ExtensionField (math 2.1.0 API)

java.lang.Object
- org.cryptimeleon.math.structures.rings.extfield.ExtensionField

All Implemented Interfaces:

RepresentationRestorer, Representable, StandaloneRepresentable, Field, Ring, Structure
```
public class ExtensionField
extends java.lang.Object
implements Field
```
An extension field using an irreducible polynomial of the form \(x^\text{extensionDegree} + \text{constant}\).

Field Summary

Fields
Modifier and Type	Field and Description
`protected FieldElement`	`constant`
`protected PolynomialRing.Polynomial`	`definingPolynomial`
`protected int`	`extensionDegree`
`protected ExtensionFieldElement[]`	`frobeniusOfXPowers` \(\text{frobeniusOfXPowers}[i] = (x^p)^i \mod (x^\text{extensionDegree} + \text{constant})\) for \(i \leq extensionDegree\)

Constructor Summary

Constructors
Constructor and Description
`ExtensionField(java.math.BigInteger p)` Instantiates the prime order finite field Zp.
`ExtensionField(FieldElement constant, int extensionDegree)` Create extension defined by polynomial \(x^\text{extensionDegree} + \text{constant}\).
`ExtensionField(Representation r)`

Method Summary

All Methods Instance Methods Concrete Methods
Modifier and Type	Method and Description
`ExtensionFieldElement`	`createElement(java.math.BigInteger b)` Map an integer b to an element in this field.
`ExtensionFieldElement`	`createElement(FieldElement... coefficients)` Create an element in this field, based on a given polynomial representation of an element in the ideal of the quotient field.
`ExtensionFieldElement`	`createElement(java.util.List<java.math.BigInteger> coefficients)`
`boolean`	`equals(java.lang.Object obj)`
`double`	`estimateCostInvPerOp()` Estimates the number of inversions that can be done per group operation for the same cost.
`double`	`estimateCostNegPerOp()` Estimates the number of negations (additive inversions) that can be done per group operation for the same cost.
`void`	`generatePrimitiveCubeRoot()` Search and set primitive cube root in this field.
`Field`	`getBaseField()`
`java.math.BigInteger`	`getCharacteristic()` Returns the characteristic of the ring.
`FieldElement`	`getConstant()`
`FieldElement`	`getCubeRoot()` Returns fixed primitive cube root in this field.
`PolynomialRing.Polynomial`	`getDefiningPolynomial()` Returns the irreducible polynomial that this field is defined over.
`ExtensionFieldElement`	`getElement(java.math.BigInteger i)` Maps the integer i into this ring \(R\), such that this map is a ring homomorphism \(\mathbb{Z}_{\text{getCharacteristic()}} \rightarrow R\).
`ExtensionFieldElement`	`getElement(long i)` Maps the integer i into this ring \(R\), such that this map is a ring homomorphism \(\mathbb{Z}_{\text{getCharacteristic()}} \rightarrow R\).
`int`	`getExtensionDegree()`
`ExtensionFieldElement`	`getOneElement()` Returns the multiplicative neutral element of this ring.
`FieldElement`	`getPrimitiveElement()` If this is a finite field, returns a generator of the field's unit group.
`Representation`	`getRepresentation()` The representation of this object.
`ExtensionFieldElement`	`getUniformlyRandomElement()` Returns an element of this structure that is drawn uniformly at random using a cryptographically strong RNG.
`java.util.Optional<java.lang.Integer>`	`getUniqueByteLength()` Returns the number of bytes returned by this structure's `UniqueByteRepresentable.getUniqueByteRepresentation()`, or an empty `Optional` if this structure's elements do not guarantee a fixed length.
`ExtensionFieldElement`	`getZeroElement()` Returns the additive neutral element of this ring.
`int`	`hashCode()`
`boolean`	`isBaseField()`
`FieldElement[]`	`reduce(FieldElement[] coefficients)`
`ExtensionFieldElement`	`restoreElement(Representation repr)` Creates an element of this structure from its representation.
`protected void`	`setCubeRoot(FieldElement cubeRoot)` Set auxiliary primitive cube root in this field.
`java.math.BigInteger`	`size()` Returns the number of elements in this structure (the size).
`java.math.BigInteger`	`sizeUnitGroup()` Returns the number of units in this ring.
`java.lang.String`	`toString()`

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

Methods inherited from interface org.cryptimeleon.math.structures.rings.Field
getUniformlyRandomUnit, isCommutative

Methods inherited from interface org.cryptimeleon.math.structures.rings.Ring
asAdditiveGroup, asUnitGroup, extendedEuclideanAlgorithm, extendedEuclideanAlgorithm, getUniformlyRandomElements, getUniformlyRandomNonzeroElement, getUniformlyRandomNonzeroElements, getUniformlyRandomUnits, restoreFromRepresentation, restoreVector

Methods inherited from interface org.cryptimeleon.math.structures.Structure
hasPrimeSize

- Field Detail
  - constant
```
protected FieldElement constant
```
  - extensionDegree
```
protected int extensionDegree
```
  - definingPolynomial
```
protected PolynomialRing.Polynomial definingPolynomial
```
  - frobeniusOfXPowers
```
protected ExtensionFieldElement[] frobeniusOfXPowers
```
    \(\text{frobeniusOfXPowers}[i] = (x^p)^i \mod (x^\text{extensionDegree} + \text{constant})\) for \(i \leq extensionDegree\)
- Constructor Detail
  - ExtensionField
```
public ExtensionField(FieldElement constant,
                      int extensionDegree)
```
    Create extension defined by polynomial \(x^\text{extensionDegree} + \text{constant}\).
  - ExtensionField
```
public ExtensionField(java.math.BigInteger p)
```
    Instantiates the prime order finite field Zp. (i.e. the special case of an extension field with extension degree 1)
    
    Parameters:
    
    p - size of the field (must be prime)
  - ExtensionField
```
public ExtensionField(Representation r)
```
- Method Detail
  - getConstant
```
public FieldElement getConstant()
```
  - isBaseField
```
public boolean isBaseField()
```
  - getRepresentation
```
public Representation getRepresentation()
```
    Description copied from interface: Representable
    
    The representation of this object. Used for serialization. A convenient way to implement this is using @link ReprUtil
    
    Specified by:
    
    getRepresentation in interface Representable
    
    Returns:
    
    a Representation or null if an equal object can be recreated without any information.
    
    See Also:
    
    Representation
  - getCubeRoot
```
public FieldElement getCubeRoot()
```
    Returns fixed primitive cube root in this field.
    
    Returns:
    
    primitive cube root
  - setCubeRoot
```
protected void setCubeRoot(FieldElement cubeRoot)
```
    Set auxiliary primitive cube root in this field.
  - generatePrimitiveCubeRoot
```
public void generatePrimitiveCubeRoot()
```
    Search and set primitive cube root in this field.
  - reduce
```
public FieldElement[] reduce(FieldElement[] coefficients)
```
  - createElement
```
public ExtensionFieldElement createElement(FieldElement... coefficients)
```
    Create an element in this field, based on a given polynomial representation of an element in the ideal of the quotient field.
  - getElement
```
public ExtensionFieldElement getElement(java.math.BigInteger i)
```
    Description copied from interface: Ring
    
    Maps the integer i into this ring \(R\), such that this map is a ring homomorphism \(\mathbb{Z}_{\text{getCharacteristic()}} \rightarrow R\).
    
    Specified by:
    
    getElement in interface Field
    
    Specified by:
    
    getElement in interface Ring
    
    Parameters:
    
    i - the integer to map
    
    Returns:
    
    an element of the ring
  - getElement
```
public ExtensionFieldElement getElement(long i)
```
    Description copied from interface: Ring
    
    Maps the integer i into this ring \(R\), such that this map is a ring homomorphism \(\mathbb{Z}_{\text{getCharacteristic()}} \rightarrow R\).
    
    Specified by:
    
    getElement in interface Field
    
    Specified by:
    
    getElement in interface Ring
    
    Parameters:
    
    i - the integer to map
    
    Returns:
    
    an element of the ring
  - estimateCostInvPerOp
```
public double estimateCostInvPerOp()
```
    Description copied from interface: Ring
    
    Estimates the number of inversions that can be done per group operation for the same cost. For example, 2 would mean that an inversion costs half as much as a group operation, on average.
    
    Specified by:
    
    estimateCostInvPerOp in interface Ring
    
    Returns:
    
    estimated number of inversions that can be done per group operation for the same cost
  - estimateCostNegPerOp
```
public double estimateCostNegPerOp()
```
    Description copied from interface: Ring
    
    Estimates the number of negations (additive inversions) that can be done per group operation for the same cost. For example, 2 would mean that a negation costs half as much as a group operation, on average.
    
    Specified by:
    
    estimateCostNegPerOp in interface Ring
    
    Returns:
    
    estimated number of inversions that can be done per group operation for the same cost
  - createElement
```
public ExtensionFieldElement createElement(java.math.BigInteger b)
```
    Map an integer b to an element in this field.
    Let this field L be a degree k extension of a field K given by \(L=K/(f(X))\). Let n be the size of K. This function computes the n-ary representation of b:
    \(b=b_0 n^0 + b_1 n^1 + b_2 n^2 ... b_t n^t\)
    Then, it maps \(b_i\) to \(c_i\) in K by recursively invoking K.createElement(b_i) and constructs the polynomial
    \(p(X) = c_0 X^0 + c_1 X^1 + c_2 X^t + ... + c_t X^t\). Then, it projects \(p(X)\) to \(K/(f(X))\) in the canonical way.
    Hence, this mapping is injective on the integers smaller than this.size().
    
    Parameters:
    
    b - the integer to map to a field element
    
    Returns:
    
    field element corresponding to the given integer
  - toString
```
public java.lang.String toString()
```
    Overrides:
    
    toString in class java.lang.Object
  - sizeUnitGroup
```
public java.math.BigInteger sizeUnitGroup()
                                   throws java.lang.UnsupportedOperationException
```
    Description copied from interface: Ring
    
    Returns the number of units in this ring.
    
    Specified by:
    
    sizeUnitGroup in interface Field
    
    Specified by:
    
    sizeUnitGroup in interface Ring
    
    Returns:
    
    size of the unit group or null if infinite
    
    Throws:
    
    java.lang.UnsupportedOperationException - if size is unknown or is computationally too expensive to compute
  - getBaseField
```
public Field getBaseField()
```
  - getCharacteristic
```
public java.math.BigInteger getCharacteristic()
                                       throws java.lang.UnsupportedOperationException
```
    Description copied from interface: Ring
    
    Returns the characteristic of the ring.
    The characteristic of a ring is defined to be the number n such that there is a ring homomorphism from Zn to the ring.
    
    Specified by:
    
    getCharacteristic in interface Ring
    
    Throws:
    
    java.lang.UnsupportedOperationException - if unknown
  - size
```
public java.math.BigInteger size()
                          throws java.lang.UnsupportedOperationException
```
    Description copied from interface: Structure
    
    Returns the number of elements in this structure (the size).
    
    Specified by:
    
    size in interface Structure
    
    Returns:
    
    the number of elements contained in this structure or null if infinite
    
    Throws:
    
    java.lang.UnsupportedOperationException - if the number of elements is unknown or too expensive to compute
  - getPrimitiveElement
```
public FieldElement getPrimitiveElement()
                                 throws java.lang.UnsupportedOperationException
```
    Description copied from interface: Field
    
    If this is a finite field, returns a generator of the field's unit group. Repeated calls may or may not always return the same element.
    
    Specified by:
    
    getPrimitiveElement in interface Field
    
    Throws:
    
    java.lang.UnsupportedOperationException - if operation is not feasible or undefined
  - getZeroElement
```
public ExtensionFieldElement getZeroElement()
```
    Description copied from interface: Ring
    
    Returns the additive neutral element of this ring.
    
    Specified by:
    
    getZeroElement in interface Field
    
    Specified by:
    
    getZeroElement in interface Ring
  - getOneElement
```
public ExtensionFieldElement getOneElement()
```
    Description copied from interface: Ring
    
    Returns the multiplicative neutral element of this ring.
    
    Specified by:
    
    getOneElement in interface Field
    
    Specified by:
    
    getOneElement in interface Ring
  - restoreElement
```
public ExtensionFieldElement restoreElement(Representation repr)
```
    Description copied from interface: Structure
    
    Creates an element of this structure from its representation.
    
    Specified by:
    
    restoreElement in interface Field
    
    Specified by:
    
    restoreElement in interface Ring
    
    Specified by:
    
    restoreElement in interface Structure
    
    Parameters:
    
    repr - the Representation returned by Representable.getRepresentation()
    
    Returns:
    
    the decoded element corresponding to the representation
  - createElement
```
public ExtensionFieldElement createElement(java.util.List<java.math.BigInteger> coefficients)
```
  - getUniformlyRandomElement
```
public ExtensionFieldElement getUniformlyRandomElement()
                                                throws java.lang.UnsupportedOperationException
```
    Description copied from interface: Structure
    
    Returns an element of this structure that is drawn uniformly at random using a cryptographically strong RNG.
    
    Specified by:
    
    getUniformlyRandomElement in interface Field
    
    Specified by:
    
    getUniformlyRandomElement in interface Ring
    
    Specified by:
    
    getUniformlyRandomElement in interface Structure
    
    Throws:
    
    java.lang.UnsupportedOperationException - if the operation is not supported
  - getDefiningPolynomial
```
public PolynomialRing.Polynomial getDefiningPolynomial()
```
    Returns the irreducible polynomial that this field is defined over. It's of the form x^extensionDegree + constant.
  - hashCode
```
public int hashCode()
```
    Overrides:
    
    hashCode in class java.lang.Object
  - equals
```
public boolean equals(java.lang.Object obj)
```
    Overrides:
    
    equals in class java.lang.Object
  - getExtensionDegree
```
public int getExtensionDegree()
```
  - getUniqueByteLength
```
public java.util.Optional<java.lang.Integer> getUniqueByteLength()
```
    Description copied from interface: Structure
    
    Returns the number of bytes returned by this structure's UniqueByteRepresentable.getUniqueByteRepresentation(), or an empty Optional if this structure's elements do not guarantee a fixed length.
    For example, elements of Zp will always be represented by ceil(ceil(log(p))/8) bytes, hence getUniqueByteLength() would return ceil(ceil(log(p))/8).
    A polynomial ring would return an empty Optional since a polynomial's unique byte representation length depends on its degree.
    
    Specified by:
    
    getUniqueByteLength in interface Structure
    
    Returns:
    
    the guaranteed fixed length of getUniqueByteRepresentation(), or an empty Optional, if no guarantee

Class ExtensionField

Field Summary

Constructor Summary

Method Summary

Methods inherited from class java.lang.Object

Methods inherited from interface org.cryptimeleon.math.structures.rings.Field

Methods inherited from interface org.cryptimeleon.math.structures.rings.Ring

Methods inherited from interface org.cryptimeleon.math.structures.Structure

Field Detail

constant

extensionDegree

definingPolynomial

frobeniusOfXPowers

Constructor Detail

ExtensionField

ExtensionField

ExtensionField

Method Detail

getConstant

isBaseField

getRepresentation

getCubeRoot

setCubeRoot

generatePrimitiveCubeRoot

reduce

createElement

getElement

getElement

estimateCostInvPerOp

estimateCostNegPerOp

createElement

toString

sizeUnitGroup

getBaseField

getCharacteristic

size

getPrimitiveElement

getZeroElement

getOneElement

restoreElement

createElement

getUniformlyRandomElement

getDefiningPolynomial

hashCode

equals

getExtensionDegree

getUniqueByteLength