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MontgomeryRepresentation Class Reference

Performs modular arithmetic in Montgomery representation for increased speed. More...

`#include <modarith.h>`

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## Public Member Functions

virtual ~MontgomeryRepresentation ()

MontgomeryRepresentation (const Integer &modulus)
Construct a MontgomeryRepresentation. More...

virtual ModularArithmeticClone () const
Clone a MontgomeryRepresentation. More...

bool IsMontgomeryRepresentation () const
Retrieves the representation. More...

Integer ConvertIn (const Integer &a) const
Reduces an element in the congruence class. More...

Integer ConvertOut (const Integer &a) const
Reduces an element in the congruence class. More...

const IntegerMultiplicativeIdentity () const
Retrieves the multiplicative identity. More...

const IntegerMultiply (const Integer &a, const Integer &b) const
Multiplies elements in the ring. More...

const IntegerSquare (const Integer &a) const
Square an element in the ring. More...

const IntegerMultiplicativeInverse (const Integer &a) const
Calculate the multiplicative inverse of an element in the ring. More...

Integer CascadeExponentiate (const Integer &x, const Integer &e1, const Integer &y, const Integer &e2) const
TODO. More...

void SimultaneousExponentiate (Element *results, const Element &base, const Integer *exponents, unsigned int exponentsCount) const
Exponentiates a base to multiple exponents in the ring. More...

Public Member Functions inherited from ModularArithmetic
virtual ~ModularArithmetic ()

ModularArithmetic (const Integer &modulus=Integer::One())
Construct a ModularArithmetic. More...

ModularArithmetic (const ModularArithmetic &ma)
Copy construct a ModularArithmetic. More...

ModularArithmetic (BufferedTransformation &bt)
Construct a ModularArithmetic. More...

void DEREncode (BufferedTransformation &bt) const
Encodes in DER format. More...

void DEREncodeElement (BufferedTransformation &out, const Element &a) const
Encodes element in DER format. More...

void BERDecodeElement (BufferedTransformation &in, Element &a) const
Decodes element in DER format. More...

const IntegerGetModulus () const
Retrieves the modulus. More...

void SetModulus (const Integer &newModulus)
Sets the modulus. More...

const IntegerHalf (const Integer &a) const
Divides an element by 2. More...

bool Equal (const Integer &a, const Integer &b) const
Compare two elements for equality. More...

const IntegerIdentity () const
Provides the Identity element. More...

const IntegerAdd (const Integer &a, const Integer &b) const
Adds elements in the ring. More...

IntegerAccumulate (Integer &a, const Integer &b) const
TODO. More...

const IntegerInverse (const Integer &a) const
Inverts the element in the ring. More...

const IntegerSubtract (const Integer &a, const Integer &b) const
Subtracts elements in the ring. More...

IntegerReduce (Integer &a, const Integer &b) const
TODO. More...

const IntegerDouble (const Integer &a) const
Doubles an element in the ring. More...

bool IsUnit (const Integer &a) const
Determines whether an element is a unit in the ring. More...

const IntegerDivide (const Integer &a, const Integer &b) const
Divides elements in the ring. More...

unsigned int MaxElementBitLength () const
Provides the maximum bit size of an element in the ring. More...

unsigned int MaxElementByteLength () const
Provides the maximum byte size of an element in the ring. More...

Element RandomElement (RandomNumberGenerator &rng, const RandomizationParameter &ignore_for_now=0) const
Provides a random element in the ring. More...

bool operator== (const ModularArithmetic &rhs) const
Compares two ModularArithmetic for equality. More...

Public Member Functions inherited from AbstractRing< Integer >
AbstractRing ()
Construct an AbstractRing. More...

AbstractRing (const AbstractRing &source)
Copy construct an AbstractRing. More...

AbstractRingoperator= (const AbstractRing &source)
Assign an AbstractRing. More...

virtual Element Exponentiate (const Element &a, const Integer &e) const
Raises a base to an exponent in the group. More...

virtual const AbstractGroup< Integer > & MultiplicativeGroup () const
Retrieves the multiplicative group. More...

Public Member Functions inherited from AbstractGroup< Integer >
virtual ~AbstractGroup ()

virtual bool InversionIsFast () const
Determine if inversion is fast. More...

virtual Element ScalarMultiply (const Element &a, const Integer &e) const
Performs a scalar multiplication. More...

virtual Element CascadeScalarMultiply (const Element &x, const Integer &e1, const Element &y, const Integer &e2) const
TODO. More...

virtual void SimultaneousMultiply (Element *results, const Element &base, const Integer *exponents, unsigned int exponentsCount) const
Multiplies a base to multiple exponents in a group. More...

## Private Attributes

Integer m_u

IntegerSecBlock m_workspace

Public Types inherited from ModularArithmetic
typedef int RandomizationParameter

typedef Integer Element

Public Types inherited from AbstractRing< Integer >
typedef Integer Element

Public Types inherited from AbstractGroup< Integer >
typedef Integer Element

Static Public Attributes inherited from ModularArithmetic
static const RandomizationParameter DefaultRandomizationParameter

Protected Attributes inherited from ModularArithmetic
Integer m_modulus

Integer m_result

Integer m_result1

## Detailed Description

Performs modular arithmetic in Montgomery representation for increased speed.

The Montgomery representation represents each congruence class `[a]` as `a*r%n`, where `r` is a convenient power of 2.

`const Element&` 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:

`    abcd = group.Add(group.Add(a,b), group.Add(c,d));`

But this should be fine:

`    abcd = group.Add(a, group.Add(b, group.Add(c,d));`

Definition at line 271 of file modarith.h.

## Constructor & Destructor Documentation

 virtual MontgomeryRepresentation::~MontgomeryRepresentation ( )
inlinevirtual

Definition at line 274 of file modarith.h.

 MontgomeryRepresentation::MontgomeryRepresentation ( const Integer & modulus )

Construct a MontgomeryRepresentation.

Parameters
 modulus congruence class modulus
Note
The modulus must be odd.

Definition at line 4572 of file integer.cpp.

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## Member Function Documentation

 Integer MontgomeryRepresentation::CascadeExponentiate ( const Integer & x, const Integer & e1, const Integer & y, const Integer & e2 ) const
inlinevirtual

TODO.

Parameters
 x first element e1 first exponent y second element e2 second exponent
Returns
TODO

Reimplemented from ModularArithmetic.

Definition at line 303 of file modarith.h.

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 virtual ModularArithmetic* MontgomeryRepresentation::Clone ( ) const
inlinevirtual

Clone a MontgomeryRepresentation.

Returns
pointer to a new MontgomeryRepresentation

Clone effectively copy constructs a new MontgomeryRepresentation. The caller is responsible for deleting the pointer returned from this method.

Reimplemented from ModularArithmetic.

Definition at line 285 of file modarith.h.

 Integer MontgomeryRepresentation::ConvertIn ( const Integer & a ) const
inlinevirtual

Reduces an element in the congruence class.

Parameters
 a element to convert
Returns
the reduced element

ConvertIn is useful for derived classes, like MontgomeryRepresentation, which must convert between representations.

Reimplemented from ModularArithmetic.

Definition at line 289 of file modarith.h.

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 Integer MontgomeryRepresentation::ConvertOut ( const Integer & a ) const
virtual

Reduces an element in the congruence class.

Parameters
 a element to convert
Returns
the reduced element

ConvertOut is useful for derived classes, like MontgomeryRepresentation, which must convert between representations.

Reimplemented from ModularArithmetic.

Definition at line 4609 of file integer.cpp.

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 bool MontgomeryRepresentation::IsMontgomeryRepresentation ( ) const
inlinevirtual

Retrieves the representation.

Returns
true if the if the modulus is in Montgomery form for multiplication, false otherwise

Reimplemented from ModularArithmetic.

Definition at line 287 of file modarith.h.

 const Integer& MontgomeryRepresentation::MultiplicativeIdentity ( ) const
inlinevirtual

Retrieves the multiplicative identity.

Returns
the multiplicative identity

the base class implementations returns 1.

Reimplemented from ModularArithmetic.

Definition at line 294 of file modarith.h.

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 const Integer & MontgomeryRepresentation::MultiplicativeInverse ( const Integer & a ) const
virtual

Calculate the multiplicative inverse of an element in the ring.

Parameters
 a the element

MultiplicativeInverse returns `a-1%n`. The element `a` must provide a InverseMod member function.

Reimplemented from ModularArithmetic.

Definition at line 4622 of file integer.cpp.

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 const Integer & MontgomeryRepresentation::Multiply ( const Integer & a, const Integer & b ) const
virtual

Multiplies elements in the ring.

Parameters
 a the multiplicand b the multiplier
Returns
the product of a and b

Multiply returns `a*b%n`.

Reimplemented from ModularArithmetic.

Definition at line 4583 of file integer.cpp.

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 void MontgomeryRepresentation::SimultaneousExponentiate ( Element * results, const Element & base, const Integer * exponents, unsigned int exponentsCount ) const
inlinevirtual

Exponentiates a base to multiple exponents in the ring.

Parameters
 results an array of Elements base the base to raise to the exponents exponents an array of exponents exponentsCount the number of exponents in the array

SimultaneousExponentiate() raises the base to each exponent in the exponents array and stores the result at the respective position in the results array.

SimultaneousExponentiate() must be implemented in a derived class.

Precondition
`COUNTOF(results) == exponentsCount`
`COUNTOF(exponents) == exponentsCount`

Reimplemented from ModularArithmetic.

Definition at line 306 of file modarith.h.

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 const Integer & MontgomeryRepresentation::Square ( const Integer & a ) const
virtual

Square an element in the ring.

Parameters
 a the element
Returns
the element squared

Square returns `a*a%n`. The element `a` must provide a Square member function.

Reimplemented from ModularArithmetic.

Definition at line 4596 of file integer.cpp.

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## Member Data Documentation

 Integer MontgomeryRepresentation::m_u
private

Definition at line 310 of file modarith.h.

 IntegerSecBlock MontgomeryRepresentation::m_workspace
mutableprivate

Definition at line 311 of file modarith.h.

The documentation for this class was generated from the following files: