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The Bouncy Castle Crypto package is a Java implementation of cryptographic algorithms. This jar contains JCE provider and lightweight API for the Bouncy Castle Cryptography APIs for JDK 1.4.
package org.bouncycastle.math.ec.tools;
import java.math.BigInteger;
import java.security.SecureRandom;
import java.util.ArrayList;
import java.util.Enumeration;
import java.util.Iterator;
import java.util.List;
import java.util.SortedSet;
import java.util.TreeSet;
import org.bouncycastle.asn1.x9.ECNamedCurveTable;
import org.bouncycastle.asn1.x9.X9ECParameters;
import org.bouncycastle.crypto.ec.CustomNamedCurves;
import org.bouncycastle.math.ec.ECAlgorithms;
import org.bouncycastle.math.ec.ECConstants;
import org.bouncycastle.math.ec.ECCurve;
import org.bouncycastle.math.ec.ECFieldElement;
import org.bouncycastle.math.ec.ECPoint;
import org.bouncycastle.util.BigIntegers;
public class DiscoverEndomorphisms
{
private static final int radix = 16;
public static void main(String[] args)
{
if (args.length > 0)
{
for (int i = 0; i < args.length; ++i)
{
discoverEndomorphisms(args[i]);
}
}
else
{
SortedSet curveNames = new TreeSet(enumToList(ECNamedCurveTable.getNames()));
curveNames.addAll(enumToList(CustomNamedCurves.getNames()));
Iterator it = curveNames.iterator();
while (it.hasNext())
{
discoverEndomorphisms((String)it.next());
}
}
}
public static void discoverEndomorphisms(X9ECParameters x9)
{
if (x9 == null)
{
throw new NullPointerException("x9");
}
discoverEndomorphisms(x9, " ");
}
private static void discoverEndomorphisms(String curveName)
{
X9ECParameters x9 = CustomNamedCurves.getByName(curveName);
if (x9 == null)
{
x9 = ECNamedCurveTable.getByName(curveName);
if (x9 == null)
{
// -DM System.err.println
System.err.println("Unknown curve: " + curveName);
return;
}
}
// System.out.println("[" + curveName + "]");
discoverEndomorphisms(x9, curveName);
}
private static void discoverEndomorphisms(X9ECParameters x9, String displayName)
{
ECCurve c = x9.getCurve();
if (ECAlgorithms.isFpCurve(c))
{
BigInteger characteristic = c.getField().getCharacteristic();
if (c.getB().isZero() && characteristic.mod(ECConstants.FOUR).equals(ECConstants.ONE))
{
// -DM System.out.println
System.out.println("Curve '" + displayName + "' has a 'GLV Type A' endomorphism with these parameters:");
printGLVTypeAParameters(x9);
}
if (c.getA().isZero() && characteristic.mod(ECConstants.THREE).equals(ECConstants.ONE))
{
// -DM System.out.println
System.out.println("Curve '" + displayName + "' has a 'GLV Type B' endomorphism with these parameters:");
printGLVTypeBParameters(x9);
}
}
}
private static void printGLVTypeAParameters(X9ECParameters x9)
{
// x^2 + 1 = 0 mod n
BigInteger[] lambdas = solveQuadraticEquation(x9.getN(),
ECConstants.ONE, ECConstants.ZERO, ECConstants.ONE);
/*
* The 'i' values are field elements of order 4. There are only two such values besides 1
* and -1, each corresponding to one choice for 'lambda'.
*/
ECFieldElement[] iValues = findNonTrivialOrder4FieldElements(x9.getCurve());
printGLVTypeAParameters(x9, lambdas[0], iValues);
// -DM System.out.println
System.out.println("OR");
printGLVTypeAParameters(x9, lambdas[1], iValues);
}
private static void printGLVTypeAParameters(X9ECParameters x9, BigInteger lambda, ECFieldElement[] iValues)
{
/*
* Check the basic premise of the endomorphism: that multiplying a point by lambda negates the x-coordinate
*/
ECPoint G = x9.getG().normalize();
ECPoint mapG = G.multiply(lambda).normalize();
if (!G.getXCoord().negate().equals(mapG.getXCoord()))
{
throw new IllegalStateException("Derivation of GLV Type A parameters failed unexpectedly");
}
/*
* Determine which of the i values corresponds with this choice of lambda, by checking that it scales
* the y-coordinate the same way a point-multiplication by lambda does.
*/
ECFieldElement i = iValues[0];
if (!G.getYCoord().multiply(i).equals(mapG.getYCoord()))
{
i = iValues[1];
if (!G.getYCoord().multiply(i).equals(mapG.getYCoord()))
{
throw new IllegalStateException("Derivation of GLV Type A parameters failed unexpectedly");
}
}
printProperty("Point map", "lambda * (x, y) = (-x, i * y)");
printProperty("i", i.toBigInteger().toString(radix));
printProperty("lambda", lambda.toString(radix));
printScalarDecompositionParameters(x9.getN(), lambda);
}
private static void printGLVTypeBParameters(X9ECParameters x9)
{
// x^2 + x + 1 = 0 mod n
BigInteger[] lambdas = solveQuadraticEquation(x9.getN(),
ECConstants.ONE, ECConstants.ONE, ECConstants.ONE);
/*
* The 'beta' values are field elements of order 3. There are only two such values besides
* 1, each corresponding to one choice for 'lambda'.
*/
ECFieldElement[] betaValues = findNonTrivialOrder3FieldElements(x9.getCurve());
printGLVTypeBParameters(x9, lambdas[0], betaValues);
// -DM System.out.println
System.out.println("OR");
printGLVTypeBParameters(x9, lambdas[1], betaValues);
}
private static void printGLVTypeBParameters(X9ECParameters x9, BigInteger lambda, ECFieldElement[] betaValues)
{
/*
* Check the basic premise of the endomorphism: that multiplying a point by lambda preserves the y-coordinate
*/
ECPoint G = x9.getG().normalize();
ECPoint mapG = G.multiply(lambda).normalize();
if (!G.getYCoord().equals(mapG.getYCoord()))
{
throw new IllegalStateException("Derivation of GLV Type B parameters failed unexpectedly");
}
/*
* Determine which of the beta values corresponds with this choice of lambda, by checking that it scales
* the x-coordinate the same way a point-multiplication by lambda does.
*/
ECFieldElement beta = betaValues[0];
if (!G.getXCoord().multiply(beta).equals(mapG.getXCoord()))
{
beta = betaValues[1];
if (!G.getXCoord().multiply(beta).equals(mapG.getXCoord()))
{
throw new IllegalStateException("Derivation of GLV Type B parameters failed unexpectedly");
}
}
printProperty("Point map", "lambda * (x, y) = (beta * x, y)");
printProperty("beta", beta.toBigInteger().toString(radix));
printProperty("lambda", lambda.toString(radix));
printScalarDecompositionParameters(x9.getN(), lambda);
}
private static void printProperty(String name, Object value)
{
StringBuffer sb = new StringBuffer(" ");
sb.append(name);
while (sb.length() < 20)
{
sb.append(' ');
}
sb.append(": ");
sb.append(value.toString());
// -DM System.out.println
System.out.println(sb.toString());
}
private static void printScalarDecompositionParameters(BigInteger n, BigInteger lambda)
{
/*
* Search for parameters to allow efficient decomposition of full-length scalars
*/
BigInteger[] v1 = null;
BigInteger[] v2 = null;
BigInteger[] rt = extEuclidGLV(n, lambda);
v1 = new BigInteger[]{ rt[2], rt[3].negate() };
v2 = chooseShortest(new BigInteger[]{ rt[0], rt[1].negate() }, new BigInteger[]{ rt[4], rt[5].negate() });
/*
* If elements of v2 are not bounded by sqrt(n), then if r1/t1 are relatively prime there
* _may_ yet be a GLV generator, so search for it. See
* "Integer Decomposition for Fast Scalar Multiplication on Elliptic Curves", D. Kim, S. Lim
* (SAC 2002)
*/
if (!isVectorBoundedBySqrt(v2, n) && areRelativelyPrime(v1[0], v1[1]))
{
BigInteger r = v1[0], t = v1[1], s = r.add(t.multiply(lambda)).divide(n);
BigInteger[] vw = extEuclidBezout(new BigInteger[]{ s.abs(), t.abs() });
if (vw != null)
{
BigInteger v = vw[0], w = vw[1];
if (s.signum() < 0)
{
v = v.negate();
}
if (t.signum() > 0)
{
w = w.negate();
}
BigInteger check = s.multiply(v).subtract(t.multiply(w));
if (!check.equals(ECConstants.ONE))
{
throw new IllegalStateException();
}
BigInteger x = w.multiply(n).subtract(v.multiply(lambda));
BigInteger base1 = v.negate();
BigInteger base2 = x.negate();
/*
* We calculate the range(s) conservatively large to avoid messy rounding issues, so
* there may be spurious candidate generators, but we won't miss any.
*/
BigInteger sqrtN = isqrt(n.subtract(ECConstants.ONE)).add(ECConstants.ONE);
BigInteger[] I1 = calculateRange(base1, sqrtN, t);
BigInteger[] I2 = calculateRange(base2, sqrtN, r);
BigInteger[] range = intersect(I1, I2);
if (range != null)
{
for (BigInteger alpha = range[0]; alpha.compareTo(range[1]) <= 0; alpha = alpha.add(ECConstants.ONE))
{
BigInteger[] candidate = new BigInteger[]{ x.add(alpha.multiply(r)), v.add(alpha.multiply(t)) };
if (isShorter(candidate, v2))
{
v2 = candidate;
}
}
}
}
}
BigInteger d = (v1[0].multiply(v2[1])).subtract(v1[1].multiply(v2[0]));
/*
* These parameters are used to avoid division when decomposing the scalar in a GLV point multiplication.
* The precision is determined by 'bits', even 2 bits is enough, but we try to get more whilst keeping it
* 8-bit aligned and limiting the possible growth of product sizes on a 32-bit machine.
*/
int bits = n.bitLength() + 16 - (n.bitLength() & 7);
BigInteger g1 = roundQuotient(v2[1].shiftLeft(bits), d);
BigInteger g2 = roundQuotient(v1[1].shiftLeft(bits), d).negate();
printProperty("v1", "{ " + v1[0].toString(radix) + ", " + v1[1].toString(radix) + " }");
printProperty("v2", "{ " + v2[0].toString(radix) + ", " + v2[1].toString(radix) + " }");
printProperty("d", d.toString(radix));
printProperty("(OPT) g1", g1.toString(radix));
printProperty("(OPT) g2", g2.toString(radix));
printProperty("(OPT) bits", Integer.toString(bits));
}
private static boolean areRelativelyPrime(BigInteger a, BigInteger b)
{
return a.gcd(b).equals(ECConstants.ONE);
}
private static BigInteger[] calculateRange(BigInteger mid, BigInteger off, BigInteger div)
{
BigInteger i1 = mid.subtract(off).divide(div);
BigInteger i2 = mid.add(off).divide(div);
return order(i1, i2);
}
private static List enumToList(Enumeration en)
{
List rv = new ArrayList();
while (en.hasMoreElements())
{
rv.add(en.nextElement());
}
return rv;
}
private static BigInteger[] extEuclidBezout(BigInteger[] ab)
{
boolean swap = ab[0].compareTo(ab[1]) < 0;
if (swap)
{
swap(ab);
}
BigInteger r0 = ab[0], r1 = ab[1];
BigInteger s0 = ECConstants.ONE, s1 = ECConstants.ZERO;
BigInteger t0 = ECConstants.ZERO, t1 = ECConstants.ONE;
while (r1.compareTo(ECConstants.ONE) > 0)
{
BigInteger[] qr = r0.divideAndRemainder(r1);
BigInteger q = qr[0], r2 = qr[1];
BigInteger s2 = s0.subtract(q.multiply(s1));
BigInteger t2 = t0.subtract(q.multiply(t1));
r0 = r1;
r1 = r2;
s0 = s1;
s1 = s2;
t0 = t1;
t1 = t2;
}
if (r1.signum() <= 0)
{
/*
* NOTE: This case occurred while testing on curves over tiny fields; probably due to a 0 input.
*/
return null;
}
BigInteger[] st = new BigInteger[]{ s1, t1 };
if (swap)
{
swap(st);
}
return st;
}
private static BigInteger[] extEuclidGLV(BigInteger n, BigInteger lambda)
{
BigInteger r0 = n, r1 = lambda;
// BigInteger s0 = ECConstants.ONE, s1 = ECConstants.ZERO;
BigInteger t0 = ECConstants.ZERO, t1 = ECConstants.ONE;
for (;;)
{
BigInteger[] qr = r0.divideAndRemainder(r1);
BigInteger q = qr[0], r2 = qr[1];
// BigInteger s2 = s0.subtract(q.multiply(s1));
BigInteger t2 = t0.subtract(q.multiply(t1));
if (isLessThanSqrt(r1, n))
{
return new BigInteger[]{ r0, t0, r1, t1, r2, t2 };
}
r0 = r1;
r1 = r2;
// s0 = s1;
// s1 = s2;
t0 = t1;
t1 = t2;
}
}
private static BigInteger[] chooseShortest(BigInteger[] u, BigInteger[] v)
{
return isShorter(u, v) ? u : v;
}
private static BigInteger[] intersect(BigInteger[] ab, BigInteger[] cd)
{
BigInteger min = ab[0].max(cd[0]);
BigInteger max = ab[1].min(cd[1]);
if (min.compareTo(max) > 0)
{
return null;
}
return new BigInteger[]{ min, max };
}
private static boolean isLessThanSqrt(BigInteger a, BigInteger b)
{
a = a.abs();
b = b.abs();
int target = b.bitLength(), maxBits = a.bitLength() * 2, minBits = maxBits - 1;
return minBits <= target && (maxBits < target || a.multiply(a).compareTo(b) < 0);
}
private static boolean isShorter(BigInteger[] u, BigInteger[] v)
{
BigInteger u1 = u[0].abs(), u2 = u[1].abs(), v1 = v[0].abs(), v2 = v[1].abs();
// TODO Check whether "shorter" just means by rectangle norm:
// return u1.max(u2).compareTo(v1.max(v2)) < 0;
boolean c1 = u1.compareTo(v1) < 0, c2 = u2.compareTo(v2) < 0;
if (c1 == c2)
{
return c1;
}
BigInteger du = u1.multiply(u1).add(u2.multiply(u2));
BigInteger dv = v1.multiply(v1).add(v2.multiply(v2));
return du.compareTo(dv) < 0;
}
private static boolean isVectorBoundedBySqrt(BigInteger[] v, BigInteger n)
{
BigInteger max = v[0].abs().max(v[1].abs());
return isLessThanSqrt(max, n);
}
private static BigInteger[] order(BigInteger a, BigInteger b)
{
if (a.compareTo(b) <= 0)
{
return new BigInteger[]{ a, b };
}
return new BigInteger[]{ b, a };
}
private static BigInteger roundQuotient(BigInteger x, BigInteger y)
{
boolean negative = (x.signum() != y.signum());
x = x.abs();
y = y.abs();
BigInteger result = x.add(y.shiftRight(1)).divide(y);
return negative ? result.negate() : result;
}
private static BigInteger[] solveQuadraticEquation(BigInteger n, BigInteger a, BigInteger b, BigInteger c)
{
BigInteger det = b.multiply(b).subtract(a.multiply(c).shiftLeft(2)).mod(n);
BigInteger root = modSqrt(det, n);
if (root == null)
{
throw new IllegalStateException("Solving quadratic equation failed unexpectedly");
}
BigInteger invDenom = a.shiftLeft(1).modInverse(n);
BigInteger s1 = root.subtract(b).multiply(invDenom).mod(n);
BigInteger s2 = root.negate().subtract(b).multiply(invDenom).mod(n);
return new BigInteger[]{ s1, s2 };
}
private static ECFieldElement[] findNonTrivialOrder3FieldElements(ECCurve c)
{
BigInteger q = c.getField().getCharacteristic();
BigInteger e = q.divide(ECConstants.THREE);
// Search for a random value that generates a non-trival cube root of 1
SecureRandom random = new SecureRandom();
BigInteger b;
do
{
BigInteger r = BigIntegers.createRandomInRange(ECConstants.TWO, q.subtract(ECConstants.TWO), random);
b = r.modPow(e, q);
}
while (b.equals(ECConstants.ONE));
ECFieldElement beta = c.fromBigInteger(b);
return new ECFieldElement[]{ beta, beta.square() };
}
private static ECFieldElement[] findNonTrivialOrder4FieldElements(ECCurve c)
{
ECFieldElement i = c.fromBigInteger(ECConstants.ONE).negate().sqrt();
if (i == null)
{
throw new IllegalStateException("Calculation of non-trivial order-4 field elements failed unexpectedly");
}
return new ECFieldElement[]{ i, i.negate() };
}
private static BigInteger isqrt(BigInteger x)
{
BigInteger g0 = x.shiftRight(x.bitLength() / 2);
for (;;)
{
BigInteger g1 = g0.add(x.divide(g0)).shiftRight(1);
if (g1.equals(g0))
{
return g1;
}
g0 = g1;
}
}
private static void swap(BigInteger[] ab)
{
BigInteger tmp = ab[0];
ab[0] = ab[1];
ab[1] = tmp;
}
private static BigInteger modSqrt(BigInteger x, BigInteger p)
{
if (!p.testBit(0))
{
throw new IllegalStateException();
}
BigInteger pSub1Halved = p.subtract(ECConstants.ONE).shiftRight(1);
BigInteger q = pSub1Halved;
if (!x.modPow(q, p).equals(ECConstants.ONE))
{
return null;
}
while (!q.testBit(0))
{
q = q.shiftRight(1);
if (!x.modPow(q, p).equals(ECConstants.ONE))
{
return modSqrtComplex(x, q, p, pSub1Halved);
}
}
q = q.add(ECConstants.ONE).shiftRight(1);
return x.modPow(q, p);
}
private static BigInteger modSqrtComplex(BigInteger x, BigInteger q, BigInteger p, BigInteger pSub1Halved)
{
BigInteger a = firstNonResidue(p, pSub1Halved);
BigInteger t = pSub1Halved;
BigInteger negPow = t;
while (!q.testBit(0))
{
q = q.shiftRight(1);
t = t.shiftRight(1);
if (!x.modPow(q, p).equals(a.modPow(t, p)))
{
t = t.add(negPow);
}
}
q = q.subtract(ECConstants.ONE).shiftRight(1);
t = t.shiftRight(1);
BigInteger invX = x.modInverse(p);
BigInteger u = invX.modPow(q, p);
BigInteger v = a.modPow(t, p);
return u.multiply(v).mod(p);
}
private static BigInteger firstNonResidue(BigInteger p, BigInteger pSub1Halved)
{
for (int a = 2; a < 1000; ++a)
{
BigInteger A = BigInteger.valueOf(a);
if (!A.modPow(pSub1Halved, p).equals(ECConstants.ONE))
{
return A;
}
}
throw new IllegalStateException();
}
}
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