SexticOverQuadraticTowerExtensionField (ECCelerate 6.2 API)

java.lang.Object
- iaik.security.ec.math.field.SexticOverQuadraticTowerExtensionField

All Implemented Interfaces:

ExtensionField, GenericField, PrimeCharacteristicField, ECField
```
public final class SexticOverQuadraticTowerExtensionField
extends Object
```
This class implements F_p¹² as sextic extension over a quadratic extension of the prime field. Non-residues are selected according to [1, Construction 6] and [1, Construction 7]. Multiplication and squaring is implemented as Karatsuba multiplication [2, Algorithm 2].
This class also offers special methods to multiply sparsely populated elements of the form v0 + v2 i^2 + v3 i^3 respectively v0 + v3 i^3 + v4 i^4. Furthermore, computation of powers in the cyclotomic subgroup is implemented [3].
References:
[1] N. Benger, M. Scott. Constructing Tower Extensions of Finite Fields for Implementation of Pairing-Based Cryptography.
[2] A. Weimerskirch, C. Paar. Generalizations of the Karatsuba Algorithm for Efficient Implementations.
[3] R. Granger, M. Scott. Faster Squaring in the Cyclotomic Subgroup of Sixth Degree Extensions

Author:

Christian Hanser, Sebastian Ramacher

Method Summary

All Methods Instance Methods Concrete Methods
Modifier and Type	Method and Description
`SexticExtensionFieldElement`	`add(GenericFieldElement a, GenericFieldElement b)` Adds `b` to `a`, i.e.
`SexticExtensionFieldElement`	`addOutOfPlace(PrimeCharacteristicFieldElement a, PrimeCharacteristicFieldElement b)` Adds two elements and returns a new object holding the result.
`SexticExtensionFieldElement`	`applyFrobenius(ExtensionFieldElement a, int i)` Compute the `i`-th Frobenius of `a`.
`SexticExtensionFieldElement`	`conjugate(ExtensionFieldElement b)` Conjugate an element.
`SexticExtensionFieldElement`	`conjugate(ExtensionFieldElement b, int i)` Conjugate an element `i` times.
`SexticExtensionFieldElement`	`divide(GenericFieldElement a, GenericFieldElement b)` Divides `a` by `b`, i.e.
`SexticExtensionFieldElement`	`divideBy2(PrimeCharacteristicFieldElement a)` Divides `a` by 2.
`boolean`	`equals(Object obj)`
`ExtensionFieldElement`	`exponentiate(GenericFieldElement a, BigInteger n)` Raises `a` to the power `n`, i.e.
`ExtensionFieldElement`	`exponentiate(GenericFieldElement a, int n)` Raises `a` to the power `n`, i.e.
`ExtensionFieldElement`	`exponentiateByPowerOf2(GenericFieldElement a, int n)` Raises `a` to the power `2ⁿ`.
`SexticExtensionFieldElement`	`exponentiateUni(SexticExtensionFieldElement a, BigInteger n)` Calculate power in cyclotomic subgroup
`AbstractPrimeField`	`getBaseField()` Returns the base field of this field.
`BigInteger`	`getCardinality()` Returns the cardinality 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.
`int`	`getFieldSize()`
`FieldTypes`	`getFieldType()` Returns the field type of the underlying implementation.
`FrobeniusSexticExtensionOverQuadraticConstants`	`getFrobeniusConstants()` Get constants used for the computation of the Frobenius map
`QuadraticExtensionFieldElement`	`getNonResidue()` Get non residue.
`PrimeFieldElement`	`getNorm(ExtensionFieldElement a)` Computes the norm of an element, which is an element of the base field.
`SexticExtensionFieldElement`	`getOne()` Returns the one element.
`int`	`getQuadraticCharacter(PrimeCharacteristicFieldElement a)` Not implemented; will throw an `UnsupportedOperationException`.
`ExtensionField`	`getSubField()` Returns `null` if the subfield equals the base field, and the subfield otherwise.
`SexticExtensionFieldElement`	`getUniformlyRandomElement()` Get a uniformly random prime field element.
`SexticExtensionFieldElement`	`getUniformlyRandomNonZeroElement()` Get a uniformly random, non-zero prime field element.
`SexticExtensionFieldElement`	`getZero()` Returns the zero element.
`int`	`hashCode()`
`SexticExtensionFieldElement`	`invert(GenericFieldElement a)` Inverts `a`, i.e.
`GenericFieldElement[]`	`invertElements(GenericFieldElement[] elements)` Simultaneously inverts an an array of field elements.
`SexticExtensionFieldElement`	`multiply(ExtensionFieldElement a, PrimeFieldElement b)` Multiply extension field element with a prime field element
`SexticExtensionFieldElement`	`multiply(GenericFieldElement a, BigInteger n)` Multiplies `a` by `n`, i.e.
`SexticExtensionFieldElement`	`multiply(GenericFieldElement a, GenericFieldElement b)` Multiplies `a` by `b`, i.e.
`SexticExtensionFieldElement`	`multiplyBy2(PrimeCharacteristicFieldElement a)` Multiplies `a` by `2`, which is a left-shift.
`SexticExtensionFieldElement`	`multiplyBy2OutOfPlace(PrimeCharacteristicFieldElement a)` Multiplies `a` by `2`, which is a left-shift and returns a new object holding the result.
`SexticExtensionFieldElement`	`multiplyBy3(PrimeCharacteristicFieldElement a)` Multiplies `a` by `3`, which is a left-shift plus an addition.
`SexticExtensionFieldElement`	`multiplyBy3OutOfPlace(PrimeCharacteristicFieldElement a)` Multiplies `a` by `3`, which is a left-shift plus an addition and returns a new object holding the result.
`SexticExtensionFieldElement`	`multiplyByAdjointRoot(ExtensionFieldElement a)` Multiply `a` by the root used to obtain the extension field.
`SexticExtensionFieldElement`	`multiplyByPowerOf2(PrimeCharacteristicFieldElement a, int exponent)` Multiplies `a` by a power of `2`, which is a left-shift.
`SexticExtensionFieldElement`	`multiplyByPowerOf2OutOfPlace(PrimeCharacteristicFieldElement a, int exponent)` Multiplies `a` by a power of `2`, which is a left-shift and returns a new object holding the result.
`SexticExtensionFieldElement`	`multiplyDenseSparse023(SexticExtensionFieldElement a, ExtensionFieldElement b)` Multiply an element with a sparsely populated element v0 + v2 * i^2 + v3 * i^3.
`SexticExtensionFieldElement`	`multiplyDenseSparse023(SexticExtensionFieldElement a, PrimeCharacteristicFieldElement b_0, PrimeCharacteristicFieldElement b_2, PrimeCharacteristicFieldElement b_3)` Multiply an element with a sparsely populated element v0 + v2 * i^2 + v3 * i^3.
`SexticExtensionFieldElement`	`multiplyDenseSparse034(SexticExtensionFieldElement a, ExtensionFieldElement b)` Multiply an element with a sparsely populated element v0 + v3 * i^3 + v4 * i^4.
`SexticExtensionFieldElement`	`multiplyDenseSparse034(SexticExtensionFieldElement a, PrimeCharacteristicFieldElement v0, PrimeCharacteristicFieldElement v3, PrimeCharacteristicFieldElement v4)` Multiply an element with a sparsely populated element v0 + v3 * i^3 + v4 * i^4.
`SexticExtensionFieldElement`	`multiplyOutOfPlace(GenericFieldElement a, BigInteger n)` Multiplies `a` by `n`, i.e.
`SexticExtensionFieldElement`	`multiplyOutOfPlace(GenericFieldElement a, GenericFieldElement b)` Multiplies `a` by `b`, i.e.
`SexticExtensionFieldElement`	`multiplySparse023(SexticExtensionFieldElement a, ExtensionFieldElement b)` Multiply a sparsely populated element element with a sparsely populated element v0 + v2 * i^2 + v3 * i^3.
`SexticExtensionFieldElement`	`multiplySparse023(SexticExtensionFieldElement a, PrimeCharacteristicFieldElement b_0, PrimeCharacteristicFieldElement b_2, PrimeCharacteristicFieldElement b_3)` Multiply a sparsely populated element element with a sparsely populated element v0 + v2 * i^2 + v3 * i^3.
`SexticExtensionFieldElement`	`multiplySparse034(SexticExtensionFieldElement a, ExtensionFieldElement b)` Multiply a sparsely populated element element with a sparsely populated element v0 + v3 * i^3 + v4 * i^4.
`SexticExtensionFieldElement`	`multiplySparse034(SexticExtensionFieldElement a, PrimeCharacteristicFieldElement v0, PrimeCharacteristicFieldElement v3, PrimeCharacteristicFieldElement v4)` Multiply a sparsely populated element element with a sparsely populated element v0 + v3 * i^3 + v4 * i^4.
`SexticExtensionFieldElement`	`negate(GenericFieldElement a)` Negates `a`, i.e.
`SexticExtensionFieldElement`	`negateOutOfPlace(GenericFieldElement a)` Negates `a`, i.e.
`SexticExtensionFieldElement`	`newElement(byte[] bytes)` Create new field element from `byte`-array representation.
`SexticExtensionFieldElement`	`newElement(Object b)` Creates a new field element from its `BigInteger`-array representation.
`SexticExtensionFieldElement`	`newElement(PrimeCharacteristicFieldElement a, PrimeCharacteristicFieldElement b, PrimeCharacteristicFieldElement c, PrimeCharacteristicFieldElement d, PrimeCharacteristicFieldElement e, PrimeCharacteristicFieldElement f)` Creates a new field element from the six coefficients of the quadratic subfield.
`SexticExtensionFieldElement`	`newElementFromBaseField(PrimeFieldElement a)` Converts a field element of the base field into an element of `this`.
`SexticExtensionFieldElement`	`newElementFromSubField(PrimeCharacteristicFieldElement a)` Converts a field element of the sub field into an element of `this`.
`SexticExtensionFieldElement`	`square(GenericFieldElement a)` Squares `a`, i.e.
`SexticExtensionFieldElement`	`squareOutOfPlace(GenericFieldElement a)` Squares `a`, i.e.
`SexticExtensionFieldElement`	`squareRoot(ExtensionFieldElement a)` Not implemented; will throw an `UnsupportedOperationException`.
`SexticExtensionFieldElement`	`squareRoot(ExtensionFieldElement a, boolean checkSquare)` Not implemented; will throw an `UnsupportedOperationException`.
`SexticExtensionFieldElement`	`squareUni(SexticExtensionFieldElement a)` Calculate square in cyclotomic subgroup.
`SexticExtensionFieldElement`	`subtract(GenericFieldElement a, GenericFieldElement b)` Subtracts `b` from `a`, i.e.
`SexticExtensionFieldElement`	`subtractOutOfPlace(GenericFieldElement a, GenericFieldElement b)` Subtracts `b` from `this` and returns a new object holding the result.
`BigInteger[][]`	`toBigIntegers(ExtensionFieldElement a)` Converts a field element to its (multi-dimensional) `BigInteger` representation.
`SexticExtensionFieldElement`	`toElement(byte[] bytes)` Create new field element from a `byte`-array.
`String`	`toString()`

Methods inherited from class java.lang.Object
clone, finalize, getClass, notify, notifyAll, wait, wait, wait

- Method Detail
  - negate
```
public SexticExtensionFieldElement negate(GenericFieldElement a)
```
    Description copied from interface: GenericField
    
    Negates a, i.e. performs the operation b = -b.
    
    Parameters:
    
    a - the field element
    
    Returns:
    
    the negative of a
  - negateOutOfPlace
```
public SexticExtensionFieldElement negateOutOfPlace(GenericFieldElement a)
```
    Description copied from interface: GenericField
    
    Negates a, i.e. performs the operation b = -b, where a new object holding the result is created.
    
    Parameters:
    
    a - the field element
    
    Returns:
    
    the negative of a
  - add
```
public SexticExtensionFieldElement add(GenericFieldElement a,
                                       GenericFieldElement b)
```
    Description copied from interface: GenericField
    
    Adds b to a, i.e. performs the operation a += b.
    
    Parameters:
    
    a - the first addend
    
    b - the second addend
    
    Returns:
    
    the sum of a and b
  - addOutOfPlace
```
public SexticExtensionFieldElement addOutOfPlace(PrimeCharacteristicFieldElement a,
                                                 PrimeCharacteristicFieldElement b)
```
    Description copied from interface: PrimeCharacteristicField
    
    Adds two elements and returns a new object holding the result.
    
    Parameters:
    
    a - first summand
    
    b - second summand
    
    Returns:
    
    the sum
  - subtract
```
public SexticExtensionFieldElement subtract(GenericFieldElement a,
                                            GenericFieldElement b)
```
    Description copied from interface: GenericField
    
    Subtracts b from a, i.e. performs the operation a -= b.
    
    Parameters:
    
    a - the minuend
    
    b - the subtrahend
    
    Returns:
    
    the difference of a and b
  - subtractOutOfPlace
```
public SexticExtensionFieldElement subtractOutOfPlace(GenericFieldElement a,
                                                      GenericFieldElement b)
```
    Description copied from interface: PrimeCharacteristicField
    
    Subtracts b from this and returns a new object holding the result.
    
    Parameters:
    
    a - the minuend
    
    b - the subtrahend
    
    Returns:
    
    the difference of a and b
  - multiply
```
public SexticExtensionFieldElement multiply(GenericFieldElement a,
                                            GenericFieldElement b)
```
    Description copied from interface: GenericField
    
    Multiplies a by b, i.e. performs the operation a *= b.
    
    Parameters:
    
    a - the first factor
    
    b - the second factor
    
    Returns:
    
    the product of a and b
  - multiplyOutOfPlace
```
public SexticExtensionFieldElement multiplyOutOfPlace(GenericFieldElement a,
                                                      GenericFieldElement b)
```
    Description copied from interface: GenericField
    
    Multiplies a by b, i.e. performs the operation a *= b, where a new object holding the result is created.
    
    Parameters:
    
    a - the first factor
    
    b - the second factor
    
    Returns:
    
    the product of a and b
  - multiply
```
public SexticExtensionFieldElement multiply(GenericFieldElement a,
                                            BigInteger n)
```
    Description copied from interface: GenericField
    
    Multiplies a by n, i.e. performs the operation a *= n.
    
    Parameters:
    
    a - the first factor
    
    n - the second factor
    
    Returns:
    
    the product of a and n
  - multiplyOutOfPlace
```
public SexticExtensionFieldElement multiplyOutOfPlace(GenericFieldElement a,
                                                      BigInteger n)
```
    Description copied from interface: GenericField
    
    Multiplies a by n, i.e. performs the operation a *= n, where a new object holding the result is created.
    
    Parameters:
    
    a - the first factor
    
    n - the second factor
    
    Returns:
    
    the product of a and n
  - multiplyBy2
```
public SexticExtensionFieldElement multiplyBy2(PrimeCharacteristicFieldElement a)
```
    Description copied from interface: PrimeCharacteristicField
    
    Multiplies a by 2, which is a left-shift. Works in-place.
    
    Parameters:
    
    a - the element to be shifted
    
    Returns:
    
    a holding the result
  - multiplyBy2OutOfPlace
```
public SexticExtensionFieldElement multiplyBy2OutOfPlace(PrimeCharacteristicFieldElement a)
```
    Description copied from interface: PrimeCharacteristicField
    
    Multiplies a by 2, which is a left-shift and returns a new object holding the result.
    
    Parameters:
    
    a - the element to be shifted
    
    Returns:
    
    a holding the result
  - multiplyBy3
```
public SexticExtensionFieldElement multiplyBy3(PrimeCharacteristicFieldElement a)
```
    Description copied from interface: PrimeCharacteristicField
    
    Multiplies a by 3, which is a left-shift plus an addition. Works in-place.
    
    Parameters:
    
    a - the element to multiplied
    
    Returns:
    
    a holding the result
  - multiplyBy3OutOfPlace
```
public SexticExtensionFieldElement multiplyBy3OutOfPlace(PrimeCharacteristicFieldElement a)
```
    Description copied from interface: PrimeCharacteristicField
    
    Multiplies a by 3, which is a left-shift plus an addition and returns a new object holding the result.
    
    Parameters:
    
    a - the element to multiplied
    
    Returns:
    
    a holding the result
  - multiplyByPowerOf2
```
public SexticExtensionFieldElement multiplyByPowerOf2(PrimeCharacteristicFieldElement a,
                                                      int exponent)
```
    Description copied from interface: PrimeCharacteristicField
    
    Multiplies a by a power of 2, which is a left-shift. Works in-place.
    
    Parameters:
    
    a - the element to be shifted
    
    exponent - the power of 2
    
    Returns:
    
    a holding the result
  - multiplyByPowerOf2OutOfPlace
```
public SexticExtensionFieldElement multiplyByPowerOf2OutOfPlace(PrimeCharacteristicFieldElement a,
                                                                int exponent)
```
    Description copied from interface: PrimeCharacteristicField
    
    Multiplies a by a power of 2, which is a left-shift and returns a new object holding the result.
    
    Parameters:
    
    a - the element to be shifted
    
    exponent - the power of 2
    
    Returns:
    
    a holding the result
  - divide
```
public SexticExtensionFieldElement divide(GenericFieldElement a,
                                          GenericFieldElement b)
```
    Description copied from interface: GenericField
    
    Divides a by b, i.e. performs the operation a /= b.
    
    Parameters:
    
    a - the dividend
    
    b - the divisor
    
    Returns:
    
    the quotient of a and b
  - divideBy2
```
public SexticExtensionFieldElement divideBy2(PrimeCharacteristicFieldElement a)
```
    Description copied from interface: PrimeCharacteristicField
    
    Divides a by 2.
    
    Parameters:
    
    a - the dividend
    
    Returns:
    
    the quotient of a and 2.
  - conjugate
```
public SexticExtensionFieldElement conjugate(ExtensionFieldElement b)
```
    Description copied from interface: ExtensionField
    
    Conjugate an element.
    
    Parameters:
    
    b - the element to be conjugated
    
    Returns:
    
    the conjugate of b
  - invert
```
public SexticExtensionFieldElement invert(GenericFieldElement a)
```
    Description copied from interface: GenericField
    
    Inverts a, i.e. performs the operation a = a^(-1).
    
    Parameters:
    
    a - the field element
    
    Returns:
    
    the inverse of a
  - square
```
public SexticExtensionFieldElement square(GenericFieldElement a)
```
    Description copied from interface: GenericField
    
    Squares a, i.e. performs the operation a *= a.
    
    Parameters:
    
    a - the field element
    
    Returns:
    
    a field element representing the square of a
  - squareOutOfPlace
```
public SexticExtensionFieldElement squareOutOfPlace(GenericFieldElement a)
```
    Description copied from interface: GenericField
    
    Squares a, i.e. performs the operation a*a, where a new object holding the result is created.
    
    Parameters:
    
    a - the field element
    
    Returns:
    
    a field element representing the square of a
  - squareRoot
```
public SexticExtensionFieldElement squareRoot(ExtensionFieldElement a,
                                              boolean checkSquare)
```
    Not implemented; will throw an UnsupportedOperationException.
    
    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
  - squareRoot
```
public SexticExtensionFieldElement squareRoot(ExtensionFieldElement a)
```
    Not implemented; will throw an UnsupportedOperationException.
    
    Parameters:
    
    a - the field element
    
    Returns:
    
    square root of a if it exists, null otherwise
  - getNorm
```
public PrimeFieldElement getNorm(ExtensionFieldElement a)
```
    Description copied from interface: ExtensionField
    
    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
```
public SexticExtensionFieldElement getOne()
```
    Description copied from interface: GenericField
    
    Returns the one element.
    
    Returns:
    
    an object holding the one element
  - getZero
```
public SexticExtensionFieldElement getZero()
```
    Description copied from interface: GenericField
    
    Returns the zero element.
    
    Returns:
    
    an object holding the zero element
  - newElement
```
public SexticExtensionFieldElement newElement(Object b)
```
    Description copied from interface: ExtensionField
    
    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
```
public SexticExtensionFieldElement newElement(PrimeCharacteristicFieldElement a,
                                              PrimeCharacteristicFieldElement b,
                                              PrimeCharacteristicFieldElement c,
                                              PrimeCharacteristicFieldElement d,
                                              PrimeCharacteristicFieldElement e,
                                              PrimeCharacteristicFieldElement f)
```
    Creates a new field element from the six coefficients of the quadratic subfield.
    
    Parameters:
    
    a - the first element
    
    b - the second element
    
    c - the third element
    
    d - the fourth element
    
    e - the fifth element
    
    f - the sixth element
    
    Returns:
    
    field element described by a,b,c,d,e,f
  - newElement
```
public SexticExtensionFieldElement newElement(byte[] bytes)
```
    Description copied from interface: GenericField
    
    Create new field element from byte-array representation.
    
    Parameters:
    
    bytes - the byte[] in little-endian order
    
    Returns:
    
    an object holding the new field element
  - toElement
```
public SexticExtensionFieldElement toElement(byte[] bytes)
```
    Description copied from interface: GenericField
    
    Create new field element from a byte-array. Unlike GenericField.newElement(byte[]), this method does not assume the input to be a valid encoding of a field element.
    
    Parameters:
    
    bytes - a byte[] in little-endian order
    
    Returns:
    
    an object holding the new field element
  - getUniformlyRandomElement
```
public SexticExtensionFieldElement getUniformlyRandomElement()
```
    Description copied from interface: PrimeCharacteristicField
    
    Get a uniformly random prime field element.
    
    Returns:
    
    element of the field.
  - getUniformlyRandomNonZeroElement
```
public SexticExtensionFieldElement getUniformlyRandomNonZeroElement()
```
    Description copied from interface: PrimeCharacteristicField
    
    Get a uniformly random, non-zero prime field element.
    
    Returns:
    
    element of the field.
  - toBigIntegers
```
public BigInteger[][] toBigIntegers(ExtensionFieldElement a)
```
    Description copied from interface: ExtensionField
    
    Converts a field element to its (multi-dimensional) BigInteger representation.
    
    Parameters:
    
    a - the field element
    
    Returns:
    
    a (multi-dimensional) BigInteger representation
  - toString
```
public String toString()
```
    Overrides:
    
    toString in class Object
  - newElementFromBaseField
```
public SexticExtensionFieldElement newElementFromBaseField(PrimeFieldElement a)
```
    Description copied from interface: ExtensionField
    
    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
```
public SexticExtensionFieldElement newElementFromSubField(PrimeCharacteristicFieldElement a)
```
    Description copied from interface: ExtensionField
    
    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
  - multiply
```
public SexticExtensionFieldElement multiply(ExtensionFieldElement a,
                                            PrimeFieldElement b)
```
    Description copied from interface: ExtensionField
    
    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.
  - getNonResidue
```
public QuadraticExtensionFieldElement getNonResidue()
```
    Get non residue.
    
    Returns:
    
    the non residue
  - getQuadraticCharacter
```
public int getQuadraticCharacter(PrimeCharacteristicFieldElement a)
```
    Not implemented; will throw an UnsupportedOperationException.
    
    Parameters:
    
    a - a field element
    
    Returns:
    
    the quadratic character (a/p)
  - multiplyByAdjointRoot
```
public SexticExtensionFieldElement multiplyByAdjointRoot(ExtensionFieldElement a)
```
    Description copied from interface: ExtensionField
    
    Multiply a by the root used to obtain the extension field.
    
    Parameters:
    
    a - extension field element
    
    Returns:
    
    a multiplied by the adjoint root.
  - multiplyDenseSparse023
```
public SexticExtensionFieldElement multiplyDenseSparse023(SexticExtensionFieldElement a,
                                                          ExtensionFieldElement b)
```
    Multiply an element with a sparsely populated element v0 + v2 * i^2 + v3 * i^3.
    
    Parameters:
    
    a - first operand
    
    b - second operand (sparse)
    
    Returns:
    
    the product of a and b
  - multiplyDenseSparse023
```
public SexticExtensionFieldElement multiplyDenseSparse023(SexticExtensionFieldElement a,
                                                          PrimeCharacteristicFieldElement b_0,
                                                          PrimeCharacteristicFieldElement b_2,
                                                          PrimeCharacteristicFieldElement b_3)
```
    Multiply an element with a sparsely populated element v0 + v2 * i^2 + v3 * i^3.
    
    Parameters:
    
    a - first operand
    
    b_0 - constant coefficient of second operand
    
    b_2 - coefficient of i^2 of second operand
    
    b_3 - coefficient of i^3 of second operand
    
    Returns:
    
    the product of a and b
  - multiplySparse023
```
public SexticExtensionFieldElement multiplySparse023(SexticExtensionFieldElement a,
                                                     ExtensionFieldElement b)
```
    Multiply a sparsely populated element element with a sparsely populated element v0 + v2 * i^2 + v3 * i^3.
    
    Parameters:
    
    a - first operand (sparse)
    
    b - second operand (sparse)
    
    Returns:
    
    the product of a and b
  - multiplySparse023
```
public SexticExtensionFieldElement multiplySparse023(SexticExtensionFieldElement a,
                                                     PrimeCharacteristicFieldElement b_0,
                                                     PrimeCharacteristicFieldElement b_2,
                                                     PrimeCharacteristicFieldElement b_3)
```
    Multiply a sparsely populated element element with a sparsely populated element v0 + v2 * i^2 + v3 * i^3.
    
    Parameters:
    
    a - first operand (sparse)
    
    b_0 - constant coefficient of second operand
    
    b_2 - coefficient of i^2 of second operand
    
    b_3 - coefficient of i^3 of second operand
    
    Returns:
    
    the product of a and b
  - multiplyDenseSparse034
```
public SexticExtensionFieldElement multiplyDenseSparse034(SexticExtensionFieldElement a,
                                                          ExtensionFieldElement b)
```
    Multiply an element with a sparsely populated element v0 + v3 * i^3 + v4 * i^4.
    
    Parameters:
    
    a - first operand
    
    b - second operand (sparse)
    
    Returns:
    
    the product of a and b
  - multiplyDenseSparse034
```
public SexticExtensionFieldElement multiplyDenseSparse034(SexticExtensionFieldElement a,
                                                          PrimeCharacteristicFieldElement v0,
                                                          PrimeCharacteristicFieldElement v3,
                                                          PrimeCharacteristicFieldElement v4)
```
    Multiply an element with a sparsely populated element v0 + v3 * i^3 + v4 * i^4.
    
    Parameters:
    
    a - first operand
    
    v0 - constant coefficient of second operand
    
    v3 - coefficient of i^3 of second operand
    
    v4 - coefficient of i^4 of second operand
    
    Returns:
    
    the product of a and b
  - multiplySparse034
```
public SexticExtensionFieldElement multiplySparse034(SexticExtensionFieldElement a,
                                                     ExtensionFieldElement b)
```
    Multiply a sparsely populated element element with a sparsely populated element v0 + v3 * i^3 + v4 * i^4.
    
    Parameters:
    
    a - first operand (sparse)
    
    b - second operand (sparse)
    
    Returns:
    
    the product of a and b
  - multiplySparse034
```
public SexticExtensionFieldElement multiplySparse034(SexticExtensionFieldElement a,
                                                     PrimeCharacteristicFieldElement v0,
                                                     PrimeCharacteristicFieldElement v3,
                                                     PrimeCharacteristicFieldElement v4)
```
    Multiply a sparsely populated element element with a sparsely populated element v0 + v3 * i^3 + v4 * i^4.
    
    Parameters:
    
    a - first operand (sparse)
    
    v0 - constant coefficient of second operand
    
    v3 - coefficient of i^3 of second operand
    
    v4 - coefficient of i^4 of second operand
    
    Returns:
    
    the product of a and b
  - conjugate
```
public SexticExtensionFieldElement conjugate(ExtensionFieldElement b,
                                             int i)
```
    Description copied from interface: ExtensionField
    
    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
  - applyFrobenius
```
public SexticExtensionFieldElement applyFrobenius(ExtensionFieldElement a,
                                                  int i)
```
    Description copied from interface: ExtensionField
    
    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.
  - getFrobeniusConstants
```
public FrobeniusSexticExtensionOverQuadraticConstants getFrobeniusConstants()
```
    Get constants used for the computation of the Frobenius map
    
    Returns:
    
    frobenius map constants
  - squareUni
```
public SexticExtensionFieldElement squareUni(SexticExtensionFieldElement a)
```
    Calculate square in cyclotomic subgroup. This method works in-place.
    
    Parameters:
    
    a - the cyclotomic subgroup element
    
    Returns:
    
    a squared
  - exponentiateUni
```
public SexticExtensionFieldElement exponentiateUni(SexticExtensionFieldElement a,
                                                   BigInteger n)
```
    Calculate power in cyclotomic subgroup
    
    Parameters:
    
    a - the cyclotomic subgroup element
    
    n - the exponent
    
    Returns:
    
    a to the n-th power
  - invertElements
```
public GenericFieldElement[] invertElements(GenericFieldElement[] elements)
```
    Description copied from interface: GenericField
    
    Simultaneously inverts an an array of field elements.
    
    Specified by:
    
    invertElements in interface GenericField
    
    Parameters:
    
    elements - the array of field elements
    
    Returns:
    
    the array holding the inverted elements (in the same order as the input elements)
  - exponentiate
```
public 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 ExtensionField
    
    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
  - exponentiate
```
public ExtensionFieldElement exponentiate(GenericFieldElement a,
                                          int 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
    
    Parameters:
    
    a - the field element
    
    n - the exponent
    
    Returns:
    
    a field element representing the nth power of a
  - exponentiateByPowerOf2
```
public ExtensionFieldElement exponentiateByPowerOf2(GenericFieldElement a,
                                                    int n)
```
    Description copied from interface: GenericField
    
    Raises a to the power 2ⁿ.
    
    Specified by:
    
    exponentiateByPowerOf2 in interface ExtensionField
    
    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
  - getCardinality
```
public BigInteger getCardinality()
```
    Description copied from interface: GenericField
    
    Returns the cardinality of this field.
    
    Specified by:
    
    getCardinality in interface GenericField
    
    Returns:
    
    a BigInteger holding the cardinality
  - getFieldType
```
public FieldTypes getFieldType()
```
    Description copied from interface: GenericField
    
    Returns the field type of the underlying implementation.
    
    Specified by:
    
    getFieldType in interface GenericField
    
    Returns:
    
    the field type
  - getFieldSize
```
public int getFieldSize()
```
    Specified by:
    
    getFieldSize in interface ECField
  - getDepth
```
public int getDepth()
```
    Description copied from interface: ExtensionField
    
    Returns the depth of the tower, i.e., the number of nested fields.
    
    Specified by:
    
    getDepth in interface ExtensionField
    
    Returns:
    
    the depth of the tower
  - getDegree
```
public int getDegree()
```
    Description copied from interface: ExtensionField
    
    Get extension degree of the extension field over its subfield.
    
    Specified by:
    
    getDegree in interface ExtensionField
    
    Returns:
    
    extension degree of this over its subfield.
  - getDegreeOverBaseField
```
public int getDegreeOverBaseField()
```
    Description copied from interface: ExtensionField
    
    Get extension degree of the extension field over the base field.
    
    Specified by:
    
    getDegreeOverBaseField in interface ExtensionField
    
    Returns:
    
    extension degree of this over its base field.
  - getBaseField
```
public AbstractPrimeField getBaseField()
```
    Description copied from interface: GenericField
    
    Returns the base field of this field.
    
    Specified by:
    
    getBaseField in interface ExtensionField
    
    Specified by:
    
    getBaseField in interface GenericField
    
    Returns:
    
    the base field
  - getSubField
```
public ExtensionField getSubField()
```
    Description copied from interface: ExtensionField
    
    Returns null if the subfield equals the base field, and the subfield otherwise.
    
    Specified by:
    
    getSubField in interface ExtensionField
    
    Returns:
    
    null if the subfield equals the base field, and the subfield otherwise.
  - hashCode
```
public int hashCode()
```
    Specified by:
    
    hashCode in interface GenericField
    
    Overrides:
    
    hashCode in class Object
  - equals
```
public boolean equals(Object obj)
```
    Specified by:
    
    equals in interface GenericField
    
    Overrides:
    
    equals in class Object

Class SexticOverQuadraticTowerExtensionField

Method Summary

Methods inherited from class java.lang.Object

Method Detail

negate

negateOutOfPlace

add

addOutOfPlace

subtract

subtractOutOfPlace

multiply

multiplyOutOfPlace

multiply

multiplyOutOfPlace

multiplyBy2

multiplyBy2OutOfPlace

multiplyBy3

multiplyBy3OutOfPlace

multiplyByPowerOf2

multiplyByPowerOf2OutOfPlace

divide

divideBy2

conjugate

invert

square

squareOutOfPlace

squareRoot

squareRoot

getNorm

getOne

getZero

newElement

newElement

newElement

toElement

getUniformlyRandomElement

getUniformlyRandomNonZeroElement

toBigIntegers

toString

newElementFromBaseField

newElementFromSubField

multiply

getNonResidue

getQuadraticCharacter

multiplyByAdjointRoot

multiplyDenseSparse023

multiplyDenseSparse023

multiplySparse023

multiplySparse023

multiplyDenseSparse034

multiplyDenseSparse034

multiplySparse034

multiplySparse034

conjugate

applyFrobenius

getFrobeniusConstants

squareUni

exponentiateUni

invertElements

exponentiate

exponentiate

exponentiateByPowerOf2

getCardinality

getFieldType

getFieldSize

getDepth

getDegree

getDegreeOverBaseField

getBaseField

getSubField

hashCode

equals