java.lang.Object
- net.i2p.crypto.eddsa.math.FieldElement
- - net.i2p.crypto.eddsa.math.ed25519.Ed25519FieldElement

All Implemented Interfaces:

Serializable
```
public class Ed25519FieldElement
extends FieldElement
```
Class to represent a field element of the finite field $p = 2^{255} - 19$ elements.
An element $t$, entries $t[0] \dots t[9]$, represents the integer $t[0]+2^{26} t[1]+2^{51} t[2]+2^{77} t[3]+2^{102} t[4]+\dots+2^{230} t[9]$. Bounds on each $t[i]$ vary depending on context.
Reviewed/commented by Bloody Rookie (nemproject@gmx.de)

See Also:

Serialized Form

Field Summary

Fields
Modifier and Type Field Description

(package private) int[] t
Variable is package private for encoding.
- Fields inherited from class net.i2p.crypto.eddsa.math.FieldElement
  f

Constructor Summary

Constructors
Constructor Description

Ed25519FieldElement(Field f, int[] t)
Creates a field element.

Method Summary

All Methods Instance Methods Concrete Methods
Modifier and Type	Method	Description
`FieldElement`	`add(FieldElement val)`	$h = f + g$
`FieldElement`	`cmov(FieldElement val, int b)`	Constant-time conditional move.
`boolean`	`equals(Object obj)`
`int`	`hashCode()`
`FieldElement`	`invert()`	Invert this field element.
`boolean`	`isNonZero()`	Gets a value indicating whether or not the field element is non-zero.
`FieldElement`	`multiply(FieldElement val)`	$h = f * g$
`FieldElement`	`negate()`	$h = -f$
`FieldElement`	`pow22523()`	Gets this field element to the power of $(2^{252} - 3)$.
`FieldElement`	`square()`	$h = f * f$
`FieldElement`	`squareAndDouble()`	$h = 2 * f * f$
`FieldElement`	`subtract(FieldElement val)`	$h = f - g$
`String`	`toString()`

Methods inherited from class net.i2p.crypto.eddsa.math.FieldElement
addOne, divide, isNegative, subtractOne, toByteArray

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

- Field Detail
  - t
```
final int[] t
```
    Variable is package private for encoding.
- Constructor Detail
  - Ed25519FieldElement
```
public Ed25519FieldElement(Field f,
                           int[] t)
```
    Creates a field element.
    
    Parameters:
    
    f - The underlying field, must be the finite field with $p = 2^{255} - 19$ elements
    
    t - The $2^{25.5}$ bit representation of the field element.
- Method Detail
  - isNonZero
```
public boolean isNonZero()
```
    Gets a value indicating whether or not the field element is non-zero.
    
    Specified by:
    
    isNonZero in class FieldElement
    
    Returns:
    
    1 if it is non-zero, 0 otherwise.
  - add
```
public FieldElement add(FieldElement val)
```
    $h = f + g$
    TODO-CR BR: $h$ is allocated via new, probably not a good idea. Do we need the copying into temp variables if we do that?
    Preconditions:
    
    $|f|$ bounded by $1.1*2^{25},1.1*2^{24},1.1*2^{25},1.1*2^{24},$ etc.
    $|g|$ bounded by $1.1*2^{25},1.1*2^{24},1.1*2^{25},1.1*2^{24},$ etc.
    Postconditions:
    
    $|h|$ bounded by $1.1*2^{26},1.1*2^{25},1.1*2^{26},1.1*2^{25},$ etc.
    Specified by:
    
    add in class FieldElement
    
    Parameters:
    
    val - The field element to add.
    
    Returns:
    
    The field element this + val.
  - subtract
```
public FieldElement subtract(FieldElement val)
```
    $h = f - g$
    Can overlap $h$ with $f$ or $g$.
    TODO-CR BR: See above.
    Preconditions:
    
    $|f|$ bounded by $1.1*2^{25},1.1*2^{24},1.1*2^{25},1.1*2^{24},$ etc.
    $|g|$ bounded by $1.1*2^{25},1.1*2^{24},1.1*2^{25},1.1*2^{24},$ etc.
    Postconditions:
    
    $|h|$ bounded by $1.1*2^{26},1.1*2^{25},1.1*2^{26},1.1*2^{25},$ etc.
    Specified by:
    
    subtract in class FieldElement
    
    Parameters:
    
    val - The field element to subtract.
    
    Returns:
    
    The field element this - val.
  - negate
```
public FieldElement negate()
```
    $h = -f$
    TODO-CR BR: see above.
    Preconditions:
    
    $|f|$ bounded by $1.1*2^{25},1.1*2^{24},1.1*2^{25},1.1*2^{24},$ etc.
    Postconditions:
    
    $|h|$ bounded by $1.1*2^{25},1.1*2^{24},1.1*2^{25},1.1*2^{24},$ etc.
    Specified by:
    
    negate in class FieldElement
    
    Returns:
    
    The field element (-1) * this.
  - multiply
```
public FieldElement multiply(FieldElement val)
```
    $h = f * g$
    Can overlap $h$ with $f$ or $g$.
    Preconditions:
    
    $|f|$ bounded by $1.65*2^{26},1.65*2^{25},1.65*2^{26},1.65*2^{25},$ etc.
    $|g|$ bounded by $1.65*2^{26},1.65*2^{25},1.65*2^{26},1.65*2^{25},$ etc.
    Postconditions:
    
    $|h|$ bounded by $1.01*2^{25},1.01*2^{24},1.01*2^{25},1.01*2^{24},$ etc.
    Notes on implementation strategy:
    Using schoolbook multiplication. Karatsuba would save a little in some cost models.
    Most multiplications by 2 and 19 are 32-bit precomputations; cheaper than 64-bit postcomputations.
    There is one remaining multiplication by 19 in the carry chain; one *19 precomputation can be merged into this, but the resulting data flow is considerably less clean.
    There are 12 carries below. 10 of them are 2-way parallelizable and vectorizable. Can get away with 11 carries, but then data flow is much deeper.
    With tighter constraints on inputs can squeeze carries into int32.
    Specified by:
    
    multiply in class FieldElement
    
    Parameters:
    
    val - The field element to multiply.
    
    Returns:
    
    The (reasonably reduced) field element this * val.
  - square
```
public FieldElement square()
```
    $h = f * f$
    Can overlap $h$ with $f$.
    Preconditions:
    
    $|f|$ bounded by $1.65*2^{26},1.65*2^{25},1.65*2^{26},1.65*2^{25},$ etc.
    Postconditions:
    
    $|h|$ bounded by $1.01*2^{25},1.01*2^{24},1.01*2^{25},1.01*2^{24},$ etc.
    See multiply(FieldElement) for discussion of implementation strategy.
    Specified by:
    
    square in class FieldElement
    
    Returns:
    
    The (reasonably reduced) square of this field element.
  - squareAndDouble
```
public FieldElement squareAndDouble()
```
    $h = 2 * f * f$
    Can overlap $h$ with $f$.
    Preconditions:
    
    $|f|$ bounded by $1.65*2^{26},1.65*2^{25},1.65*2^{26},1.65*2^{25},$ etc.
    Postconditions:
    
    $|h|$ bounded by $1.01*2^{25},1.01*2^{24},1.01*2^{25},1.01*2^{24},$ etc.
    See multiply(FieldElement) for discussion of implementation strategy.
    Specified by:
    
    squareAndDouble in class FieldElement
    
    Returns:
    
    The (reasonably reduced) square of this field element times 2.
  - invert
```
public FieldElement invert()
```
    Invert this field element.
    The inverse is found via Fermat's little theorem:
    $a^p \cong a \mod p$ and therefore $a^{(p-2)} \cong a^{-1} \mod p$
    
    Specified by:
    
    invert in class FieldElement
    
    Returns:
    
    The inverse of this field element.
  - pow22523
```
public FieldElement pow22523()
```
    Gets this field element to the power of $(2^{252} - 3)$. This is a helper function for calculating the square root.
    TODO-CR BR: I think it makes sense to have a sqrt function.
    
    Specified by:
    
    pow22523 in class FieldElement
    
    Returns:
    
    This field element to the power of $(2^{252} - 3)$.
  - cmov
```
public FieldElement cmov(FieldElement val,
                         int b)
```
    Constant-time conditional move. Well, actually it is a conditional copy. Logic is inspired by the SUPERCOP implementation at: https://github.com/floodyberry/supercop/blob/master/crypto_sign/ed25519/ref10/fe_cmov.c
    
    Specified by:
    
    cmov in class FieldElement
    
    Parameters:
    
    val - the other field element.
    
    b - must be 0 or 1, otherwise results are undefined.
    
    Returns:
    
    a copy of this if $b == 0$, or a copy of val if $b == 1$.
    
    Since:
    
    0.9.36
  - hashCode
```
public int hashCode()
```
    Overrides:
    
    hashCode in class Object
  - equals
```
public boolean equals(Object obj)
```
    Overrides:
    
    equals in class Object
  - toString
```
public String toString()
```
    Overrides:
    
    toString in class Object

Class Ed25519FieldElement

Field Summary

Fields inherited from class net.i2p.crypto.eddsa.math.FieldElement

Constructor Summary

Method Summary

Methods inherited from class net.i2p.crypto.eddsa.math.FieldElement

Methods inherited from class java.lang.Object

Field Detail

t

Constructor Detail

Ed25519FieldElement

Method Detail

isNonZero

add

subtract

negate

multiply

square

squareAndDouble

invert

pow22523

cmov

hashCode

equals

toString