QuadraticExtensionField (ECCelerate 6.2 API)

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

All Implemented Interfaces:

ExtensionField, GenericField, PrimeCharacteristicField, ECField
```
public final class QuadraticExtensionField
extends Object
```
Implements quadratic extension fields of prime fields. For multiplication Karatsuba's algorithm is used [1]. Square roots are computed according to the complex method [2, Algorithm 8]. Note that [2, Algorithm 8] does not handle certain cases correctly. References:
[1] Multiplication and squaring in pairing friendly fields.
[2] G. Adj, F. Rodrigues-Henriquez. Square root computation over even extension fields.

Author:

Christian Hanser, Sebastian Ramacher

Method Summary

All Methods Instance Methods Concrete Methods
Modifier and Type	Method and Description
`QuadraticExtensionFieldElement`	`add(GenericFieldElement a, GenericFieldElement b)` Adds `b` to `a`, i.e.
`QuadraticExtensionFieldElement`	`addOutOfPlace(PrimeCharacteristicFieldElement a, PrimeCharacteristicFieldElement b)` Adds two elements and returns a new object holding the result.
`QuadraticExtensionFieldElement`	`applyFrobenius(ExtensionFieldElement a, int i)` Compute the `i`-th Frobenius of `a`.
`QuadraticExtensionFieldElement`	`conjugate(ExtensionFieldElement b)` Conjugate an element.
`QuadraticExtensionFieldElement`	`conjugate(ExtensionFieldElement b, int i)` Conjugate an element `i` times.
`QuadraticExtensionFieldElement`	`divide(GenericFieldElement a, GenericFieldElement b)` Divides `a` by `b`, i.e.
`QuadraticExtensionFieldElement`	`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`.
`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.
`GenericFieldElement`	`getNonResidue()` Get non residue.
`PrimeFieldElement`	`getNorm(ExtensionFieldElement a)` Computes the norm of an element, which is an element of the base field.
`QuadraticExtensionFieldElement`	`getOne()` Returns the one element.
`int`	`getQuadraticCharacter(PrimeCharacteristicFieldElement e)` Implements the Legendre-Kronecker-Jacobi symbol according to [2, Algorithm 11.69], which returns -1,0,1 depending on whether the given value `a`is a quadratic non-residue, a multiple of the modulus, or a quadratic residue.
`ExtensionField`	`getSubField()` Returns `null` if the subfield equals the base field, and the subfield otherwise.
`QuadraticExtensionFieldElement`	`getUniformlyRandomElement()` Get a uniformly random prime field element.
`QuadraticExtensionFieldElement`	`getUniformlyRandomNonZeroElement()` Get a uniformly random, non-zero prime field element.
`QuadraticExtensionFieldElement`	`getZero()` Returns the zero element.
`int`	`hashCode()`
`QuadraticExtensionFieldElement`	`invert(GenericFieldElement a)` Inverts `a`, i.e.
`GenericFieldElement[]`	`invertElements(GenericFieldElement[] elements)` Simultaneously inverts an an array of field elements.
`QuadraticExtensionFieldElement`	`multiply(ExtensionFieldElement a, PrimeFieldElement b)` Multiply extension field element with a prime field element
`QuadraticExtensionFieldElement`	`multiply(GenericFieldElement a, BigInteger n)` Multiplies `a` by `n`, i.e.
`QuadraticExtensionFieldElement`	`multiply(GenericFieldElement a, GenericFieldElement b)` Multiplies `a` by `b`, i.e.
`QuadraticExtensionFieldElement`	`multiplyBy2(PrimeCharacteristicFieldElement a)` Multiplies `a` by `2`, which is a left-shift.
`QuadraticExtensionFieldElement`	`multiplyBy2OutOfPlace(PrimeCharacteristicFieldElement a)` Multiplies `a` by `2`, which is a left-shift and returns a new object holding the result.
`QuadraticExtensionFieldElement`	`multiplyBy3(PrimeCharacteristicFieldElement a)` Multiplies `a` by `3`, which is a left-shift plus an addition.
`QuadraticExtensionFieldElement`	`multiplyBy3OutOfPlace(PrimeCharacteristicFieldElement a)` Multiplies `a` by `3`, which is a left-shift plus an addition and returns a new object holding the result.
`QuadraticExtensionFieldElement`	`multiplyByAdjointRoot(ExtensionFieldElement a)` Multiply `a` by the root used to obtain the extension field.
`QuadraticExtensionFieldElement`	`multiplyByAdjointRootOutOfPlace(ExtensionFieldElement a)`
`QuadraticExtensionFieldElement`	`multiplyByPowerOf2(PrimeCharacteristicFieldElement a, int exponent)` Multiplies `a` by a power of `2`, which is a left-shift.
`QuadraticExtensionFieldElement`	`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.
`QuadraticExtensionFieldElement`	`multiplyOutOfPlace(ExtensionFieldElement a, PrimeFieldElement b)`
`QuadraticExtensionFieldElement`	`multiplyOutOfPlace(GenericFieldElement a, BigInteger n)` Multiplies `a` by `n`, i.e.
`QuadraticExtensionFieldElement`	`multiplyOutOfPlace(GenericFieldElement a, GenericFieldElement b)` Multiplies `a` by `b`, i.e.
`QuadraticExtensionFieldElement`	`negate(GenericFieldElement a)` Negates `a`, i.e.
`QuadraticExtensionFieldElement`	`negateOutOfPlace(GenericFieldElement a)` Negates `a`, i.e.
`QuadraticExtensionFieldElement`	`newElement(byte[] bytes)` Create new field element from `byte`-array representation.
`QuadraticExtensionFieldElement`	`newElement(Object b)` Creates a new field element from its `BigInteger`-array representation.
`QuadraticExtensionFieldElement`	`newElement(PrimeFieldElement a, PrimeFieldElement b)` Create new element from two prime field elements
`QuadraticExtensionFieldElement`	`newElementFromBaseField(PrimeFieldElement a)` Converts a field element of the base field into an element of `this`.
`QuadraticExtensionFieldElement`	`newElementFromSubField(PrimeCharacteristicFieldElement a)` Converts a field element of the sub field into an element of `this`.
`QuadraticExtensionFieldElement`	`square(GenericFieldElement a)` Squares `a`, i.e.
`QuadraticExtensionFieldElement`	`squareOutOfPlace(GenericFieldElement a)` Squares `a`, i.e.
`QuadraticExtensionFieldElement`	`squareRoot(ExtensionFieldElement a)` Return the square root (if it exists).
`QuadraticExtensionFieldElement`	`squareRoot(ExtensionFieldElement a, boolean checkSquare)` Return the square root (if it exists).
`QuadraticExtensionFieldElement`	`subtract(GenericFieldElement a, GenericFieldElement b)` Subtracts `b` from `a`, i.e.
`QuadraticExtensionFieldElement`	`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.
`QuadraticExtensionFieldElement`	`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
  - getQuadraticCharacter
```
public int getQuadraticCharacter(PrimeCharacteristicFieldElement e)
```
    Implements the Legendre-Kronecker-Jacobi symbol according to [2, Algorithm 11.69], which returns -1,0,1 depending on whether the given value ais a quadratic non-residue, a multiple of the modulus, or a quadratic residue.
    
    Parameters:
    
    e - a field element
    
    Returns:
    
    the Legendre-Kronecker-Jacobi symbol of e
  - multiply
```
public QuadraticExtensionFieldElement multiply(ExtensionFieldElement a,
                                               PrimeFieldElement b)
```
    Description copied from interface: ExtensionField
    
    Multiply extension field element with a prime field element
    
    Specified by:
    
    multiply in interface ExtensionField
    
    Parameters:
    
    a - extension field element
    
    b - prime field element
    
    Returns:
    
    a multiplied by the adjoint root.
  - multiplyOutOfPlace
```
public QuadraticExtensionFieldElement multiplyOutOfPlace(ExtensionFieldElement a,
                                                         PrimeFieldElement b)
```
  - multiply
```
public QuadraticExtensionFieldElement 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 ExtensionField
    
    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
  - multiplyOutOfPlace
```
public QuadraticExtensionFieldElement 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.
    
    Specified by:
    
    multiplyOutOfPlace in interface GenericField
    
    Parameters:
    
    a - the first factor
    
    b - the second factor
    
    Returns:
    
    the product of a and b
  - multiply
```
public QuadraticExtensionFieldElement 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 ExtensionField
    
    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
  - multiplyOutOfPlace
```
public QuadraticExtensionFieldElement 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.
    
    Specified by:
    
    multiplyOutOfPlace in interface GenericField
    
    Parameters:
    
    a - the first factor
    
    n - the second factor
    
    Returns:
    
    the product of a and n
  - square
```
public QuadraticExtensionFieldElement square(GenericFieldElement a)
```
    Description copied from interface: GenericField
    
    Squares a, i.e. performs the operation a *= a.
    
    Specified by:
    
    square in interface ExtensionField
    
    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
  - squareOutOfPlace
```
public QuadraticExtensionFieldElement 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.
    
    Specified by:
    
    squareOutOfPlace in interface GenericField
    
    Parameters:
    
    a - the field element
    
    Returns:
    
    a field element representing the square of a
  - squareRoot
```
public QuadraticExtensionFieldElement squareRoot(ExtensionFieldElement a)
```
    Description copied from interface: ExtensionField
    
    Return the square root (if it exists).
    
    Specified by:
    
    squareRoot in interface ExtensionField
    
    Parameters:
    
    a - the field element
    
    Returns:
    
    square root of a if it exists, null otherwise
  - squareRoot
```
public QuadraticExtensionFieldElement squareRoot(ExtensionFieldElement a,
                                                 boolean checkSquare)
```
    Description copied from interface: ExtensionField
    
    Return the square root (if it exists).
    
    Specified by:
    
    squareRoot in interface ExtensionField
    
    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
  - multiplyBy2
```
public QuadraticExtensionFieldElement multiplyBy2(PrimeCharacteristicFieldElement a)
```
    Description copied from interface: PrimeCharacteristicField
    
    Multiplies a by 2, which is a left-shift. Works in-place.
    
    Specified by:
    
    multiplyBy2 in interface PrimeCharacteristicField
    
    Parameters:
    
    a - the element to be shifted
    
    Returns:
    
    a holding the result
  - multiplyBy2OutOfPlace
```
public QuadraticExtensionFieldElement 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.
    
    Specified by:
    
    multiplyBy2OutOfPlace in interface PrimeCharacteristicField
    
    Parameters:
    
    a - the element to be shifted
    
    Returns:
    
    a holding the result
  - multiplyBy3
```
public QuadraticExtensionFieldElement multiplyBy3(PrimeCharacteristicFieldElement a)
```
    Description copied from interface: PrimeCharacteristicField
    
    Multiplies a by 3, which is a left-shift plus an addition. Works in-place.
    
    Specified by:
    
    multiplyBy3 in interface PrimeCharacteristicField
    
    Parameters:
    
    a - the element to multiplied
    
    Returns:
    
    a holding the result
  - multiplyBy3OutOfPlace
```
public QuadraticExtensionFieldElement 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.
    
    Specified by:
    
    multiplyBy3OutOfPlace in interface PrimeCharacteristicField
    
    Parameters:
    
    a - the element to multiplied
    
    Returns:
    
    a holding the result
  - multiplyByPowerOf2
```
public QuadraticExtensionFieldElement 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.
    
    Specified by:
    
    multiplyByPowerOf2 in interface PrimeCharacteristicField
    
    Parameters:
    
    a - the element to be shifted
    
    exponent - the power of 2
    
    Returns:
    
    a holding the result
  - multiplyByPowerOf2OutOfPlace
```
public QuadraticExtensionFieldElement 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.
    
    Specified by:
    
    multiplyByPowerOf2OutOfPlace in interface PrimeCharacteristicField
    
    Parameters:
    
    a - the element to be shifted
    
    exponent - the power of 2
    
    Returns:
    
    a holding the result
  - invert
```
public QuadraticExtensionFieldElement invert(GenericFieldElement a)
```
    Description copied from interface: GenericField
    
    Inverts a, i.e. performs the operation a = a^(-1).
    
    Specified by:
    
    invert in interface ExtensionField
    
    Specified by:
    
    invert in interface GenericField
    
    Specified by:
    
    invert in interface PrimeCharacteristicField
    
    Parameters:
    
    a - the field element
    
    Returns:
    
    the inverse of a
  - 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
  - newElement
```
public QuadraticExtensionFieldElement 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 QuadraticExtensionFieldElement 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
  - newElement
```
public QuadraticExtensionFieldElement newElement(PrimeFieldElement a,
                                                 PrimeFieldElement b)
```
    Create new element from two prime field elements
    
    Parameters:
    
    a - a prime field element
    
    b - a prime field element
    
    Returns:
    
    a quadratic field element representing a + i b
  - toElement
```
public QuadraticExtensionFieldElement 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 QuadraticExtensionFieldElement getUniformlyRandomElement()
```
    Description copied from interface: PrimeCharacteristicField
    
    Get a uniformly random prime field element.
    
    Returns:
    
    element of the field.
  - getUniformlyRandomNonZeroElement
```
public QuadraticExtensionFieldElement 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
  - newElementFromBaseField
```
public QuadraticExtensionFieldElement 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 QuadraticExtensionFieldElement 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
  - multiplyByAdjointRoot
```
public QuadraticExtensionFieldElement multiplyByAdjointRoot(ExtensionFieldElement a)
```
    Description copied from interface: ExtensionField
    
    Multiply a by the root used to obtain the extension field.
    
    Specified by:
    
    multiplyByAdjointRoot in interface ExtensionField
    
    Parameters:
    
    a - extension field element
    
    Returns:
    
    a multiplied by the adjoint root.
  - multiplyByAdjointRootOutOfPlace
```
public QuadraticExtensionFieldElement multiplyByAdjointRootOutOfPlace(ExtensionFieldElement a)
```
  - applyFrobenius
```
public QuadraticExtensionFieldElement applyFrobenius(ExtensionFieldElement a,
                                                     int i)
```
    Description copied from interface: ExtensionField
    
    Compute the i-th Frobenius of a.
    
    Specified by:
    
    applyFrobenius in interface ExtensionField
    
    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.
  - negate
```
public QuadraticExtensionFieldElement 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 QuadraticExtensionFieldElement 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 QuadraticExtensionFieldElement 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 QuadraticExtensionFieldElement 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 QuadraticExtensionFieldElement 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 QuadraticExtensionFieldElement 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
  - divide
```
public QuadraticExtensionFieldElement 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 QuadraticExtensionFieldElement 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 QuadraticExtensionFieldElement conjugate(ExtensionFieldElement b)
```
    Description copied from interface: ExtensionField
    
    Conjugate an element.
    
    Parameters:
    
    b - the element to be conjugated
    
    Returns:
    
    the conjugate of b
  - conjugate
```
public QuadraticExtensionFieldElement 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
  - getOne
```
public QuadraticExtensionFieldElement getOne()
```
    Description copied from interface: GenericField
    
    Returns the one element.
    
    Returns:
    
    an object holding the one element
  - getZero
```
public QuadraticExtensionFieldElement getZero()
```
    Description copied from interface: GenericField
    
    Returns the zero element.
    
    Returns:
    
    an object holding the zero element
  - toString
```
public String toString()
```
    Overrides:
    
    toString in class Object
  - getNonResidue
```
public GenericFieldElement getNonResidue()
```
    Get non residue.
    
    Returns:
    
    the non residue
  - 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 QuadraticExtensionField

Method Summary

Methods inherited from class java.lang.Object

Method Detail

getQuadraticCharacter

multiply

multiplyOutOfPlace

multiply

multiplyOutOfPlace

multiply

multiplyOutOfPlace

square

squareOutOfPlace

squareRoot

squareRoot

multiplyBy2

multiplyBy2OutOfPlace

multiplyBy3

multiplyBy3OutOfPlace

multiplyByPowerOf2

multiplyByPowerOf2OutOfPlace

invert

getNorm

newElement

newElement

newElement

toElement

getUniformlyRandomElement

getUniformlyRandomNonZeroElement

toBigIntegers

newElementFromBaseField

newElementFromSubField

multiplyByAdjointRoot

multiplyByAdjointRootOutOfPlace

applyFrobenius

negate

negateOutOfPlace

add

addOutOfPlace

subtract

subtractOutOfPlace

divide

divideBy2

conjugate

conjugate

getOne

getZero

toString

getNonResidue

invertElements

exponentiate

exponentiate

exponentiateByPowerOf2

getCardinality

getFieldType

getFieldSize

getDepth

getDegree

getDegreeOverBaseField

getBaseField

getSubField

hashCode

equals