de.flexiprovider.common.math.ellipticcurves
Class Point

java.lang.Object
  |
  +--de.flexiprovider.common.math.ellipticcurves.Point

Direct Known Subclasses:

PointGF2n, PointGFP

public abstract class Point

extends java.lang.Object

This abstract class implements points and their arithmetic on elliptic curves over finite prime fields as well as over finite fields with characteristic 2. There is a variety of methods for point multiplication. See the references for complexity and details.

MvV97	A. J. Menezes, P. C. van Oorschot, and S. A. Vanstone. Handbook of Applied Cryptography. CRC Press, Boca Raton, Florida, 1997.
LL94	C. H. Lim and P. J Lee. More flexible exponentiation with precompu-tation. In Yvo G. Desmedt, editor, Advances in Cryptology CRYP-TO 94, volume 839 of Lecture Notes in Computer Science, pages 108 113. Springer-Verlag, August 1994.
YLL94	S.-M. Yen, C.-S. Laih, and A. Lenstra. Multi-exponentiation. In IEE Proceedings - Computers and Digital Techniques, volume 141, pages 325 326, 1994.
MOC97	Atsuko Miyaji, Takatoshi Ono, Henri Cohen. Efficient elliptic curve exponentiation.
Moe01	Bodo Moeller. Algorithms for multi-exponentiation. In Serge Vau-denay and Amr M. Youssef, editors, Selected Areas in Cryptography - SAC 2001, pages 165 180. Springer-Verlag, 2001.
Moe04	Bodo Moeller. Fractional Windows Revisited:Improved Signed-Digit Representations for Efficient Exponentiation.
OSST04	Katsuyuki Okeya, Katja Schmidt-Samoa, Christian Spahn, Tsuyoshi Takagi. Signed Binary Representations Revisited.
DOT07	Erik Dahmen, Katsuyuki Okeya, Tsuyoshi Takagi. A new upper bound for the minimal density of joint representations in elliptic curve cryptosystems.
PRO03	John Proos. Joint Sparse Forms and Generating Zero Columns when Combing.

Author:

Birgit Henhapl

See Also:

EllipticCurve, PointGFP, PointGF2n

Field Summary

static int	ENCODING_TYPE_COMPRESSED Encoding type 'compressed'
static int	ENCODING_TYPE_HYBRID Encoding type 'hybrid'
static int	ENCODING_TYPE_UNCOMPRESSED Encoding type 'uncompressed'
protected EllipticCurve	mE The elliptic curve this point is on.
protected FlexiBigInt	mP Characteristic of underlying field.

Constructor Summary

Point()

Method Summary

abstract Point	add(Point other) Adds to this point the point other.
abstract Point	addAffine(Point other) Adds in affine coordinates to this point the point other.
abstract void	addToThis(Point other) Adds to this point the point other.
abstract java.lang.Object	clone()
byte[]	EC2OSP(int type) Encode this point into a byte array (octet string) using the specified encoding format (one of ENCODING_TYPE_UNCOMPRESSED, ENCODING_TYPE_COMPRESSED, and ENCODING_TYPE_HYBRID).
(package private) abstract byte[]	encodeCompressed() Returns this point in affin, compressed form as a byte array.
(package private) abstract byte[]	encodeHybrid() Returns this point in affin, hybrid form as a byte array.
(package private) abstract byte[]	encodeUncompressed() Returns this point in affin, decompressed form as a byte array.
abstract boolean	equals(java.lang.Object other) Compare this point with another object.
abstract Point	getAffin() Returns this point with affin coordinates.
EllipticCurve	getE()
abstract GFElement	getX()
abstract GFElement	getXAffin()
abstract GFElement	getY()
abstract GFElement	getYAffin()
abstract GFElement	getZ()
abstract int	hashCode()
boolean	isNegativeOf(Point other) Tests whether this point is the negative of other.
abstract boolean	isZero() Tests, whether this point is at infinity.
abstract Point	multiplyBy2() Doubles this point.
abstract Point	multiplyBy2Affine() Doubles this point in affine coordinates.
abstract void	multiplyThisBy2() Doubles this point.
abstract Point	negate()
abstract void	negateThis() Additively invert this point.
abstract boolean	onCurve() Test whether this point is on curve E.
static Point	OS2ECP(byte[] encoded, CurveParams params) Decode the encoded point given as byte array using the given EC domain parameters.
abstract Point	subtract(Point other) Subtracts the point other from this point.
abstract void	subtractFromThis(Point other) Subtracts the point other from this point.

Methods inherited from class java.lang.Object

, finalize, getClass, notify, notifyAll, toString, wait, wait, wait

Field Detail

mE

protected EllipticCurve mE

The elliptic curve this point is on.

See Also:

EllipticCurve

mP

protected FlexiBigInt mP

Characteristic of underlying field.

ENCODING_TYPE_UNCOMPRESSED

public static final int ENCODING_TYPE_UNCOMPRESSED

Encoding type 'uncompressed'

ENCODING_TYPE_COMPRESSED

public static final int ENCODING_TYPE_COMPRESSED

Encoding type 'compressed'

ENCODING_TYPE_HYBRID

public static final int ENCODING_TYPE_HYBRID

Encoding type 'hybrid'

Constructor Detail

Point

public Point()

Method Detail

clone

public abstract java.lang.Object clone()

Overrides:

clone in class java.lang.Object

Returns:

a copy of this Point

equals

public abstract boolean equals(java.lang.Object other)

Compare this point with another object.

Overrides:

equals in class java.lang.Object

Parameters:

other - the other object

Returns:

the result of the comparison

hashCode

public abstract int hashCode()

Overrides:

hashCode in class java.lang.Object

Returns:

the hash code of this point

getE

public final EllipticCurve getE()

Returns:

the underlying elliptic curve

getX

public abstract GFElement getX()

Returns:

the x-coordinate of this point

getY

public abstract GFElement getY()

Returns:

the y-coordinate of this point

getZ

public abstract GFElement getZ()

Returns:

the z-coordinate of this point

getXAffin

public abstract GFElement getXAffin()

Returns:

the x-coordinate of this point in affine representation

getYAffin

public abstract GFElement getYAffin()

Returns:

the y-coordinate of this point in affine representation

getAffin

public abstract Point getAffin()

Returns this point with affin coordinates.

Returns:

this

onCurve

public abstract boolean onCurve()

Test whether this point is on curve E.

Returns:

true if this == O or if (mX, mY) is on mE
false if (mX, mY) is not on mE

isZero

public abstract boolean isZero()

Tests, whether this point is at infinity.

Returns:

true if this == O
false if this != O

isNegativeOf

public final boolean isNegativeOf(Point other)

Tests whether this point is the negative of other.

Parameters:

other - point to compare this point with

Returns:

true if this == -other, false otherwise.

add

public abstract Point add(Point other)
                   throws DifferentCurvesException

Adds to this point the point other.

Parameters:

other - point to add to this point

Returns:

this + other

Throws:

DifferentCurvesException - when other is defined over another curve

addToThis

public abstract void addToThis(Point other)
                        throws DifferentCurvesException

Adds to this point the point other.

Parameters:

other - point to add to this point

Throws:

DifferentCurvesException - when other is defined over another curve

addAffine

public abstract Point addAffine(Point other)
                         throws DifferentCurvesException

Adds in affine coordinates to this point the point other.

Parameters:

other - point to add to this point

Returns:

this + other in affine coordinates

Throws:

DifferentCurvesException - when other is defined over another curve

subtract

public abstract Point subtract(Point other)
                        throws DifferentCurvesException

Subtracts the point other from this point.

Parameters:

other - point to subtract from this point

Returns:

this - other

Throws:

DifferentCurvesException - if other is defined over a different curve

subtractFromThis

public abstract void subtractFromThis(Point other)
                               throws DifferentCurvesException

Subtracts the point other from this point.

Parameters:

other - point to subtract from this point

Throws:

DifferentCurvesException - if other is defined over a different curve

multiplyBy2

public abstract Point multiplyBy2()

Doubles this point.

Returns:

2*this

multiplyThisBy2

public abstract void multiplyThisBy2()

Doubles this point.

multiplyBy2Affine

public abstract Point multiplyBy2Affine()

Doubles this point in affine coordinates.

Returns:

2*this in affine coordinates

negate

public abstract Point negate()

Returns:

the inverse of this point. returns -this

negateThis

public abstract void negateThis()

Additively invert this point.

EC2OSP

public final byte[] EC2OSP(int type)

Encode this point into a byte array (octet string) using the specified encoding format (one of ENCODING_TYPE_UNCOMPRESSED, ENCODING_TYPE_COMPRESSED, and ENCODING_TYPE_HYBRID).

Parameters:

type - the encoding format

Returns:

the encoded point, or null if the encoding format is unknown.

OS2ECP

public static Point OS2ECP(byte[] encoded,
                           CurveParams params)
                    throws InvalidPointException,
                           InvalidFormatException,
                           InvalidParameterSpecException

Decode the encoded point given as byte array using the given EC domain parameters.

Parameters:

encoded - the encoded point

params - the EC domain parameters

Returns:

the decoded point

Throws:

InvalidPointException - if the point is encoded in hybrid format and the given and the decoded y-coordinates don't match.

InvalidFormatException - if the encoded point is given in an invalid encoding format.

InvalidParameterSpecException - if the parameters are defined neither over GF(2^n) nor over GF(p).

encodeUncompressed

abstract byte[] encodeUncompressed()

Returns this point in affin, decompressed form as a byte array. The first byte keeps the value 4, to indicate, that this point is stored in an uncompressed format. The rest of the returned array is split in two halves, the first holds the x-coordinate mX and the second they-coordinate mY.

Returns:

this as byte array

encodeCompressed

abstract byte[] encodeCompressed()

Returns this point in affin, compressed form as a byte array. The first byte keeps the value 2 or 3, to indicate, that this point is stored in a compressed format. The rest of the returned array is the x-coordinate mX.

Returns:

this as byte array

encodeHybrid

abstract byte[] encodeHybrid()

Returns this point in affin, hybrid form as a byte array. The first byte keeps the value 6 or 7, to indicate, that this point is stored in a hybrid format. The rest of the returned array is split in two halves, the first holds the x-coordinate mX and the second they-coordinate mY.

Returns:

this as byte array