/** * Calculates a.modInverse(p) Based on: Savas, E; Koc, C "The Montgomery Modular * Inverse - Revised" */ internal static BigInteger modInverseMontgomery(BigInteger a, BigInteger p) { if (a.sign == 0) { // ZERO hasn't inverse // math.19: BigInteger not invertible throw new ArithmeticException("BigInteger not invertible"); } if (!p.testBit(0)) { // montgomery inverse require even modulo return(modInverseHars(a, p)); } int m = p.numberLength * 32; // PRE: a \in [1, p - 1] BigInteger u, v, r, s; u = p.copy(); // make copy to use inplace method v = a.copy(); int max = java.lang.Math.max(v.numberLength, u.numberLength); r = new BigInteger(1, 1, new int[max + 1]); s = new BigInteger(1, 1, new int[max + 1]); s.digits[0] = 1; // s == 1 && v == 0 int k = 0; int lsbu = u.getLowestSetBit(); int lsbv = v.getLowestSetBit(); int toShift; if (lsbu > lsbv) { BitLevel.inplaceShiftRight(u, lsbu); BitLevel.inplaceShiftRight(v, lsbv); BitLevel.inplaceShiftLeft(r, lsbv); k += lsbu - lsbv; } else { BitLevel.inplaceShiftRight(u, lsbu); BitLevel.inplaceShiftRight(v, lsbv); BitLevel.inplaceShiftLeft(s, lsbu); k += lsbv - lsbu; } r.sign = 1; while (v.signum() > 0) { // INV v >= 0, u >= 0, v odd, u odd (except last iteration when v is even (0)) while (u.compareTo(v) > BigInteger.EQUALS) { Elementary.inplaceSubtract(u, v); toShift = u.getLowestSetBit(); BitLevel.inplaceShiftRight(u, toShift); Elementary.inplaceAdd(r, s); BitLevel.inplaceShiftLeft(s, toShift); k += toShift; } while (u.compareTo(v) <= BigInteger.EQUALS) { Elementary.inplaceSubtract(v, u); if (v.signum() == 0) { break; } toShift = v.getLowestSetBit(); BitLevel.inplaceShiftRight(v, toShift); Elementary.inplaceAdd(s, r); BitLevel.inplaceShiftLeft(r, toShift); k += toShift; } } if (!u.isOne()) { // in u is stored the gcd // math.19: BigInteger not invertible. throw new ArithmeticException("BigInteger not invertible"); } if (r.compareTo(p) >= BigInteger.EQUALS) { Elementary.inplaceSubtract(r, p); } r = p.subtract(r); // Have pair: ((BigInteger)r, (Integer)k) where r == a^(-1) * 2^k mod (module) int n1 = calcN(p); if (k > m) { r = monPro(r, BigInteger.ONE, p, n1); k = k - m; } r = monPro(r, BigInteger.getPowerOfTwo(m - k), p, n1); return(r); }
/** * @param m a positive modulus * Return the greatest common divisor of op1 and op2, * * @param op1 * must be greater than zero * @param op2 * must be greater than zero * @see BigInteger#gcd(BigInteger) * @return {@code GCD(op1, op2)} */ internal static BigInteger gcdBinary(BigInteger op1, BigInteger op2) { // PRE: (op1 > 0) and (op2 > 0) /* * Divide both number the maximal possible times by 2 without rounding * gcd(2*a, 2*b) = 2 * gcd(a,b) */ int lsb1 = op1.getLowestSetBit(); int lsb2 = op2.getLowestSetBit(); int pow2Count = java.lang.Math.min(lsb1, lsb2); BitLevel.inplaceShiftRight(op1, lsb1); BitLevel.inplaceShiftRight(op2, lsb2); BigInteger swap; // I want op2 > op1 if (op1.compareTo(op2) == BigInteger.GREATER) { swap = op1; op1 = op2; op2 = swap; } do // INV: op2 >= op1 && both are odd unless op1 = 0 // Optimization for small operands // (op2.bitLength() < 64) implies by INV (op1.bitLength() < 64) { if ((op2.numberLength == 1) || ((op2.numberLength == 2) && (op2.digits[1] > 0))) { op2 = BigInteger.valueOf(Division.gcdBinary(op1.longValue(), op2.longValue())); break; } // Implements one step of the Euclidean algorithm // To reduce one operand if it's much smaller than the other one if (op2.numberLength > op1.numberLength * 1.2) { op2 = op2.remainder(op1); if (op2.signum() != 0) { BitLevel.inplaceShiftRight(op2, op2.getLowestSetBit()); } } else { // Use Knuth's algorithm of successive subtract and shifting do { Elementary.inplaceSubtract(op2, op1); // both are odd BitLevel.inplaceShiftRight(op2, op2.getLowestSetBit()); // op2 is even } while (op2.compareTo(op1) >= BigInteger.EQUALS); } // now op1 >= op2 swap = op2; op2 = op1; op1 = swap; } while (op1.sign != 0); return(op2.shiftLeft(pow2Count)); }