io.nem.core.crypto.ed25519.arithmetic

Class Ed25519FieldElement

java.lang.Object

io.nem.core.crypto.ed25519.arithmetic.Ed25519FieldElement

public class Ed25519FieldElement
extends java.lang.Object

Represents a element of the finite field with p=2^255-19 elements.

values[0] ... values[9], represent the integer
values[0] + 2^26 * values[1] + 2^51 * values[2] + 2^77 * values[3] + 2^102 * values[4] + ... + 2^230 * values[9].
Bounds on each values[i] vary depending on context.

This implementation is based on the ref10 implementation of SUPERCOP.

Constructor Summary

Constructors

Constructor and Description
Ed25519FieldElement(int[] values) Creates a field element.

Method Summary

Modifier and Type	Method and Description
Ed25519FieldElement	add(Ed25519FieldElement g) Adds the given field element to this and returns the result.
Ed25519EncodedFieldElement	encode() Encodes a given field element in its 32 byte 2^8 bit representation.
boolean	equals(java.lang.Object obj)
int[]	getRaw() Gets the underlying int array.
int	hashCode()
Ed25519FieldElement	invert() Invert this field element and return the result.
boolean	isNegative() Return true if this is in {1,3,5,...,q-2} Return false if this is in {0,2,4,...,q-1}
boolean	isNonZero() Gets a value indicating whether or not the field element is non-zero.
Ed25519FieldElement	multiply(Ed25519FieldElement g) Multiplies this field element with the given field element and returns the result.
Ed25519FieldElement	negate() Negates this field element and return the result.
static Ed25519FieldElement	sqrt(Ed25519FieldElement u, Ed25519FieldElement v) Calculates and returns one of the square roots of u / v.
Ed25519FieldElement	square() Squares this field element and returns the result.
Ed25519FieldElement	squareAndDouble() Squares this field element, multiplies by two and returns the result.
Ed25519FieldElement	subtract(Ed25519FieldElement g) Subtract the given field element from this and returns the result.
java.lang.String	toString()

Methods inherited from class java.lang.Object

clone, finalize, getClass, notify, notifyAll, wait, wait, wait

Constructor Detail

Ed25519FieldElement

public Ed25519FieldElement(int[] values)

Creates a field element.

Parameters:

values - The 2^25.5 bit representation of the field element.

Method Detail

sqrt

public static Ed25519FieldElement sqrt(Ed25519FieldElement u,
                                       Ed25519FieldElement v)

Calculates and returns one of the square roots of u / v.

 x = (u * v^3) * (u * v^7)^((p - 5) / 8) ==> x^2 = +-(u / v).
 

Note that this means x can be sqrt(u / v), -sqrt(u / v), +i * sqrt(u / v), -i * sqrt(u / v).

Parameters:

u - The nominator of the fraction.

v - The denominator of the fraction.

Returns:

The square root of u / v.

getRaw

public int[] getRaw()

Gets the underlying int array.

Returns:

The int array.

isNonZero

public boolean isNonZero()

Gets a value indicating whether or not the field element is non-zero.

Returns:

1 if it is non-zero, 0 otherwise.

add

public Ed25519FieldElement add(Ed25519FieldElement g)

Adds the given field element to this and returns the result. h = this + g

 Preconditions:
     |this| 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.
 

Parameters:

g - The field element to add.

Returns:

The field element this + val.

subtract

public Ed25519FieldElement subtract(Ed25519FieldElement g)

Subtract the given field element from this and returns the result. h = this - g

 Preconditions:
     |this| 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.
 

Parameters:

g - The field element to subtract.

Returns:

The field element this - val.

negate

public Ed25519FieldElement negate()

Negates this field element and return the result. h = -this

 Preconditions:
     |this| 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.
 

Returns:

The field element (-1) * this.

multiply

public Ed25519FieldElement multiply(Ed25519FieldElement g)

Multiplies this field element with the given field element and returns the result. h = this * g Preconditions:

     |this| 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.

Parameters:

g - The field element to multiply.

Returns:

The (reasonably reduced) field element this * val.

square

public Ed25519FieldElement square()

Squares this field element and returns the result. h = this * this

 Preconditions:
     |this| 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 for discussion of implementation strategy.

Returns:

The square of this field element.

squareAndDouble

public Ed25519FieldElement squareAndDouble()

Squares this field element, multiplies by two and returns the result. h = 2 * this * this

 Preconditions:
     |this| 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 for discussion of implementation strategy.

Returns:

The square of this field element times 2.

invert

public Ed25519FieldElement invert()

Invert this field element and return the result. The inverse is found via Fermat's little theorem: a^p congruent a mod p and therefore a^(p-2) congruent a^-1 mod p

Returns:

The inverse of this field element.

encode

public Ed25519EncodedFieldElement encode()

Encodes a given field element in its 32 byte 2^8 bit representation. This is done in two steps. Step 1: Reduce the value of the field element modulo p. Step 2: Convert the field element to the 32 byte representation.

Returns:

Encoded field element (32 bytes).

isNegative

public boolean isNegative()

Return true if this is in {1,3,5,...,q-2} Return false if this is in {0,2,4,...,q-1}

 Preconditions:
     |x| bounded by 1.1*2^26,1.1*2^25,1.1*2^26,1.1*2^25,etc.
 

Returns:

true if this is in {1,3,5,...,q-2}, false otherwise.

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

toString

public java.lang.String toString()

Overrides:

toString in class java.lang.Object