iaik.security.ec.math.field

Interface ExtensionField

All Superinterfaces:

ECField, GenericField, PrimeCharacteristicField

All Known Implementing Classes:

QuadraticExtensionField, SexticOverQuadraticTowerExtensionField

public interface ExtensionField
extends PrimeCharacteristicField

This interface specifies important (arithmetical) methods that all extension fields have in common. This interface is used to describe extension fields of prime characteristic.

Author:

Christian Hanser, Sebastian Ramacher

See Also:

FieldElement, BinaryField, PrimeField

Method Summary

Modifier and Type	Method and Description
ExtensionFieldElement	add(GenericFieldElement a, GenericFieldElement b) Adds b to a, i.e.
ExtensionFieldElement	applyFrobenius(ExtensionFieldElement a, int i) Compute the i-th Frobenius of a.
ExtensionFieldElement	conjugate(ExtensionFieldElement b) Conjugate an element.
ExtensionFieldElement	conjugate(ExtensionFieldElement b, int i) Conjugate an element i times.
ExtensionFieldElement	divide(GenericFieldElement a, GenericFieldElement b) Divides a by b, i.e.
ExtensionFieldElement	divideBy2(PrimeCharacteristicFieldElement a) Divides a by 2.
ExtensionFieldElement	exponentiate(GenericFieldElement a, BigInteger n) Raises a to the power n, i.e.
ExtensionFieldElement	exponentiateByPowerOf2(GenericFieldElement a, int n) Raises a to the power 2n.
AbstractPrimeField	getBaseField() Returns the base field of this field.
int	getDegree() Get extension degree of the extension field over its subfield.
int	getDegreeOverBaseField() Get extension degree of the extension field over the base field.
int	getDepth() Returns the depth of the tower, i.e., the number of nested fields.
PrimeFieldElement	getNorm(ExtensionFieldElement a) Computes the norm of an element, which is an element of the base field.
ExtensionFieldElement	getOne() Returns the one element.
ExtensionField	getSubField() Returns null if the subfield equals the base field, and the subfield otherwise.
ExtensionFieldElement	getUniformlyRandomElement() Get a uniformly random prime field element.
ExtensionFieldElement	getUniformlyRandomNonZeroElement() Get a uniformly random, non-zero prime field element.
ExtensionFieldElement	getZero() Returns the zero element.
ExtensionFieldElement	invert(GenericFieldElement a) Inverts a, i.e.
ExtensionFieldElement	multiply(ExtensionFieldElement a, PrimeFieldElement b) Multiply extension field element with a prime field element
ExtensionFieldElement	multiply(GenericFieldElement a, BigInteger n) Multiplies a by n, i.e.
ExtensionFieldElement	multiply(GenericFieldElement a, GenericFieldElement b) Multiplies a by b, i.e.
ExtensionFieldElement	multiplyByAdjointRoot(ExtensionFieldElement a) Multiply a by the root used to obtain the extension field.
ExtensionFieldElement	negate(GenericFieldElement a) Negates a, i.e.
ExtensionFieldElement	newElement(byte[] bytes) Create new field element from byte-array representation.
ExtensionFieldElement	newElement(Object b) Creates a new field element from its BigInteger-array representation.
ExtensionFieldElement	newElementFromBaseField(PrimeFieldElement a) Converts a field element of the base field into an element of this.
ExtensionFieldElement	newElementFromSubField(PrimeCharacteristicFieldElement a) Converts a field element of the sub field into an element of this.
ExtensionFieldElement	square(GenericFieldElement a) Squares a, i.e.
ExtensionFieldElement	squareRoot(ExtensionFieldElement a) Return the square root (if it exists).
ExtensionFieldElement	squareRoot(ExtensionFieldElement a, boolean checkSquare) Return the square root (if it exists).
ExtensionFieldElement	subtract(GenericFieldElement a, GenericFieldElement b) Subtracts b from a, i.e.
Object	toBigIntegers(ExtensionFieldElement a) Converts a field element to its (multi-dimensional) BigInteger representation.

Methods inherited from interface iaik.security.ec.math.field.PrimeCharacteristicField

addOutOfPlace, getQuadraticCharacter, multiplyBy2, multiplyBy2OutOfPlace, multiplyBy3, multiplyBy3OutOfPlace, multiplyByPowerOf2, multiplyByPowerOf2OutOfPlace, subtractOutOfPlace, toElement

Methods inherited from interface iaik.security.ec.math.field.GenericField

equals, exponentiate, getCardinality, getFieldType, hashCode, invertElements, multiplyOutOfPlace, multiplyOutOfPlace, negateOutOfPlace, squareOutOfPlace

Methods inherited from interface java.security.spec.ECField

getFieldSize

Method Detail

getDepth

int getDepth()

Returns the depth of the tower, i.e., the number of nested fields.

Returns:

the depth of the tower

getDegree

int getDegree()

Get extension degree of the extension field over its subfield.

Returns:

extension degree of this over its subfield.

getDegreeOverBaseField

int getDegreeOverBaseField()

Get extension degree of the extension field over the base field.

Returns:

extension degree of this over its base field.

negate

ExtensionFieldElement negate(GenericFieldElement a)

Description copied from interface: GenericField

Negates a, i.e. performs the operation b = -b.

Specified by:

negate in interface GenericField

Specified by:

negate in interface PrimeCharacteristicField

Parameters:

a - the field element

Returns:

the negative of a

add

ExtensionFieldElement add(GenericFieldElement a,
                          GenericFieldElement b)

Description copied from interface: GenericField

Adds b to a, i.e. performs the operation a += b.

Specified by:

add in interface GenericField

Specified by:

add in interface PrimeCharacteristicField

Parameters:

a - the first addend

b - the second addend

Returns:

the sum of a and b

subtract

ExtensionFieldElement subtract(GenericFieldElement a,
                               GenericFieldElement b)

Description copied from interface: GenericField

Subtracts b from a, i.e. performs the operation a -= b.

Specified by:

subtract in interface GenericField

Specified by:

subtract in interface PrimeCharacteristicField

Parameters:

a - the minuend

b - the subtrahend

Returns:

the difference of a and b

multiply

ExtensionFieldElement multiply(GenericFieldElement a,
                               GenericFieldElement b)

Description copied from interface: GenericField

Multiplies a by b, i.e. performs the operation a *= b.

Specified by:

multiply in interface GenericField

Specified by:

multiply in interface PrimeCharacteristicField

Parameters:

a - the first factor

b - the second factor

Returns:

the product of a and b

multiply

ExtensionFieldElement multiply(GenericFieldElement a,
                               BigInteger n)

Description copied from interface: GenericField

Multiplies a by n, i.e. performs the operation a *= n.

Specified by:

multiply in interface GenericField

Specified by:

multiply in interface PrimeCharacteristicField

Parameters:

a - the first factor

n - the second factor

Returns:

the product of a and n

multiply

ExtensionFieldElement multiply(ExtensionFieldElement a,
                               PrimeFieldElement b)

Multiply extension field element with a prime field element

Parameters:

a - extension field element

b - prime field element

Returns:

a multiplied by the adjoint root.

multiplyByAdjointRoot

ExtensionFieldElement multiplyByAdjointRoot(ExtensionFieldElement a)

Multiply a by the root used to obtain the extension field.

Parameters:

a - extension field element

Returns:

a multiplied by the adjoint root.

divide

ExtensionFieldElement divide(GenericFieldElement a,
                             GenericFieldElement b)

Description copied from interface: GenericField

Divides a by b, i.e. performs the operation a /= b.

Specified by:

divide in interface GenericField

Specified by:

divide in interface PrimeCharacteristicField

Parameters:

a - the dividend

b - the divisor

Returns:

the quotient of a and b

divideBy2

ExtensionFieldElement divideBy2(PrimeCharacteristicFieldElement a)

Description copied from interface: PrimeCharacteristicField

Divides a by 2.

Specified by:

divideBy2 in interface PrimeCharacteristicField

Parameters:

a - the dividend

Returns:

the quotient of a and 2.

conjugate

ExtensionFieldElement conjugate(ExtensionFieldElement b)

Conjugate an element.

Parameters:

b - the element to be conjugated

Returns:

the conjugate of b

conjugate

ExtensionFieldElement conjugate(ExtensionFieldElement b,
                                int i)

Conjugate an element i times.

Parameters:

b - the element to be conjugated

i - number of times the element should be conjugated

Returns:

b conjugated i times

invert

ExtensionFieldElement invert(GenericFieldElement a)

Description copied from interface: GenericField

Inverts a, i.e. performs the operation a = a^(-1).

Specified by:

invert in interface GenericField

Specified by:

invert in interface PrimeCharacteristicField

Parameters:

a - the field element

Returns:

the inverse of a

square

ExtensionFieldElement square(GenericFieldElement a)

Description copied from interface: GenericField

Squares a, i.e. performs the operation a *= a.

Specified by:

square in interface GenericField

Specified by:

square in interface PrimeCharacteristicField

Parameters:

a - the field element

Returns:

a field element representing the square of a

exponentiate

ExtensionFieldElement exponentiate(GenericFieldElement a,
                                   BigInteger n)

Description copied from interface: GenericField

Raises a to the power n, i.e. performs the operation a = a^n.

Specified by:

exponentiate in interface GenericField

Specified by:

exponentiate in interface PrimeCharacteristicField

Parameters:

a - the field element

n - the exponent

Returns:

a field element representing the nth power of a

applyFrobenius

ExtensionFieldElement applyFrobenius(ExtensionFieldElement a,
                                     int i)

Compute the i-th Frobenius of a.

Parameters:

a - the element on which the Frobenius map should be applied.

i - number of times the Frobenius map should be applied.

Returns:

the Frobenius map applied i times to a.

exponentiateByPowerOf2

ExtensionFieldElement exponentiateByPowerOf2(GenericFieldElement a,
                                             int n)

Description copied from interface: GenericField

Raises a to the power 2n.

Specified by:

exponentiateByPowerOf2 in interface GenericField

Specified by:

exponentiateByPowerOf2 in interface PrimeCharacteristicField

Parameters:

a - the field element

n - the exponent

Returns:

the resulting field element

squareRoot

ExtensionFieldElement squareRoot(ExtensionFieldElement a)

Return the square root (if it exists).

Parameters:

a - the field element

Returns:

square root of a if it exists, null otherwise

squareRoot

ExtensionFieldElement squareRoot(ExtensionFieldElement a,
                                 boolean checkSquare)

Return the square root (if it exists).

Parameters:

a - the field element

checkSquare - check if a is a square. If true, the return value is undefined for non-squares.

Returns:

square root of a if it exists, null otherwise

getNorm

PrimeFieldElement getNorm(ExtensionFieldElement a)

Computes the norm of an element, which is an element of the base field.

Parameters:

a - the field element

Returns:

the norm of a

getOne

ExtensionFieldElement getOne()

Description copied from interface: GenericField

Returns the one element.

Specified by:

getOne in interface GenericField

Specified by:

getOne in interface PrimeCharacteristicField

Returns:

an object holding the one element

getZero

ExtensionFieldElement getZero()

Description copied from interface: GenericField

Returns the zero element.

Specified by:

getZero in interface GenericField

Specified by:

getZero in interface PrimeCharacteristicField

Returns:

an object holding the zero element

getBaseField

AbstractPrimeField getBaseField()

Description copied from interface: GenericField

Returns the base field of this field.

Specified by:

getBaseField in interface GenericField

Returns:

the base field

getSubField

ExtensionField getSubField()

Returns null if the subfield equals the base field, and the subfield otherwise.

Returns:

null if the subfield equals the base field, and the subfield otherwise.

newElement

ExtensionFieldElement newElement(Object b)

Creates a new field element from its BigInteger-array representation. Every component in the canonical representation of the field's elements corresponds to an array dimension, in ascending order.

Parameters:

b - the BigInteger[]

Returns:

the field element

newElement

ExtensionFieldElement newElement(byte[] bytes)

Description copied from interface: GenericField

Create new field element from byte-array representation.

Specified by:

newElement in interface GenericField

Specified by:

newElement in interface PrimeCharacteristicField

Parameters:

bytes - the byte[] in little-endian order

Returns:

an object holding the new field element

toBigIntegers

Object toBigIntegers(ExtensionFieldElement a)

Converts a field element to its (multi-dimensional) BigInteger representation.

Parameters:

a - the field element

Returns:

a (multi-dimensional) BigInteger representation

newElementFromBaseField

ExtensionFieldElement newElementFromBaseField(PrimeFieldElement a)

Converts a field element of the base field into an element of this.

Parameters:

a - a prime field element

Returns:

a as element of this

newElementFromSubField

ExtensionFieldElement newElementFromSubField(PrimeCharacteristicFieldElement a)

Converts a field element of the sub field into an element of this.

Parameters:

a - a sub field element

Returns:

a as element of this

getUniformlyRandomElement

ExtensionFieldElement getUniformlyRandomElement()

Description copied from interface: PrimeCharacteristicField

Get a uniformly random prime field element.

Specified by:

getUniformlyRandomElement in interface PrimeCharacteristicField

Returns:

element of the field.

getUniformlyRandomNonZeroElement

ExtensionFieldElement getUniformlyRandomNonZeroElement()

Description copied from interface: PrimeCharacteristicField

Get a uniformly random, non-zero prime field element.

Specified by:

getUniformlyRandomNonZeroElement in interface PrimeCharacteristicField

Returns:

element of the field.