ******************************************************** |
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/// \brief Base and exponent |
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/// \tparam T base class or type |
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/// \tparam E exponent class or type |
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template <class T, class E = Integer> |
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struct BaseAndExponent |
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{ |
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public: |
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BaseAndExponent() {} |
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BaseAndExponent(const T &base, const E &exponent) : base(base), exponent(exponent) {} |
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bool operator<(const BaseAndExponent<T, E> &rhs) const {return exponent < rhs.exponent;} |
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T base; |
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E exponent; |
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}; |
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// VC60 workaround: incomplete member template support |
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template <class Element, class Iterator> |
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Element GeneralCascadeMultiplication(const AbstractGroup<Element> &group, Iterator begin, Iterator end); |
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template <class Element, class Iterator> |
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Element GeneralCascadeExponentiation(const AbstractRing<Element> &ring, Iterator begin, Iterator end); |
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// ******************************************************** |
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/// \brief Abstract Euclidean domain |
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/// \tparam T element class or type |
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/// \details <tt>const Element&</tt> returned by member functions are references |
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/// to internal data members. Since each object may have only |
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/// one such data member for holding results, the following code |
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/// will produce incorrect results: |
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/// <pre> abcd = group.Add(group.Add(a,b), group.Add(c,d));</pre> |
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/// But this should be fine: |
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/// <pre> abcd = group.Add(a, group.Add(b, group.Add(c,d));</pre> |
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template <class T> class CRYPTOPP_NO_VTABLE AbstractEuclideanDomain : public AbstractRing<T> |
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{ |
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public: |
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typedef T Element; |
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/// \brief Performs the division algorithm on two elements in the ring |
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/// \param r the remainder |
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/// \param q the quotient |
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/// \param a the dividend |
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/// \param d the divisor |
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virtual void DivisionAlgorithm(Element &r, Element &q, const Element &a, const Element &d) const =0; |
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/// \brief Performs a modular reduction in the ring |
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/// \param a the element |
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/// \param b the modulus |
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/// \returns the result of <tt>a%b</tt>. |
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virtual const Element& Mod(const Element &a, const Element &b) const =0; |
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/// \brief Calculates the greatest common denominator in the ring |
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/// \param a the first element |
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/// \param b the second element |
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/// \returns the the greatest common denominator of a and b. |
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virtual const Element& Gcd(const Element &a, const Element &b) const; |
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protected: |
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mutable Element result; |
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}; |
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// ******************************************************** |
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/// \brief Euclidean domain |
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/// \tparam T element class or type |
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/// \details <tt>const Element&</tt> returned by member functions are references |
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/// to internal data members. Since each object may have only |
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/// one such data member for holding results, the following code |
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/// will produce incorrect results: |
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/// <pre> abcd = group.Add(group.Add(a,b), group.Add(c,d));</pre> |
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/// But this should be fine: |
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/// <pre> abcd = group.Add(a, group.Add(b, group.Add(c,d));</pre> |
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template <class T> class EuclideanDomainOf : public AbstractEuclideanDomain<T> |
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{ |
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public: |
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typedef T Element; |
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EuclideanDomainOf() {} |
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bool Equal(const Element &a, const Element &b) const |
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{return a==b;} |
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const Element& Identity() const |
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{return Element::Zero();} |
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const Element& Add(const Element &a, const Element &b) const |
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{return result = a+b;} |
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Element& Accumulate(Element &a, const Element &b) const |
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{return a+=b;} |
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const Element& Inverse(const Element &a) const |
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{return result = -a;} |
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const Element& Subtract(const Element &a, const Element &b) const |
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{return result = a-b;} |
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Element& Reduce(Element &a, const Element &b) const |
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{return a-=b;} |
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const Element& Double(const Element &a) const |
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{return result = a.Doubled();} |
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const Element& MultiplicativeIdentity() const |
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{return Element::One();} |
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const Element& Multiply(const Element &a, const Element &b) const |
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{return result = a*b;} |
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const Element& Square(const Element &a) const |
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{return result = a.Squared();} |
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bool IsUnit(const Element &a) const |
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{return a.IsUnit();} |
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const Element& MultiplicativeInverse(const Element &a) const |
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{return result = a.MultiplicativeInverse();} |
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const Element& Divide(const Element &a, const Element &b) const |
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{return result = a/b;} |
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const Element& Mod(const Element &a, const Element &b) const |
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{return result = a%b;} |
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void DivisionAlgorithm(Element &r, Element &q, const Element &a, const Element &d) const |
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{Element::Divide(r, q, a, d);} |
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bool operator==(const EuclideanDomainOf<T> &rhs) const |
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{CRYPTOPP_UNUSED(rhs); return true;} |
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private: |
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mutable Element result; |
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}; |
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/// \brief Quotient ring |
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/// \tparam T element class or type |
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/// \details <tt>const Element&</tt> returned by member functions are references |
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/// to internal data members. Since each object may have only |
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/// one such data member for holding results, the following code |
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/// will produce incorrect results: |
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/// <pre> abcd = group.Add(group.Add(a,b), group.Add(c,d));</pre> |
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/// But this should be fine: |
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/// <pre> abcd = group.Add(a, group.Add(b, group.Add(c,d));</pre> |
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template <class T> class QuotientRing : public AbstractRing<typename T::Element> |
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{ |
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public: |
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typedef T EuclideanDomain; |
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typedef typename T::Element Element; |
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QuotientRing(const EuclideanDomain &domain, const Element &modulus) |
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: m_domain(domain), m_modulus(modulus) {} |
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const EuclideanDomain & GetDomain() const |
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{return m_domain;} |
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const Element& GetModulus() const |
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{return m_modulus;} |
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bool Equal(const Element &a, const Element &b) const |
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{return m_domain.Equal(m_domain.Mod(m_domain.Subtract(a, b), m_modulus), m_domain.Identity());} |
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const Element& Identity() const |
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{return m_domain.Identity();} |
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const Element& Add(const Element &a, const Element &b) const |
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{return m_domain.Add(a, b);} |
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Element& Accumulate(Element &a, const Element &b) const |
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{return m_domain.Accumulate(a, b);} |
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const Element& Inverse(const Element &a) const |
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{return m_domain.Inverse(a);} |
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const Element& Subtract(const Element &a, const Element &b) const |
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{return m_domain.Subtract(a, b);} |
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Element& Reduce(Element &a, const Element &b) const |
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{return m_domain.Reduce(a, b);} |
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const Element& Double(const Element &a) const |
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{return m_domain.Double(a);} |
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bool IsUnit(const Element &a) const |
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{return m_domain.IsUnit(m_domain.Gcd(a, m_modulus));} |
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const Element& MultiplicativeIdentity() const |
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{return m_domain.MultiplicativeIdentity();} |
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const Element& Multiply(const Element &a, const Element &b) const |
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{return m_domain.Mod(m_domain.Multiply(a, b), m_modulus);} |
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const Element& Square(const Element &a) const |
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{return m_domain.Mod(m_domain.Square(a), m_modulus);} |
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const Element& MultiplicativeInverse(const Element &a) const; |
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bool operator==(const QuotientRing<T> &rhs) const |
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{return m_domain == rhs.m_domain && m_modulus == rhs.m_modulus;} |
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protected: |
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EuclideanDomain m_domain; |
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Element m_modulus; |
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}; |
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NAMESPACE_END |
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#ifdef CRYPTOPP_MANUALLY_INSTANTIATE_TEMPLATES |
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#include "algebra.cpp" |
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#endif |
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#endif
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