/* * It uses the sieve of Eratosthenes to discard several composite numbers in * some appropriate range (at the moment {@code [this, this + 1024]}). After * this process it applies the Miller-Rabin test to the numbers that were * not discarded in the sieve. * * @see BigInteger#nextProbablePrime() * @see #millerRabin(BigInteger, int) */ internal static BigInteger nextProbablePrime(BigInteger n) { // PRE: n >= 0 int i, j; int certainty; int gapSize = 1024; // for searching of the next probable prime number int [] modules = new int[primes.Length]; bool[] isDivisible = new bool[gapSize]; BigInteger startPoint; BigInteger probPrime; // If n < "last prime of table" searches next prime in the table if ((n.numberLength == 1) && (n.digits[0] >= 0) && (n.digits[0] < primes[primes.Length - 1])) { for (i = 0; n.digits[0] >= primes[i]; i++) { ; } return(BIprimes[i]); } /* * Creates a "N" enough big to hold the next probable prime Note that: N < * "next prime" < 2*N */ startPoint = new BigInteger(1, n.numberLength, new int[n.numberLength + 1]); java.lang.SystemJ.arraycopy(n.digits, 0, startPoint.digits, 0, n.numberLength); // To fix N to the "next odd number" if (n.testBit(0)) { Elementary.inplaceAdd(startPoint, 2); } else { startPoint.digits[0] |= 1; } // To set the improved certainly of Miller-Rabin j = startPoint.bitLength(); for (certainty = 2; j < BITS[certainty]; certainty++) { ; } // To calculate modules: N mod p1, N mod p2, ... for first primes. for (i = 0; i < primes.Length; i++) { modules[i] = Division.remainder(startPoint, primes[i]) - gapSize; } while (true) { // At this point, all numbers in the gap are initialized as // probably primes java.util.Arrays <Object> .fill(isDivisible, false); // To discard multiples of first primes for (i = 0; i < primes.Length; i++) { modules[i] = (modules[i] + gapSize) % primes[i]; j = (modules[i] == 0) ? 0 : (primes[i] - modules[i]); for (; j < gapSize; j += primes[i]) { isDivisible[j] = true; } } // To execute Miller-Rabin for non-divisible numbers by all first // primes for (j = 0; j < gapSize; j++) { if (!isDivisible[j]) { probPrime = startPoint.copy(); Elementary.inplaceAdd(probPrime, j); if (millerRabin(probPrime, certainty)) { return(probPrime); } } } Elementary.inplaceAdd(startPoint, gapSize); } }
/* * 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); }