com.oceanbase.jdbc.internal.com.send.authentication.ed25519.math.ed25519

Class Ed25519FieldElement

java.lang.Object

com.oceanbase.jdbc.internal.com.send.authentication.ed25519.math.FieldElement

com.oceanbase.jdbc.internal.com.send.authentication.ed25519.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 inherited from class com.oceanbase.jdbc.internal.com.send.authentication.ed25519.math.FieldElement

f

Constructor Summary

Constructors

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

Method Summary

Modifier and Type	Method and 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 com.oceanbase.jdbc.internal.com.send.authentication.ed25519.math.FieldElement

addOne, divide, isNegative, subtractOne, toByteArray

Methods inherited from class java.lang.Object

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

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 $b == 1$.

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