org.opendaylight.netconf.shaded.eddsa.math.GroupElement Maven / Gradle / Ivy
/**
* EdDSA-Java by str4d
*
* To the extent possible under law, the person who associated CC0 with
* EdDSA-Java has waived all copyright and related or neighboring rights
* to EdDSA-Java.
*
* You should have received a copy of the CC0 legalcode along with this
* work. If not, see
* Reviewed/commented by Bloody Rookie ([email protected])
*
* Literature:
* [1] Daniel J. Bernstein, Niels Duif, Tanja Lange, Peter Schwabe and Bo-Yin Yang : High-speed high-security signatures
* [2] Huseyin Hisil, Kenneth Koon-Ho Wong, Gary Carter, Ed Dawson: Twisted Edwards Curves Revisited
* [3] Daniel J. Bernsteina, Tanja Lange: A complete set of addition laws for incomplete Edwards curves
* [4] Daniel J. Bernstein, Peter Birkner, Marc Joye, Tanja Lange and Christiane Peters: Twisted Edwards Curves
* [5] Christiane Pascale Peters: Curves, Codes, and Cryptography (PhD thesis)
* [6] Daniel J. Bernstein, Peter Birkner, Tanja Lange and Christiane Peters: Optimizing double-base elliptic-curve single-scalar multiplication
*
* @author str4d
*/
public class GroupElement implements Serializable {
private static final long serialVersionUID = 2395879087349587L;
/**
* Available representations for a group element.
*
* - P2: Projective representation $(X:Y:Z)$ satisfying $x=X/Z, y=Y/Z$.
*
- P3: Extended projective representation $(X:Y:Z:T)$ satisfying $x=X/Z, y=Y/Z, XY=ZT$.
*
- P3PrecomputedDouble: P3 but with dblPrecmp populated.
*
- P1P1: Completed representation $((X:Z), (Y:T))$ satisfying $x=X/Z, y=Y/T$.
*
- PRECOMP: Precomputed representation $(y+x, y-x, 2dxy)$.
*
- CACHED: Cached representation $(Y+X, Y-X, Z, 2dT)$
*
*/
public enum Representation {
/** Projective ($P^2$): $(X:Y:Z)$ satisfying $x=X/Z, y=Y/Z$ */
P2,
/** Extended ($P^3$): $(X:Y:Z:T)$ satisfying $x=X/Z, y=Y/Z, XY=ZT$ */
P3,
/** Can only be requested. Results in P3 representation but also populates dblPrecmp. */
P3PrecomputedDouble,
/** Completed ($P \times P$): $((X:Z),(Y:T))$ satisfying $x=X/Z, y=Y/T$ */
P1P1,
/** Precomputed (Duif): $(y+x,y-x,2dxy)$ */
PRECOMP,
/** Cached: $(Y+X,Y-X,Z,2dT)$ */
CACHED
}
/**
* Creates a new group element in P2 representation.
*
* @param curve The curve.
* @param X The $X$ coordinate.
* @param Y The $Y$ coordinate.
* @param Z The $Z$ coordinate.
* @return The group element in P2 representation.
*/
public static GroupElement p2(
final Curve curve,
final FieldElement X,
final FieldElement Y,
final FieldElement Z) {
return new GroupElement(curve, Representation.P2, X, Y, Z, null);
}
/**
* Creates a new group element in P3 representation, without pre-computation.
*
* @param curve The curve.
* @param X The $X$ coordinate.
* @param Y The $Y$ coordinate.
* @param Z The $Z$ coordinate.
* @param T The $T$ coordinate.
* @return The group element in P3 representation.
*/
public static GroupElement p3(
final Curve curve,
final FieldElement X,
final FieldElement Y,
final FieldElement Z,
final FieldElement T) {
return p3(curve, X, Y, Z, T, false);
}
/**
* Creates a new group element in P3 representation, potentially with pre-computation.
*
* @param curve The curve.
* @param X The $X$ coordinate.
* @param Y The $Y$ coordinate.
* @param Z The $Z$ coordinate.
* @param T The $T$ coordinate.
* @param precomputeDoubleOnly If true, populate dblPrecmp, else set to null.
* @return The group element in P3 representation.
*/
public static GroupElement p3(
final Curve curve,
final FieldElement X,
final FieldElement Y,
final FieldElement Z,
final FieldElement T,
final boolean precomputeDoubleOnly) {
return new GroupElement(curve, Representation.P3, X, Y, Z, T, precomputeDoubleOnly);
}
/**
* Creates a new group element in P1P1 representation.
*
* @param curve The curve.
* @param X The $X$ coordinate.
* @param Y The $Y$ coordinate.
* @param Z The $Z$ coordinate.
* @param T The $T$ coordinate.
* @return The group element in P1P1 representation.
*/
public static GroupElement p1p1(
final Curve curve,
final FieldElement X,
final FieldElement Y,
final FieldElement Z,
final FieldElement T) {
return new GroupElement(curve, Representation.P1P1, X, Y, Z, T);
}
/**
* Creates a new group element in PRECOMP representation.
*
* @param curve The curve.
* @param ypx The $y + x$ value.
* @param ymx The $y - x$ value.
* @param xy2d The $2 * d * x * y$ value.
* @return The group element in PRECOMP representation.
*/
public static GroupElement precomp(
final Curve curve,
final FieldElement ypx,
final FieldElement ymx,
final FieldElement xy2d) {
return new GroupElement(curve, Representation.PRECOMP, ypx, ymx, xy2d, null);
}
/**
* Creates a new group element in CACHED representation.
*
* @param curve The curve.
* @param YpX The $Y + X$ value.
* @param YmX The $Y - X$ value.
* @param Z The $Z$ coordinate.
* @param T2d The $2 * d * T$ value.
* @return The group element in CACHED representation.
*/
public static GroupElement cached(
final Curve curve,
final FieldElement YpX,
final FieldElement YmX,
final FieldElement Z,
final FieldElement T2d) {
return new GroupElement(curve, Representation.CACHED, YpX, YmX, Z, T2d);
}
/**
* Variable is package private only so that tests run.
*/
final Curve curve;
/**
* Variable is package private only so that tests run.
*/
final Representation repr;
/**
* Variable is package private only so that tests run.
*/
final FieldElement X;
/**
* Variable is package private only so that tests run.
*/
final FieldElement Y;
/**
* Variable is package private only so that tests run.
*/
final FieldElement Z;
/**
* Variable is package private only so that tests run.
*/
final FieldElement T;
/**
* Precomputed table for {@link #scalarMultiply(byte[])},
* filled if necessary.
*
* Variable is package private only so that tests run.
*/
final GroupElement[][] precmp;
/**
* Precomputed table for {@link #doubleScalarMultiplyVariableTime(GroupElement, byte[], byte[])},
* filled if necessary.
*
* Variable is package private only so that tests run.
*/
final GroupElement[] dblPrecmp;
/**
* Creates a group element for a curve, without any pre-computation.
*
* @param curve The curve.
* @param repr The representation used to represent the group element.
* @param X The $X$ coordinate.
* @param Y The $Y$ coordinate.
* @param Z The $Z$ coordinate.
* @param T The $T$ coordinate.
*/
public GroupElement(
final Curve curve,
final Representation repr,
final FieldElement X,
final FieldElement Y,
final FieldElement Z,
final FieldElement T) {
this(curve, repr, X, Y, Z, T, false);
}
/**
* Creates a group element for a curve, with optional pre-computation.
*
* @param curve The curve.
* @param repr The representation used to represent the group element.
* @param X The $X$ coordinate.
* @param Y The $Y$ coordinate.
* @param Z The $Z$ coordinate.
* @param T The $T$ coordinate.
* @param precomputeDouble If true, populate dblPrecmp, else set to null.
*/
public GroupElement(
final Curve curve,
final Representation repr,
final FieldElement X,
final FieldElement Y,
final FieldElement Z,
final FieldElement T,
final boolean precomputeDouble) {
this.curve = curve;
this.repr = repr;
this.X = X;
this.Y = Y;
this.Z = Z;
this.T = T;
this.precmp = null;
this.dblPrecmp = precomputeDouble ? precomputeDouble() : null;
}
/**
* Creates a group element for a curve from a given encoded point. No pre-computation.
*
* A point $(x,y)$ is encoded by storing $y$ in bit 0 to bit 254 and the sign of $x$ in bit 255.
* $x$ is recovered in the following way:
*
* - $x = sign(x) * \sqrt{(y^2 - 1) / (d * y^2 + 1)} = sign(x) * \sqrt{u / v}$ with $u = y^2 - 1$ and $v = d * y^2 + 1$.
*
- Setting $β = (u * v^3) * (u * v^7)^{((q - 5) / 8)}$ one has $β^2 = \pm(u / v)$.
*
- If $v * β = -u$ multiply $β$ with $i=\sqrt{-1}$.
*
- Set $x := β$.
*
- If $sign(x) \ne$ bit 255 of $s$ then negate $x$.
*
*
* @param curve The curve.
* @param s The encoded point.
*/
public GroupElement(final Curve curve, final byte[] s) {
this(curve, s, false);
}
/**
* Creates a group element for a curve from a given encoded point. With optional pre-computation.
*
* A point $(x,y)$ is encoded by storing $y$ in bit 0 to bit 254 and the sign of $x$ in bit 255.
* $x$ is recovered in the following way:
*
* - $x = sign(x) * \sqrt{(y^2 - 1) / (d * y^2 + 1)} = sign(x) * \sqrt{u / v}$ with $u = y^2 - 1$ and $v = d * y^2 + 1$.
*
- Setting $β = (u * v^3) * (u * v^7)^{((q - 5) / 8)}$ one has $β^2 = \pm(u / v)$.
*
- If $v * β = -u$ multiply $β$ with $i=\sqrt{-1}$.
*
- Set $x := β$.
*
- If $sign(x) \ne$ bit 255 of $s$ then negate $x$.
*
*
* @param curve The curve.
* @param s The encoded point.
* @param precomputeSingleAndDouble If true, populate both precmp and dblPrecmp, else set both to null.
*/
public GroupElement(final Curve curve, final byte[] s, boolean precomputeSingleAndDouble) {
FieldElement x, y, yy, u, v, v3, vxx, check;
y = curve.getField().fromByteArray(s);
yy = y.square();
// u = y^2-1
u = yy.subtractOne();
// v = dy^2+1
v = yy.multiply(curve.getD()).addOne();
// v3 = v^3
v3 = v.square().multiply(v);
// x = (v3^2)vu, aka x = uv^7
x = v3.square().multiply(v).multiply(u);
// x = (uv^7)^((q-5)/8)
x = x.pow22523();
// x = uv^3(uv^7)^((q-5)/8)
x = v3.multiply(u).multiply(x);
vxx = x.square().multiply(v);
check = vxx.subtract(u); // vx^2-u
if (check.isNonZero()) {
check = vxx.add(u); // vx^2+u
if (check.isNonZero())
throw new IllegalArgumentException("not a valid GroupElement");
x = x.multiply(curve.getI());
}
if ((x.isNegative() ? 1 : 0) != Utils.bit(s, curve.getField().getb()-1)) {
x = x.negate();
}
this.curve = curve;
this.repr = Representation.P3;
this.X = x;
this.Y = y;
this.Z = curve.getField().ONE;
this.T = this.X.multiply(this.Y);
if(precomputeSingleAndDouble) {
precmp = precomputeSingle();
dblPrecmp = precomputeDouble();
} else {
precmp = null;
dblPrecmp = null;
}
}
/**
* Gets the curve of the group element.
*
* @return The curve.
*/
public Curve getCurve() {
return this.curve;
}
/**
* Gets the representation of the group element.
*
* @return The representation.
*/
public Representation getRepresentation() {
return this.repr;
}
/**
* Gets the $X$ value of the group element.
* This is for most representation the projective $X$ coordinate.
*
* @return The $X$ value.
*/
public FieldElement getX() {
return this.X;
}
/**
* Gets the $Y$ value of the group element.
* This is for most representation the projective $Y$ coordinate.
*
* @return The $Y$ value.
*/
public FieldElement getY() {
return this.Y;
}
/**
* Gets the $Z$ value of the group element.
* This is for most representation the projective $Z$ coordinate.
*
* @return The $Z$ value.
*/
public FieldElement getZ() {
return this.Z;
}
/**
* Gets the $T$ value of the group element.
* This is for most representation the projective $T$ coordinate.
*
* @return The $T$ value.
*/
public FieldElement getT() {
return this.T;
}
/**
* Converts the group element to an encoded point on the curve.
*
* @return The encoded point as byte array.
*/
public byte[] toByteArray() {
switch (this.repr) {
case P2:
case P3:
FieldElement recip = Z.invert();
FieldElement x = X.multiply(recip);
FieldElement y = Y.multiply(recip);
byte[] s = y.toByteArray();
s[s.length-1] |= (x.isNegative() ? (byte) 0x80 : 0);
return s;
default:
return toP2().toByteArray();
}
}
/**
* Converts the group element to the P2 representation.
*
* @return The group element in the P2 representation.
*/
public GroupElement toP2() {
return toRep(Representation.P2);
}
/**
* Converts the group element to the P3 representation.
*
* @return The group element in the P3 representation.
*/
public GroupElement toP3() {
return toRep(Representation.P3);
}
/**
* Converts the group element to the P3 representation, with dblPrecmp populated.
*
* @return The group element in the P3 representation.
*/
public GroupElement toP3PrecomputeDouble() {
return toRep(Representation.P3PrecomputedDouble);
}
/**
* Converts the group element to the CACHED representation.
*
* @return The group element in the CACHED representation.
*/
public GroupElement toCached() {
return toRep(Representation.CACHED);
}
/**
* Convert a GroupElement from one Representation to another.
* TODO-CR: Add additional conversion?
* $r = p$
*
* Supported conversions:
*
* - P3 $\rightarrow$ P2
*
- P3 $\rightarrow$ CACHED (1 multiply, 1 add, 1 subtract)
*
- P1P1 $\rightarrow$ P2 (3 multiply)
*
- P1P1 $\rightarrow$ P3 (4 multiply)
*
* @param repr The representation to convert to.
* @return A new group element in the given representation.
*/
private GroupElement toRep(final Representation repr) {
switch (this.repr) {
case P2:
switch (repr) {
case P2:
return p2(this.curve, this.X, this.Y, this.Z);
default:
throw new IllegalArgumentException();
}
case P3:
switch (repr) {
case P2:
return p2(this.curve, this.X, this.Y, this.Z);
case P3:
return p3(this.curve, this.X, this.Y, this.Z, this.T);
case CACHED:
return cached(this.curve, this.Y.add(this.X), this.Y.subtract(this.X), this.Z, this.T.multiply(this.curve.get2D()));
default:
throw new IllegalArgumentException();
}
case P1P1:
switch (repr) {
case P2:
return p2(this.curve, this.X.multiply(this.T), Y.multiply(this.Z), this.Z.multiply(this.T));
case P3:
return p3(this.curve, this.X.multiply(this.T), Y.multiply(this.Z), this.Z.multiply(this.T), this.X.multiply(this.Y), false);
case P3PrecomputedDouble:
return p3(this.curve, this.X.multiply(this.T), Y.multiply(this.Z), this.Z.multiply(this.T), this.X.multiply(this.Y), true);
case P1P1:
return p1p1(this.curve, this.X, this.Y, this.Z, this.T);
default:
throw new IllegalArgumentException();
}
case PRECOMP:
switch (repr) {
case PRECOMP:
return precomp(this.curve, this.X, this.Y, this.Z);
default:
throw new IllegalArgumentException();
}
case CACHED:
switch (repr) {
case CACHED:
return cached(this.curve, this.X, this.Y, this.Z, this.T);
default:
throw new IllegalArgumentException();
}
default:
throw new UnsupportedOperationException();
}
}
/**
* Precomputes table for {@link #scalarMultiply(byte[])}.
*/
private GroupElement[][] precomputeSingle() {
// Precomputation for single scalar multiplication.
GroupElement[][] precmp = new GroupElement[32][8];
// TODO-CR BR: check that this == base point when the method is called.
GroupElement Bi = this;
for (int i = 0; i < 32; i++) {
GroupElement Bij = Bi;
for (int j = 0; j < 8; j++) {
final FieldElement recip = Bij.Z.invert();
final FieldElement x = Bij.X.multiply(recip);
final FieldElement y = Bij.Y.multiply(recip);
precmp[i][j] = precomp(this.curve, y.add(x), y.subtract(x), x.multiply(y).multiply(this.curve.get2D()));
Bij = Bij.add(Bi.toCached()).toP3();
}
// Only every second summand is precomputed (16^2 = 256)
for (int k = 0; k < 8; k++) {
Bi = Bi.add(Bi.toCached()).toP3();
}
}
return precmp;
}
/**
* Precomputes table for {@link #doubleScalarMultiplyVariableTime(GroupElement, byte[], byte[])}.
*/
private GroupElement[] precomputeDouble() {
// Precomputation for double scalar multiplication.
// P,3P,5P,7P,9P,11P,13P,15P
GroupElement[] dblPrecmp = new GroupElement[8];
GroupElement Bi = this;
for (int i = 0; i < 8; i++) {
final FieldElement recip = Bi.Z.invert();
final FieldElement x = Bi.X.multiply(recip);
final FieldElement y = Bi.Y.multiply(recip);
dblPrecmp[i] = precomp(this.curve, y.add(x), y.subtract(x), x.multiply(y).multiply(this.curve.get2D()));
// Bi = edwards(B,edwards(B,Bi))
Bi = this.add(this.add(Bi.toCached()).toP3().toCached()).toP3();
}
return dblPrecmp;
}
/**
* Doubles a given group element $p$ in $P^2$ or $P^3$ representation and returns the result in $P \times P$ representation.
* $r = 2 * p$ where $p = (X : Y : Z)$ or $p = (X : Y : Z : T)$
*
* $r$ in $P \times P$ representation:
*
* $r = ((X' : Z'), (Y' : T'))$ where
*
* - $X' = (X + Y)^2 - (Y^2 + X^2)$
*
- $Y' = Y^2 + X^2$
*
- $Z' = y^2 - X^2$
*
- $T' = 2 * Z^2 - (y^2 - X^2)$
*
* $r$ converted from $P \times P$ to $P^2$ representation:
*
* $r = (X'' : Y'' : Z'')$ where
*
* - $X'' = X' * Z' = ((X + Y)^2 - Y^2 - X^2) * (2 * Z^2 - (y^2 - X^2))$
*
- $Y'' = Y' * T' = (Y^2 + X^2) * (2 * Z^2 - (y^2 - X^2))$
*
- $Z'' = Z' * T' = (y^2 - X^2) * (2 * Z^2 - (y^2 - X^2))$
*
* Formula for the $P^2$ representation is in agreement with the formula given in [4] page 12 (with $a = -1$)
* up to a common factor -1 which does not matter:
*
* $$
* B = (X + Y)^2; C = X^2; D = Y^2; E = -C = -X^2; F := E + D = Y^2 - X^2; H = Z^2; J = F − 2 * H; \\
* X3 = (B − C − D) · J = X' * (-T'); \\
* Y3 = F · (E − D) = Z' * (-Y'); \\
* Z3 = F · J = Z' * (-T').
* $$
*
* @return The P1P1 representation
*/
public GroupElement dbl() {
switch (this.repr) {
case P2:
case P3: // Ignore T for P3 representation
FieldElement XX, YY, B, A, AA, Yn, Zn;
XX = this.X.square();
YY = this.Y.square();
B = this.Z.squareAndDouble();
A = this.X.add(this.Y);
AA = A.square();
Yn = YY.add(XX);
Zn = YY.subtract(XX);
return p1p1(this.curve, AA.subtract(Yn), Yn, Zn, B.subtract(Zn));
default:
throw new UnsupportedOperationException();
}
}
/**
* GroupElement addition using the twisted Edwards addition law with
* extended coordinates (Hisil2008).
*
* this must be in $P^3$ representation and $q$ in PRECOMP representation.
* $r = p + q$ where $p = this = (X1 : Y1 : Z1 : T1), q = (q.X, q.Y, q.Z) = (Y2/Z2 + X2/Z2, Y2/Z2 - X2/Z2, 2 * d * X2/Z2 * Y2/Z2)$
*
* $r$ in $P \times P$ representation:
*
* $r = ((X' : Z'), (Y' : T'))$ where
*
* - $X' = (Y1 + X1) * q.X - (Y1 - X1) * q.Y = ((Y1 + X1) * (Y2 + X2) - (Y1 - X1) * (Y2 - X2)) * 1/Z2$
*
- $Y' = (Y1 + X1) * q.X + (Y1 - X1) * q.Y = ((Y1 + X1) * (Y2 + X2) + (Y1 - X1) * (Y2 - X2)) * 1/Z2$
*
- $Z' = 2 * Z1 + T1 * q.Z = 2 * Z1 + T1 * 2 * d * X2 * Y2 * 1/Z2^2 = (2 * Z1 * Z2 + 2 * d * T1 * T2) * 1/Z2$
*
- $T' = 2 * Z1 - T1 * q.Z = 2 * Z1 - T1 * 2 * d * X2 * Y2 * 1/Z2^2 = (2 * Z1 * Z2 - 2 * d * T1 * T2) * 1/Z2$
*
* Setting $A = (Y1 - X1) * (Y2 - X2), B = (Y1 + X1) * (Y2 + X2), C = 2 * d * T1 * T2, D = 2 * Z1 * Z2$ we get
*
* - $X' = (B - A) * 1/Z2$
*
- $Y' = (B + A) * 1/Z2$
*
- $Z' = (D + C) * 1/Z2$
*
- $T' = (D - C) * 1/Z2$
*
* $r$ converted from $P \times P$ to $P^2$ representation:
*
* $r = (X'' : Y'' : Z'' : T'')$ where
*
* - $X'' = X' * Z' = (B - A) * (D + C) * 1/Z2^2$
*
- $Y'' = Y' * T' = (B + A) * (D - C) * 1/Z2^2$
*
- $Z'' = Z' * T' = (D + C) * (D - C) * 1/Z2^2$
*
- $T'' = X' * Y' = (B - A) * (B + A) * 1/Z2^2$
*
* TODO-CR BR: Formula for the $P^2$ representation is not in agreement with the formula given in [2] page 6
* TODO-CR BR: (the common factor $1/Z2^2$ does not matter):
* $$
* E = B - A, F = D - C, G = D + C, H = B + A \\
* X3 = E * F = (B - A) * (D - C); \\
* Y3 = G * H = (D + C) * (B + A); \\
* Z3 = F * G = (D - C) * (D + C); \\
* T3 = E * H = (B - A) * (B + A);
* $$
*
* @param q the PRECOMP representation of the GroupElement to add.
* @return the P1P1 representation of the result.
*/
private GroupElement madd(GroupElement q) {
if (this.repr != Representation.P3)
throw new UnsupportedOperationException();
if (q.repr != Representation.PRECOMP)
throw new IllegalArgumentException();
FieldElement YpX, YmX, A, B, C, D;
YpX = this.Y.add(this.X);
YmX = this.Y.subtract(this.X);
A = YpX.multiply(q.X); // q->y+x
B = YmX.multiply(q.Y); // q->y-x
C = q.Z.multiply(this.T); // q->2dxy
D = this.Z.add(this.Z);
return p1p1(this.curve, A.subtract(B), A.add(B), D.add(C), D.subtract(C));
}
/**
* GroupElement subtraction using the twisted Edwards addition law with
* extended coordinates (Hisil2008).
*
* this must be in $P^3$ representation and $q$ in PRECOMP representation.
* $r = p - q$ where $p = this = (X1 : Y1 : Z1 : T1), q = (q.X, q.Y, q.Z) = (Y2/Z2 + X2/Z2, Y2/Z2 - X2/Z2, 2 * d * X2/Z2 * Y2/Z2)$
*
* Negating $q$ means negating the value of $X2$ and $T2$ (the latter is irrelevant here).
* The formula is in accordance to {@link #madd the above addition}.
*
* @param q the PRECOMP representation of the GroupElement to subtract.
* @return the P1P1 representation of the result.
*/
private GroupElement msub(GroupElement q) {
if (this.repr != Representation.P3)
throw new UnsupportedOperationException();
if (q.repr != Representation.PRECOMP)
throw new IllegalArgumentException();
FieldElement YpX, YmX, A, B, C, D;
YpX = this.Y.add(this.X);
YmX = this.Y.subtract(this.X);
A = YpX.multiply(q.Y); // q->y-x
B = YmX.multiply(q.X); // q->y+x
C = q.Z.multiply(this.T); // q->2dxy
D = this.Z.add(this.Z);
return p1p1(this.curve, A.subtract(B), A.add(B), D.subtract(C), D.add(C));
}
/**
* GroupElement addition using the twisted Edwards addition law with
* extended coordinates (Hisil2008).
*
* this must be in $P^3$ representation and $q$ in CACHED representation.
* $r = p + q$ where $p = this = (X1 : Y1 : Z1 : T1), q = (q.X, q.Y, q.Z, q.T) = (Y2 + X2, Y2 - X2, Z2, 2 * d * T2)$
*
* $r$ in $P \times P$ representation:
*
* - $X' = (Y1 + X1) * (Y2 + X2) - (Y1 - X1) * (Y2 - X2)$
*
- $Y' = (Y1 + X1) * (Y2 + X2) + (Y1 - X1) * (Y2 - X2)$
*
- $Z' = 2 * Z1 * Z2 + 2 * d * T1 * T2$
*
- $T' = 2 * Z1 * T2 - 2 * d * T1 * T2$
*
* Setting $A = (Y1 - X1) * (Y2 - X2), B = (Y1 + X1) * (Y2 + X2), C = 2 * d * T1 * T2, D = 2 * Z1 * Z2$ we get
*
* - $X' = (B - A)$
*
- $Y' = (B + A)$
*
- $Z' = (D + C)$
*
- $T' = (D - C)$
*
* Same result as in {@link #madd} (up to a common factor which does not matter).
*
* @param q the CACHED representation of the GroupElement to add.
* @return the P1P1 representation of the result.
*/
public GroupElement add(GroupElement q) {
if (this.repr != Representation.P3)
throw new UnsupportedOperationException();
if (q.repr != Representation.CACHED)
throw new IllegalArgumentException();
FieldElement YpX, YmX, A, B, C, ZZ, D;
YpX = this.Y.add(this.X);
YmX = this.Y.subtract(this.X);
A = YpX.multiply(q.X); // q->Y+X
B = YmX.multiply(q.Y); // q->Y-X
C = q.T.multiply(this.T); // q->2dT
ZZ = this.Z.multiply(q.Z);
D = ZZ.add(ZZ);
return p1p1(this.curve, A.subtract(B), A.add(B), D.add(C), D.subtract(C));
}
/**
* GroupElement subtraction using the twisted Edwards addition law with
* extended coordinates (Hisil2008).
*
* $r = p - q$
*
* Negating $q$ means negating the value of the coordinate $X2$ and $T2$.
* The formula is in accordance to {@link #add the above addition}.
*
* @param q the PRECOMP representation of the GroupElement to subtract.
* @return the P1P1 representation of the result.
*/
public GroupElement sub(GroupElement q) {
if (this.repr != Representation.P3)
throw new UnsupportedOperationException();
if (q.repr != Representation.CACHED)
throw new IllegalArgumentException();
FieldElement YpX, YmX, A, B, C, ZZ, D;
YpX = Y.add(X);
YmX = Y.subtract(X);
A = YpX.multiply(q.Y); // q->Y-X
B = YmX.multiply(q.X); // q->Y+X
C = q.T.multiply(T); // q->2dT
ZZ = Z.multiply(q.Z);
D = ZZ.add(ZZ);
return p1p1(curve, A.subtract(B), A.add(B), D.subtract(C), D.add(C));
}
/**
* Negates this group element by subtracting it from the neutral group element.
*
* TODO-CR BR: why not simply negate the coordinates $X$ and $T$?
*
* @return The negative of this group element.
*/
public GroupElement negate() {
if (this.repr != Representation.P3)
throw new UnsupportedOperationException();
return this.curve.getZero(Representation.P3).sub(toCached()).toP3PrecomputeDouble();
}
@Override
public int hashCode() {
return Arrays.hashCode(this.toByteArray());
}
@Override
public boolean equals(Object obj) {
if (obj == this)
return true;
if (!(obj instanceof GroupElement))
return false;
GroupElement ge = (GroupElement) obj;
if (!this.repr.equals(ge.repr)) {
try {
ge = ge.toRep(this.repr);
} catch (RuntimeException e) {
return false;
}
}
switch (this.repr) {
case P2:
case P3:
// Try easy way first
if (this.Z.equals(ge.Z))
return this.X.equals(ge.X) && this.Y.equals(ge.Y);
// X1/Z1 = X2/Z2 --> X1*Z2 = X2*Z1
final FieldElement x1 = this.X.multiply(ge.Z);
final FieldElement y1 = this.Y.multiply(ge.Z);
final FieldElement x2 = ge.X.multiply(this.Z);
final FieldElement y2 = ge.Y.multiply(this.Z);
return x1.equals(x2) && y1.equals(y2);
case P1P1:
return toP2().equals(ge);
case PRECOMP:
// Compare directly, PRECOMP is derived directly from x and y
return this.X.equals(ge.X) && this.Y.equals(ge.Y) && this.Z.equals(ge.Z);
case CACHED:
// Try easy way first
if (this.Z.equals(ge.Z))
return this.X.equals(ge.X) && this.Y.equals(ge.Y) && this.T.equals(ge.T);
// (Y+X)/Z = y+x etc.
final FieldElement x3 = this.X.multiply(ge.Z);
final FieldElement y3 = this.Y.multiply(ge.Z);
final FieldElement t3 = this.T.multiply(ge.Z);
final FieldElement x4 = ge.X.multiply(this.Z);
final FieldElement y4 = ge.Y.multiply(this.Z);
final FieldElement t4 = ge.T.multiply(this.Z);
return x3.equals(x4) && y3.equals(y4) && t3.equals(t4);
default:
return false;
}
}
/**
* Convert a to radix 16.
*
* Method is package private only so that tests run.
*
* @param a $= a[0]+256*a[1]+...+256^{31} a[31]$
* @return 64 bytes, each between -8 and 7
*/
static byte[] toRadix16(final byte[] a) {
final byte[] e = new byte[64];
int i;
// Radix 16 notation
for (i = 0; i < 32; i++) {
e[2*i+0] = (byte) (a[i] & 15);
e[2*i+1] = (byte) ((a[i] >> 4) & 15);
}
/* each e[i] is between 0 and 15 */
/* e[63] is between 0 and 7 */
int carry = 0;
for (i = 0; i < 63; i++) {
e[i] += carry;
carry = e[i] + 8;
carry >>= 4;
e[i] -= carry << 4;
}
e[63] += carry;
/* each e[i] is between -8 and 7 */
return e;
}
/**
* Constant-time conditional move.
*
* Replaces this with $u$ if $b == 1$.
* Replaces this with this if $b == 0$.
*
* Method is package private only so that tests run.
*
* @param u The group element to return if $b == 1$.
* @param b in $\{0, 1\}$
* @return $u$ if $b == 1$; this if $b == 0$. Results undefined if $b$ is not in $\{0, 1\}$.
*/
GroupElement cmov(final GroupElement u, final int b) {
return precomp(curve, X.cmov(u.X, b), Y.cmov(u.Y, b), Z.cmov(u.Z, b));
}
/**
* Look up $16^i r_i B$ in the precomputed table.
*
* No secret array indices, no secret branching.
* Constant time.
*
* Must have previously precomputed.
*
* Method is package private only so that tests run.
*
* @param pos $= i/2$ for $i$ in $\{0, 2, 4,..., 62\}$
* @param b $= r_i$
* @return the GroupElement
*/
GroupElement select(final int pos, final int b) {
// Is r_i negative?
final int bnegative = Utils.negative(b);
// |r_i|
final int babs = b - (((-bnegative) & b) << 1);
// 16^i |r_i| B
final GroupElement t = this.curve.getZero(Representation.PRECOMP)
.cmov(this.precmp[pos][0], Utils.equal(babs, 1))
.cmov(this.precmp[pos][1], Utils.equal(babs, 2))
.cmov(this.precmp[pos][2], Utils.equal(babs, 3))
.cmov(this.precmp[pos][3], Utils.equal(babs, 4))
.cmov(this.precmp[pos][4], Utils.equal(babs, 5))
.cmov(this.precmp[pos][5], Utils.equal(babs, 6))
.cmov(this.precmp[pos][6], Utils.equal(babs, 7))
.cmov(this.precmp[pos][7], Utils.equal(babs, 8));
// -16^i |r_i| B
final GroupElement tminus = precomp(curve, t.Y, t.X, t.Z.negate());
// 16^i r_i B
return t.cmov(tminus, bnegative);
}
/**
* $h = a * B$ where $a = a[0]+256*a[1]+\dots+256^{31} a[31]$ and
* $B$ is this point. If its lookup table has not been precomputed, it
* will be at the start of the method (and cached for later calls).
* Constant time.
*
* Preconditions: (TODO: Check this applies here)
* $a[31] \le 127$
* @param a $= a[0]+256*a[1]+\dots+256^{31} a[31]$
* @return the GroupElement
*/
public GroupElement scalarMultiply(final byte[] a) {
GroupElement t;
int i;
final byte[] e = toRadix16(a);
GroupElement h = this.curve.getZero(Representation.P3);
for (i = 1; i < 64; i += 2) {
t = select(i/2, e[i]);
h = h.madd(t).toP3();
}
h = h.dbl().toP2().dbl().toP2().dbl().toP2().dbl().toP3();
for (i = 0; i < 64; i += 2) {
t = select(i/2, e[i]);
h = h.madd(t).toP3();
}
return h;
}
/**
* Calculates a sliding-windows base 2 representation for a given value $a$.
* To learn more about it see [6] page 8.
*
* Output: $r$ which satisfies
* $a = r0 * 2^0 + r1 * 2^1 + \dots + r255 * 2^{255}$ with $ri$ in $\{-15, -13, -11, -9, -7, -5, -3, -1, 0, 1, 3, 5, 7, 9, 11, 13, 15\}$
*
* Method is package private only so that tests run.
*
* @param a $= a[0]+256*a[1]+\dots+256^{31} a[31]$.
* @return The byte array $r$ in the above described form.
*/
static byte[] slide(final byte[] a) {
byte[] r = new byte[256];
// Put each bit of 'a' into a separate byte, 0 or 1
for (int i = 0; i < 256; ++i) {
r[i] = (byte) (1 & (a[i >> 3] >> (i & 7)));
}
// Note: r[i] will always be odd.
for (int i = 0; i < 256; ++i) {
if (r[i] != 0) {
for (int b = 1; b <= 6 && i + b < 256; ++b) {
// Accumulate bits if possible
if (r[i + b] != 0) {
if (r[i] + (r[i + b] << b) <= 15) {
r[i] += r[i + b] << b;
r[i + b] = 0;
} else if (r[i] - (r[i + b] << b) >= -15) {
r[i] -= r[i + b] << b;
for (int k = i + b; k < 256; ++k) {
if (r[k] == 0) {
r[k] = 1;
break;
}
r[k] = 0;
}
} else
break;
}
}
}
}
return r;
}
/**
* $r = a * A + b * B$ where $a = a[0]+256*a[1]+\dots+256^{31} a[31]$,
* $b = b[0]+256*b[1]+\dots+256^{31} b[31]$ and $B$ is this point.
*
* $A$ must have been previously precomputed.
*
* @param A in P3 representation.
* @param a $= a[0]+256*a[1]+\dots+256^{31} a[31]$
* @param b $= b[0]+256*b[1]+\dots+256^{31} b[31]$
* @return the GroupElement
*/
public GroupElement doubleScalarMultiplyVariableTime(final GroupElement A, final byte[] a, final byte[] b) {
// TODO-CR BR: A check that this is the base point is needed.
final byte[] aslide = slide(a);
final byte[] bslide = slide(b);
GroupElement r = this.curve.getZero(Representation.P2);
int i;
for (i = 255; i >= 0; --i) {
if (aslide[i] != 0 || bslide[i] != 0) break;
}
for (; i >= 0; --i) {
GroupElement t = r.dbl();
if (aslide[i] > 0) {
t = t.toP3().madd(A.dblPrecmp[aslide[i]/2]);
} else if(aslide[i] < 0) {
t = t.toP3().msub(A.dblPrecmp[(-aslide[i])/2]);
}
if (bslide[i] > 0) {
t = t.toP3().madd(this.dblPrecmp[bslide[i]/2]);
} else if(bslide[i] < 0) {
t = t.toP3().msub(this.dblPrecmp[(-bslide[i])/2]);
}
r = t.toP2();
}
return r;
}
/**
* Verify that a point is on its curve.
* @return true if the point lies on its curve.
*/
public boolean isOnCurve() {
return isOnCurve(curve);
}
/**
* Verify that a point is on the curve.
* @param curve The curve to check.
* @return true if the point lies on the curve.
*/
public boolean isOnCurve(Curve curve) {
switch (repr) {
case P2:
case P3:
FieldElement recip = Z.invert();
FieldElement x = X.multiply(recip);
FieldElement y = Y.multiply(recip);
FieldElement xx = x.square();
FieldElement yy = y.square();
FieldElement dxxyy = curve.getD().multiply(xx).multiply(yy);
return curve.getField().ONE.add(dxxyy).add(xx).equals(yy);
default:
return toP2().isOnCurve(curve);
}
}
@Override
public String toString() {
return "[GroupElement\nX="+X+"\nY="+Y+"\nZ="+Z+"\nT="+T+"\n]";
}
}