/// <summary> /// </summary> /// <param name="thisVal"> /// </param> /// <returns> /// </returns> private bool LucasStrongTestHelper(BigInteger thisVal) { // Do the test (selects D based on Selfridge) // Let D be the first element of the sequence // 5, -7, 9, -11, 13, ... for which J(D,n) = -1 // Let P = 1, Q = (1-D) / 4 long D = 5, sign = -1, dCount = 0; bool done = false; while (!done) { int Jresult = Jacobi(D, thisVal); if (Jresult == -1) { done = true; // J(D, this) = 1 } else { if (Jresult == 0 && Math.Abs(D) < thisVal) { // divisor found return false; } if (dCount == 20) { // check for square BigInteger root = thisVal.sqrt(); if (root * root == thisVal) { return false; } } // Console.WriteLine(D); D = (Math.Abs(D) + 2) * sign; sign = -sign; } dCount++; } long Q = (1 - D) >> 2; /* Console.WriteLine("D = " + D); Console.WriteLine("Q = " + Q); Console.WriteLine("(n,D) = " + thisVal.gcd(D)); Console.WriteLine("(n,Q) = " + thisVal.gcd(Q)); Console.WriteLine("J(D|n) = " + BigInteger.Jacobi(D, thisVal)); */ BigInteger p_add1 = thisVal + 1; int s = 0; for (int index = 0; index < p_add1.dataLength; index++) { uint mask = 0x01; for (int i = 0; i < 32; i++) { if ((p_add1.data[index] & mask) != 0) { index = p_add1.dataLength; // to break the outer loop break; } mask <<= 1; s++; } } BigInteger t = p_add1 >> s; // calculate constant = b^(2k) / m // for Barrett Reduction BigInteger constant = new BigInteger(); int nLen = thisVal.dataLength << 1; constant.data[nLen] = 0x00000001; constant.dataLength = nLen + 1; constant = constant / thisVal; BigInteger[] lucas = LucasSequenceHelper(1, Q, t, thisVal, constant, 0); bool isPrime = false; if ((lucas[0].dataLength == 1 && lucas[0].data[0] == 0) || (lucas[1].dataLength == 1 && lucas[1].data[0] == 0)) { // u(t) = 0 or V(t) = 0 isPrime = true; } for (int i = 1; i < s; i++) { if (!isPrime) { // doubling of index lucas[1] = thisVal.BarrettReduction(lucas[1] * lucas[1], thisVal, constant); lucas[1] = (lucas[1] - (lucas[2] << 1)) % thisVal; // lucas[1] = ((lucas[1] * lucas[1]) - (lucas[2] << 1)) % thisVal; if (lucas[1].dataLength == 1 && lucas[1].data[0] == 0) { isPrime = true; } } lucas[2] = thisVal.BarrettReduction(lucas[2] * lucas[2], thisVal, constant); // Q^k } if (isPrime) { // additional checks for composite numbers // If n is prime and gcd(n, Q) == 1, then // Q^((n+1)/2) = Q * Q^((n-1)/2) is congruent to (Q * J(Q, n)) mod n BigInteger g = thisVal.gcd(Q); if (g.dataLength == 1 && g.data[0] == 1) { // gcd(this, Q) == 1 if ((lucas[2].data[maxLength - 1] & 0x80000000) != 0) { lucas[2] += thisVal; } BigInteger temp = (Q * Jacobi(Q, thisVal)) % thisVal; if ((temp.data[maxLength - 1] & 0x80000000) != 0) { temp += thisVal; } if (lucas[2] != temp) { isPrime = false; } } } return isPrime; }
// *********************************************************************** // Returns the k_th number in the Lucas Sequence reduced modulo n. // Uses index doubling to speed up the process. For example, to calculate V(k), // we maintain two numbers in the sequence V(n) and V(n+1). // To obtain V(2n), we use the identity // V(2n) = (V(n) * V(n)) - (2 * Q^n) // To obtain V(2n+1), we first write it as // V(2n+1) = V((n+1) + n) // and use the identity // V(m+n) = V(m) * V(n) - Q * V(m-n) // Hence, // V((n+1) + n) = V(n+1) * V(n) - Q^n * V((n+1) - n) // = V(n+1) * V(n) - Q^n * V(1) // = V(n+1) * V(n) - Q^n * P // We use k in its binary expansion and perform index doubling for each // bit position. For each bit position that is set, we perform an // index doubling followed by an index addition. This means that for V(n), // we need to update it to V(2n+1). For V(n+1), we need to update it to // V((2n+1)+1) = V(2*(n+1)) // This function returns // [0] = U(k) // [1] = V(k) // [2] = Q^n // Where U(0) = 0 % n, U(1) = 1 % n // V(0) = 2 % n, V(1) = P % n // *********************************************************************** // *********************************************************************** // Performs the calculation of the kth term in the Lucas Sequence. // For details of the algorithm, see reference [9]. // k must be odd. i.e LSB == 1 // *********************************************************************** #region Methods /// <summary> /// </summary> /// <param name="P"> /// </param> /// <param name="Q"> /// </param> /// <param name="k"> /// </param> /// <param name="n"> /// </param> /// <param name="constant"> /// </param> /// <param name="s"> /// </param> /// <returns> /// </returns> /// <exception cref="ArgumentException"> /// </exception> private static BigInteger[] LucasSequenceHelper( BigInteger P, BigInteger Q, BigInteger k, BigInteger n, BigInteger constant, int s) { BigInteger[] result = new BigInteger[3]; if ((k.data[0] & 0x00000001) == 0) { throw new ArgumentException("Argument k must be odd."); } int numbits = k.bitCount(); uint mask = (uint)0x1 << ((numbits & 0x1F) - 1); // v = v0, v1 = v1, u1 = u1, Q_k = Q^0 BigInteger v = 2 % n, Q_k = 1 % n, v1 = P % n, u1 = Q_k; bool flag = true; for (int i = k.dataLength - 1; i >= 0; i--) { // iterate on the binary expansion of k // Console.WriteLine("round"); while (mask != 0) { if (i == 0 && mask == 0x00000001) { // last bit break; } if ((k.data[i] & mask) != 0) { // bit is set // index doubling with addition u1 = (u1 * v1) % n; v = ((v * v1) - (P * Q_k)) % n; v1 = n.BarrettReduction(v1 * v1, n, constant); v1 = (v1 - ((Q_k * Q) << 1)) % n; if (flag) { flag = false; } else { Q_k = n.BarrettReduction(Q_k * Q_k, n, constant); } Q_k = (Q_k * Q) % n; } else { // index doubling u1 = ((u1 * v) - Q_k) % n; v1 = ((v * v1) - (P * Q_k)) % n; v = n.BarrettReduction(v * v, n, constant); v = (v - (Q_k << 1)) % n; if (flag) { Q_k = Q % n; flag = false; } else { Q_k = n.BarrettReduction(Q_k * Q_k, n, constant); } } mask >>= 1; } mask = 0x80000000; } // at this point u1 = u(n+1) and v = v(n) // since the last bit always 1, we need to transform u1 to u(2n+1) and v to v(2n+1) u1 = ((u1 * v) - Q_k) % n; v = ((v * v1) - (P * Q_k)) % n; if (flag) { flag = false; } else { Q_k = n.BarrettReduction(Q_k * Q_k, n, constant); } Q_k = (Q_k * Q) % n; for (int i = 0; i < s; i++) { // index doubling u1 = (u1 * v) % n; v = ((v * v) - (Q_k << 1)) % n; if (flag) { Q_k = Q % n; flag = false; } else { Q_k = n.BarrettReduction(Q_k * Q_k, n, constant); } } result[0] = u1; result[1] = v; result[2] = Q_k; return result; }