/// <summary> /// Decrypts Login Key from Client /// </summary> /// <param name="LoginKey"> /// Login Key from Client /// </param> /// <param name="UserName"> /// Username stored in Login Key /// </param> /// <param name="ServerSalt"> /// Server Salt stored in Login Key /// </param> /// <param name="Password"> /// Password stored in Login Key /// </param> public void DecryptLoginKey(string LoginKey, out string UserName, out string ServerSalt, out string Password) { string[] LoginKeySplit = LoginKey.Split('-'); BigInteger ClientPublicKey = new BigInteger(LoginKeySplit[0], 16); string EncryptedBlock = LoginKeySplit[1]; // These should really be in a config file, but for now hardcoded BigInteger ServerPrivateKey = new BigInteger("7ad852c6494f664e8df21446285ecd6f400cf20e1d872ee96136d7744887424b", 16); BigInteger Prime = new BigInteger( "eca2e8c85d863dcdc26a429a71a9815ad052f6139669dd659f98ae159d313d13c6bf2838e10a69b6478b64a24bd054ba8248e8fa778703b418408249440b2c1edd28853e240d8a7e49540b76d120d3b1ad2878b1b99490eb4a2a5e84caa8a91cecbdb1aa7c816e8be343246f80c637abc653b893fd91686cf8d32d6cfe5f2a6f", 16); string TeaKey = ClientPublicKey.modPow(ServerPrivateKey, Prime).ToString(16).ToLower(); if (TeaKey.Length < 32) { // If TeaKey is not at least 128bits, pad to the left with 0x00 TeaKey.PadLeft(32, '0'); } else { // If TeaKey is more than 128bits, truncate TeaKey = TeaKey.Substring(0, 32); } string DecryptedBlock = this.DecryptTea(EncryptedBlock, TeaKey); DecryptedBlock = DecryptedBlock.Substring(8); // Strip first 8 bytes of padding int DataLength = this.ConvertStringToIntSwapEndian(DecryptedBlock.Substring(0, 4)); DecryptedBlock = DecryptedBlock.Substring(4); string[] BlockParts = DecryptedBlock.Split(new[] { '|' }, 2); UserName = BlockParts[0]; ServerSalt = string.Empty; for (int i = 0; i < 32; i += 4) { ServerSalt += string.Format("{0:x8}", this.ConvertStringToIntSwapEndian(BlockParts[1].Substring(i, 4))); } Password = BlockParts[1].Substring(33, DataLength - 34 - UserName.Length); }
/// <summary> /// </summary> /// <param name="bi1"> /// </param> /// <param name="shiftVal"> /// </param> /// <returns> /// </returns> public static BigInteger operator >>(BigInteger bi1, int shiftVal) { BigInteger result = new BigInteger(bi1); result.dataLength = shiftRight(result.data, shiftVal); if ((bi1.data[maxLength - 1] & 0x80000000) != 0) { // negative for (int i = maxLength - 1; i >= result.dataLength; i--) { result.data[i] = 0xFFFFFFFF; } uint mask = 0x80000000; for (int i = 0; i < 32; i++) { if ((result.data[result.dataLength - 1] & mask) != 0) { break; } result.data[result.dataLength - 1] |= mask; mask >>= 1; } result.dataLength = maxLength; } return result; }
/// <summary> /// </summary> /// <param name="bits"> /// </param> /// <param name="confidence"> /// </param> /// <param name="rand"> /// </param> /// <returns> /// </returns> public static BigInteger genPseudoPrime(int bits, int confidence, Random rand) { BigInteger result = new BigInteger(); bool done = false; while (!done) { result.genRandomBits(bits, rand); result.data[0] |= 0x01; // make it odd // prime test done = result.isProbablePrime(confidence); } return result; }
// *********************************************************************** // Tests the correct implementation of the modulo exponential and // inverse modulo functions using RSA encryption and decryption. The two // pseudoprimes p and q are fixed, but the two RSA keys are generated // for each round of testing. // *********************************************************************** /// <summary> /// </summary> /// <param name="rounds"> /// </param> public static void RSATest2(int rounds) { Random rand = new Random(); byte[] val = new byte[64]; byte[] pseudoPrime1 = { 0x85, 0x84, 0x64, 0xFD, 0x70, 0x6A, 0x9F, 0xF0, 0x94, 0x0C, 0x3E, 0x2C, 0x74, 0x34, 0x05, 0xC9, 0x55, 0xB3, 0x85, 0x32, 0x98, 0x71, 0xF9, 0x41, 0x21, 0x5F, 0x02, 0x9E, 0xEA, 0x56, 0x8D, 0x8C, 0x44, 0xCC, 0xEE, 0xEE, 0x3D, 0x2C, 0x9D, 0x2C, 0x12, 0x41, 0x1E, 0xF1, 0xC5, 0x32, 0xC3, 0xAA, 0x31, 0x4A, 0x52, 0xD8, 0xE8, 0xAF, 0x42, 0xF4, 0x72, 0xA1, 0x2A, 0x0D, 0x97, 0xB1, 0x31, 0xB3, }; byte[] pseudoPrime2 = { 0x99, 0x98, 0xCA, 0xB8, 0x5E, 0xD7, 0xE5, 0xDC, 0x28, 0x5C, 0x6F, 0x0E, 0x15, 0x09, 0x59, 0x6E, 0x84, 0xF3, 0x81, 0xCD, 0xDE, 0x42, 0xDC, 0x93, 0xC2, 0x7A, 0x62, 0xAC, 0x6C, 0xAF, 0xDE, 0x74, 0xE3, 0xCB, 0x60, 0x20, 0x38, 0x9C, 0x21, 0xC3, 0xDC, 0xC8, 0xA2, 0x4D, 0xC6, 0x2A, 0x35, 0x7F, 0xF3, 0xA9, 0xE8, 0x1D, 0x7B, 0x2C, 0x78, 0xFA, 0xB8, 0x02, 0x55, 0x80, 0x9B, 0xC2, 0xA5, 0xCB, }; BigInteger bi_p = new BigInteger(pseudoPrime1); BigInteger bi_q = new BigInteger(pseudoPrime2); BigInteger bi_pq = (bi_p - 1) * (bi_q - 1); BigInteger bi_n = bi_p * bi_q; for (int count = 0; count < rounds; count++) { // generate private and public key BigInteger bi_e = bi_pq.genCoPrime(512, rand); BigInteger bi_d = bi_e.modInverse(bi_pq); Console.WriteLine("\ne =\n" + bi_e.ToString(10)); Console.WriteLine("\nd =\n" + bi_d.ToString(10)); Console.WriteLine("\nn =\n" + bi_n.ToString(10) + "\n"); // generate data of random length int t1 = 0; while (t1 == 0) { t1 = (int)(rand.NextDouble() * 65); } bool done = false; while (!done) { for (int i = 0; i < 64; i++) { if (i < t1) { val[i] = (byte)(rand.NextDouble() * 256); } else { val[i] = 0; } if (val[i] != 0) { done = true; } } } while (val[0] == 0) { val[0] = (byte)(rand.NextDouble() * 256); } Console.Write("Round = " + count); // encrypt and decrypt data BigInteger bi_data = new BigInteger(val, t1); BigInteger bi_encrypted = bi_data.modPow(bi_e, bi_n); BigInteger bi_decrypted = bi_encrypted.modPow(bi_d, bi_n); // compare if (bi_decrypted != bi_data) { Console.WriteLine("\nError at round " + count); Console.WriteLine(bi_data + "\n"); return; } Console.WriteLine(" <PASSED>."); } }
/// <summary> /// </summary> /// <param name="rounds"> /// </param> public static void MulDivTest(int rounds) { Random rand = new Random(); byte[] val = new byte[64]; byte[] val2 = new byte[64]; for (int count = 0; count < rounds; count++) { // generate 2 numbers of random length int t1 = 0; while (t1 == 0) { t1 = (int)(rand.NextDouble() * 65); } int t2 = 0; while (t2 == 0) { t2 = (int)(rand.NextDouble() * 65); } bool done = false; while (!done) { for (int i = 0; i < 64; i++) { if (i < t1) { val[i] = (byte)(rand.NextDouble() * 256); } else { val[i] = 0; } if (val[i] != 0) { done = true; } } } done = false; while (!done) { for (int i = 0; i < 64; i++) { if (i < t2) { val2[i] = (byte)(rand.NextDouble() * 256); } else { val2[i] = 0; } if (val2[i] != 0) { done = true; } } } while (val[0] == 0) { val[0] = (byte)(rand.NextDouble() * 256); } while (val2[0] == 0) { val2[0] = (byte)(rand.NextDouble() * 256); } Console.WriteLine(count); BigInteger bn1 = new BigInteger(val, t1); BigInteger bn2 = new BigInteger(val2, t2); // Determine the quotient and remainder by dividing // the first number by the second. BigInteger bn3 = bn1 / bn2; BigInteger bn4 = bn1 % bn2; // Recalculate the number BigInteger bn5 = (bn3 * bn2) + bn4; // Make sure they're the same if (bn5 != bn1) { Console.WriteLine("Error at " + count); Console.WriteLine(bn1 + "\n"); Console.WriteLine(bn2 + "\n"); Console.WriteLine(bn3 + "\n"); Console.WriteLine(bn4 + "\n"); Console.WriteLine(bn5 + "\n"); return; } } }
/// <summary> /// </summary> /// <param name="a"> /// </param> /// <param name="b"> /// </param> /// <returns> /// </returns> /// <exception cref="ArgumentException"> /// </exception> public static int Jacobi(BigInteger a, BigInteger b) { // Jacobi defined only for odd integers if ((b.data[0] & 0x1) == 0) { throw new ArgumentException("Jacobi defined only for odd integers."); } if (a >= b) { a %= b; } if (a.dataLength == 1 && a.data[0] == 0) { return 0; // a == 0 } if (a.dataLength == 1 && a.data[0] == 1) { return 1; // a == 1 } if (a < 0) { if (((b - 1).data[0] & 0x2) == 0) { // if( (((b-1) >> 1).data[0] & 0x1) == 0) return Jacobi(-a, b); } else { return -Jacobi(-a, b); } } int e = 0; for (int index = 0; index < a.dataLength; index++) { uint mask = 0x01; for (int i = 0; i < 32; i++) { if ((a.data[index] & mask) != 0) { index = a.dataLength; // to break the outer loop break; } mask <<= 1; e++; } } BigInteger a1 = a >> e; int s = 1; if ((e & 0x1) != 0 && ((b.data[0] & 0x7) == 3 || (b.data[0] & 0x7) == 5)) { s = -1; } if ((b.data[0] & 0x3) == 3 && (a1.data[0] & 0x3) == 3) { s = -s; } if (a1.dataLength == 1 && a1.data[0] == 1) { return s; } else { return s * Jacobi(b % a1, a1); } }
/// <summary> /// </summary> /// <param name="x"> /// </param> /// <param name="n"> /// </param> /// <param name="constant"> /// </param> /// <returns> /// </returns> private BigInteger BarrettReduction(BigInteger x, BigInteger n, BigInteger constant) { int k = n.dataLength, kPlusOne = k + 1, kMinusOne = k - 1; BigInteger q1 = new BigInteger(); // q1 = x / b^(k-1) for (int i = kMinusOne, j = 0; i < x.dataLength; i++, j++) { q1.data[j] = x.data[i]; } q1.dataLength = x.dataLength - kMinusOne; if (q1.dataLength <= 0) { q1.dataLength = 1; } BigInteger q2 = q1 * constant; BigInteger q3 = new BigInteger(); // q3 = q2 / b^(k+1) for (int i = kPlusOne, j = 0; i < q2.dataLength; i++, j++) { q3.data[j] = q2.data[i]; } q3.dataLength = q2.dataLength - kPlusOne; if (q3.dataLength <= 0) { q3.dataLength = 1; } // r1 = x mod b^(k+1) // i.e. keep the lowest (k+1) words BigInteger r1 = new BigInteger(); int lengthToCopy = (x.dataLength > kPlusOne) ? kPlusOne : x.dataLength; for (int i = 0; i < lengthToCopy; i++) { r1.data[i] = x.data[i]; } r1.dataLength = lengthToCopy; // r2 = (q3 * n) mod b^(k+1) // partial multiplication of q3 and n BigInteger r2 = new BigInteger(); for (int i = 0; i < q3.dataLength; i++) { if (q3.data[i] == 0) { continue; } ulong mcarry = 0; int t = i; for (int j = 0; j < n.dataLength && t < kPlusOne; j++, t++) { // t = i + j ulong val = (q3.data[i] * (ulong)n.data[j]) + r2.data[t] + mcarry; r2.data[t] = (uint)(val & 0xFFFFFFFF); mcarry = val >> 32; } if (t < kPlusOne) { r2.data[t] = (uint)mcarry; } } r2.dataLength = kPlusOne; while (r2.dataLength > 1 && r2.data[r2.dataLength - 1] == 0) { r2.dataLength--; } r1 -= r2; if ((r1.data[maxLength - 1] & 0x80000000) != 0) { // negative BigInteger val = new BigInteger(); val.data[kPlusOne] = 0x00000001; val.dataLength = kPlusOne + 1; r1 += val; } while (r1 >= n) { r1 -= n; } return r1; }
/// <summary> /// </summary> /// <param name="bi1"> /// </param> /// <param name="bi2"> /// </param> /// <param name="outQuotient"> /// </param> /// <param name="outRemainder"> /// </param> private static void multiByteDivide( BigInteger bi1, BigInteger bi2, BigInteger outQuotient, BigInteger outRemainder) { uint[] result = new uint[maxLength]; int remainderLen = bi1.dataLength + 1; uint[] remainder = new uint[remainderLen]; uint mask = 0x80000000; uint val = bi2.data[bi2.dataLength - 1]; int shift = 0, resultPos = 0; while (mask != 0 && (val & mask) == 0) { shift++; mask >>= 1; } // Console.WriteLine("shift = {0}", shift); // Console.WriteLine("Before bi1 Len = {0}, bi2 Len = {1}", bi1.dataLength, bi2.dataLength); for (int i = 0; i < bi1.dataLength; i++) { remainder[i] = bi1.data[i]; } shiftLeft(remainder, shift); bi2 = bi2 << shift; /* Console.WriteLine("bi1 Len = {0}, bi2 Len = {1}", bi1.dataLength, bi2.dataLength); Console.WriteLine("dividend = " + bi1 + "\ndivisor = " + bi2); for(int q = remainderLen - 1; q >= 0; q--) Console.Write("{0:x2}", remainder[q]); Console.WriteLine(); */ int j = remainderLen - bi2.dataLength; int pos = remainderLen - 1; ulong firstDivisorByte = bi2.data[bi2.dataLength - 1]; ulong secondDivisorByte = bi2.data[bi2.dataLength - 2]; int divisorLen = bi2.dataLength + 1; uint[] dividendPart = new uint[divisorLen]; while (j > 0) { ulong dividend = ((ulong)remainder[pos] << 32) + remainder[pos - 1]; // Console.WriteLine("dividend = {0}", dividend); ulong q_hat = dividend / firstDivisorByte; ulong r_hat = dividend % firstDivisorByte; // Console.WriteLine("q_hat = {0:X}, r_hat = {1:X}", q_hat, r_hat); bool done = false; while (!done) { done = true; if (q_hat == 0x100000000 || (q_hat * secondDivisorByte) > ((r_hat << 32) + remainder[pos - 2])) { q_hat--; r_hat += firstDivisorByte; if (r_hat < 0x100000000) { done = false; } } } for (int h = 0; h < divisorLen; h++) { dividendPart[h] = remainder[pos - h]; } BigInteger kk = new BigInteger(dividendPart); BigInteger ss = bi2 * (long)q_hat; // Console.WriteLine("ss before = " + ss); while (ss > kk) { q_hat--; ss -= bi2; // Console.WriteLine(ss); } BigInteger yy = kk - ss; // Console.WriteLine("ss = " + ss); // Console.WriteLine("kk = " + kk); // Console.WriteLine("yy = " + yy); for (int h = 0; h < divisorLen; h++) { remainder[pos - h] = yy.data[bi2.dataLength - h]; } /* Console.WriteLine("dividend = "); for(int q = remainderLen - 1; q >= 0; q--) Console.Write("{0:x2}", remainder[q]); Console.WriteLine("\n************ q_hat = {0:X}\n", q_hat); */ result[resultPos++] = (uint)q_hat; pos--; j--; } outQuotient.dataLength = resultPos; int y = 0; for (int x = outQuotient.dataLength - 1; x >= 0; x--, y++) { outQuotient.data[y] = result[x]; } for (; y < maxLength; y++) { outQuotient.data[y] = 0; } while (outQuotient.dataLength > 1 && outQuotient.data[outQuotient.dataLength - 1] == 0) { outQuotient.dataLength--; } if (outQuotient.dataLength == 0) { outQuotient.dataLength = 1; } outRemainder.dataLength = shiftRight(remainder, shift); for (y = 0; y < outRemainder.dataLength; y++) { outRemainder.data[y] = remainder[y]; } for (; y < maxLength; y++) { outRemainder.data[y] = 0; } }
// *********************************************************************** // Returns a string representing the BigInteger in sign-and-magnitude // format in the specified radix. // Example // ------- // If the value of BigInteger is -255 in base 10, then // ToString(16) returns "-FF" // *********************************************************************** /// <summary> /// </summary> /// <param name="radix"> /// </param> /// <returns> /// </returns> /// <exception cref="ArgumentException"> /// </exception> public string ToString(int radix) { if (radix < 2 || radix > 36) { throw new ArgumentException("Radix must be >= 2 and <= 36"); } string charSet = "ABCDEFGHIJKLMNOPQRSTUVWXYZ"; string result = string.Empty; BigInteger a = this; bool negative = false; if ((a.data[maxLength - 1] & 0x80000000) != 0) { negative = true; try { a = -a; } catch (Exception) { } } BigInteger quotient = new BigInteger(); BigInteger remainder = new BigInteger(); BigInteger biRadix = new BigInteger(radix); if (a.dataLength == 1 && a.data[0] == 0) { result = "0"; } else { while (a.dataLength > 1 || (a.dataLength == 1 && a.data[0] != 0)) { singleByteDivide(a, biRadix, quotient, remainder); if (remainder.data[0] < 10) { result = remainder.data[0] + result; } else { result = charSet[(int)remainder.data[0] - 10] + result; } a = quotient; } if (negative) { result = "-" + result; } } return result; }
// *********************************************************************** // Probabilistic prime test based on Solovay-Strassen (Euler Criterion) // p is probably prime if for any a < p (a is not multiple of p), // a^((p-1)/2) mod p = J(a, p) // where J is the Jacobi symbol. // Otherwise, p is composite. // Returns // ------- // True if "this" is a Euler pseudoprime to randomly chosen // bases. The number of chosen bases is given by the "confidence" // parameter. // False if "this" is definitely NOT prime. // *********************************************************************** /// <summary> /// </summary> /// <param name="confidence"> /// </param> /// <returns> /// </returns> public bool SolovayStrassenTest(int confidence) { BigInteger thisVal; if ((this.data[maxLength - 1] & 0x80000000) != 0) { // negative thisVal = -this; } else { thisVal = this; } if (thisVal.dataLength == 1) { // test small numbers if (thisVal.data[0] == 0 || thisVal.data[0] == 1) { return false; } else if (thisVal.data[0] == 2 || thisVal.data[0] == 3) { return true; } } if ((thisVal.data[0] & 0x1) == 0) { // even numbers return false; } int bits = thisVal.bitCount(); BigInteger a = new BigInteger(); BigInteger p_sub1 = thisVal - 1; BigInteger p_sub1_shift = p_sub1 >> 1; Random rand = new Random(); for (int round = 0; round < confidence; round++) { bool done = false; while (!done) { // generate a < n int testBits = 0; // make sure "a" has at least 2 bits while (testBits < 2) { testBits = (int)(rand.NextDouble() * bits); } a.genRandomBits(testBits, rand); int byteLen = a.dataLength; // make sure "a" is not 0 if (byteLen > 1 || (byteLen == 1 && a.data[0] != 1)) { done = true; } } // check whether a factor exists (fix for version 1.03) BigInteger gcdTest = a.gcd(thisVal); if (gcdTest.dataLength == 1 && gcdTest.data[0] != 1) { return false; } // calculate a^((p-1)/2) mod p BigInteger expResult = a.modPow(p_sub1_shift, thisVal); if (expResult == p_sub1) { expResult = -1; } // calculate Jacobi symbol BigInteger jacob = Jacobi(a, thisVal); // Console.WriteLine("a = " + a.ToString(10) + " b = " + thisVal.ToString(10)); // Console.WriteLine("expResult = " + expResult.ToString(10) + " Jacob = " + jacob.ToString(10)); // if they are different then it is not prime if (expResult != jacob) { return false; } } return true; }
// *********************************************************************** // Constructor (Default value provided by BigInteger) // *********************************************************************** /// <summary> /// </summary> /// <param name="bi"> /// </param> public BigInteger(BigInteger bi) { this.data = new uint[maxLength]; this.dataLength = bi.dataLength; for (int i = 0; i < this.dataLength; i++) { this.data[i] = bi.data[i]; } }
/// <summary> /// </summary> /// <param name="confidence"> /// </param> /// <returns> /// </returns> public bool RabinMillerTest(int confidence) { BigInteger thisVal; if ((this.data[maxLength - 1] & 0x80000000) != 0) { // negative thisVal = -this; } else { thisVal = this; } if (thisVal.dataLength == 1) { // test small numbers if (thisVal.data[0] == 0 || thisVal.data[0] == 1) { return false; } else if (thisVal.data[0] == 2 || thisVal.data[0] == 3) { return true; } } if ((thisVal.data[0] & 0x1) == 0) { // even numbers return false; } // calculate values of s and t BigInteger p_sub1 = thisVal - (new BigInteger(1)); int s = 0; for (int index = 0; index < p_sub1.dataLength; index++) { uint mask = 0x01; for (int i = 0; i < 32; i++) { if ((p_sub1.data[index] & mask) != 0) { index = p_sub1.dataLength; // to break the outer loop break; } mask <<= 1; s++; } } BigInteger t = p_sub1 >> s; int bits = thisVal.bitCount(); BigInteger a = new BigInteger(); Random rand = new Random(); for (int round = 0; round < confidence; round++) { bool done = false; while (!done) { // generate a < n int testBits = 0; // make sure "a" has at least 2 bits while (testBits < 2) { testBits = (int)(rand.NextDouble() * bits); } a.genRandomBits(testBits, rand); int byteLen = a.dataLength; // make sure "a" is not 0 if (byteLen > 1 || (byteLen == 1 && a.data[0] != 1)) { done = true; } } // check whether a factor exists (fix for version 1.03) BigInteger gcdTest = a.gcd(thisVal); if (gcdTest.dataLength == 1 && gcdTest.data[0] != 1) { return false; } BigInteger b = a.modPow(t, thisVal); /* Console.WriteLine("a = " + a.ToString(10)); Console.WriteLine("b = " + b.ToString(10)); Console.WriteLine("t = " + t.ToString(10)); Console.WriteLine("s = " + s); */ bool result = false; if (b.dataLength == 1 && b.data[0] == 1) { // a^t mod p = 1 result = true; } for (int j = 0; result == false && j < s; j++) { if (b == p_sub1) { // a^((2^j)*t) mod p = p-1 for some 0 <= j <= s-1 result = true; break; } b = (b * b) % thisVal; } if (result == false) { return false; } } return true; }
/// <summary> /// </summary> /// <param name="confidence"> /// </param> /// <returns> /// </returns> public bool FermatLittleTest(int confidence) { BigInteger thisVal; if ((this.data[maxLength - 1] & 0x80000000) != 0) { // negative thisVal = -this; } else { thisVal = this; } if (thisVal.dataLength == 1) { // test small numbers if (thisVal.data[0] == 0 || thisVal.data[0] == 1) { return false; } else if (thisVal.data[0] == 2 || thisVal.data[0] == 3) { return true; } } if ((thisVal.data[0] & 0x1) == 0) { // even numbers return false; } int bits = thisVal.bitCount(); BigInteger a = new BigInteger(); BigInteger p_sub1 = thisVal - (new BigInteger(1)); Random rand = new Random(); for (int round = 0; round < confidence; round++) { bool done = false; while (!done) { // generate a < n int testBits = 0; // make sure "a" has at least 2 bits while (testBits < 2) { testBits = (int)(rand.NextDouble() * bits); } a.genRandomBits(testBits, rand); int byteLen = a.dataLength; // make sure "a" is not 0 if (byteLen > 1 || (byteLen == 1 && a.data[0] != 1)) { done = true; } } // check whether a factor exists (fix for version 1.03) BigInteger gcdTest = a.gcd(thisVal); if (gcdTest.dataLength == 1 && gcdTest.data[0] != 1) { return false; } // calculate a^(p-1) mod p BigInteger expResult = a.modPow(p_sub1, thisVal); int resultLen = expResult.dataLength; // is NOT prime is a^(p-1) mod p != 1 if (resultLen > 1 || (resultLen == 1 && expResult.data[0] != 1)) { // Console.WriteLine("a = " + a.ToString()); return false; } } return true; }
// *********************************************************************** // Overloading of the NEGATE operator (2's complement) // *********************************************************************** /// <summary> /// </summary> /// <param name="bi1"> /// </param> /// <returns> /// </returns> /// <exception cref="ArithmeticException"> /// </exception> public static BigInteger operator -(BigInteger bi1) { // handle neg of zero separately since it'll cause an overflow // if we proceed. if (bi1.dataLength == 1 && bi1.data[0] == 0) { return new BigInteger(); } BigInteger result = new BigInteger(bi1); // 1's complement for (int i = 0; i < maxLength; i++) { result.data[i] = ~bi1.data[i]; } // add one to result of 1's complement long val, carry = 1; int index = 0; while (carry != 0 && index < maxLength) { val = result.data[index]; val++; result.data[index] = (uint)(val & 0xFFFFFFFF); carry = val >> 32; index++; } if ((bi1.data[maxLength - 1] & 0x80000000) == (result.data[maxLength - 1] & 0x80000000)) { throw new ArithmeticException("Overflow in negation.\n"); } result.dataLength = maxLength; while (result.dataLength > 1 && result.data[result.dataLength - 1] == 0) { result.dataLength--; } return result; }
/// <summary> /// </summary> /// <param name="bi1"> /// </param> /// <param name="bi2"> /// </param> /// <returns> /// </returns> /// <exception cref="ArithmeticException"> /// </exception> public static BigInteger operator -(BigInteger bi1, BigInteger bi2) { BigInteger result = new BigInteger(); result.dataLength = (bi1.dataLength > bi2.dataLength) ? bi1.dataLength : bi2.dataLength; long carryIn = 0; for (int i = 0; i < result.dataLength; i++) { long diff; diff = bi1.data[i] - (long)bi2.data[i] - carryIn; result.data[i] = (uint)(diff & 0xFFFFFFFF); if (diff < 0) { carryIn = 1; } else { carryIn = 0; } } // roll over to negative if (carryIn != 0) { for (int i = result.dataLength; i < maxLength; i++) { result.data[i] = 0xFFFFFFFF; } result.dataLength = maxLength; } // fixed in v1.03 to give correct datalength for a - (-b) while (result.dataLength > 1 && result.data[result.dataLength - 1] == 0) { result.dataLength--; } // overflow check int lastPos = maxLength - 1; if ((bi1.data[lastPos] & 0x80000000) != (bi2.data[lastPos] & 0x80000000) && (result.data[lastPos] & 0x80000000) != (bi1.data[lastPos] & 0x80000000)) { throw new ArithmeticException(); } return result; }
// *********************************************************************** // Sets the value of the specified bit to 0 // The Least Significant Bit position is 0. // *********************************************************************** // *********************************************************************** // Returns a value that is equivalent to the integer square root // of the BigInteger. // The integer square root of "this" is defined as the largest integer n // such that (n * n) <= this // *********************************************************************** /// <summary> /// </summary> /// <returns> /// </returns> public BigInteger sqrt() { uint numBits = (uint)this.bitCount(); if ((numBits & 0x1) != 0) { // odd number of bits numBits = (numBits >> 1) + 1; } else { numBits = numBits >> 1; } uint bytePos = numBits >> 5; byte bitPos = (byte)(numBits & 0x1F); uint mask; BigInteger result = new BigInteger(); if (bitPos == 0) { mask = 0x80000000; } else { mask = (uint)1 << bitPos; bytePos++; } result.dataLength = (int)bytePos; for (int i = (int)bytePos - 1; i >= 0; i--) { while (mask != 0) { // guess result.data[i] ^= mask; // undo the guess if its square is larger than this if ((result * result) > this) { result.data[i] ^= mask; } mask >>= 1; } mask = 0x80000000; } return result; }
// *********************************************************************** // 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; }
// *********************************************************************** // Constructor (Default value provided by a string of digits of the // specified base) // Example (base 10) // ----------------- // To initialize "a" with the default value of 1234 in base 10 // BigInteger a = new BigInteger("1234", 10) // To initialize "a" with the default value of -1234 // BigInteger a = new BigInteger("-1234", 10) // Example (base 16) // ----------------- // To initialize "a" with the default value of 0x1D4F in base 16 // BigInteger a = new BigInteger("1D4F", 16) // To initialize "a" with the default value of -0x1D4F // BigInteger a = new BigInteger("-1D4F", 16) // Note that string values are specified in the <sign><magnitude> // format. // *********************************************************************** /// <summary> /// </summary> /// <param name="value"> /// </param> /// <param name="radix"> /// </param> /// <exception cref="ArithmeticException"> /// </exception> public BigInteger(string value, int radix) { BigInteger multiplier = new BigInteger(1); BigInteger result = new BigInteger(); value = value.ToUpper().Trim(); int limit = 0; if (value[0] == '-') { limit = 1; } for (int i = value.Length - 1; i >= limit; i--) { int posVal = value[i]; if (posVal >= '0' && posVal <= '9') { posVal -= '0'; } else if (posVal >= 'A' && posVal <= 'Z') { posVal = (posVal - 'A') + 10; } else { posVal = 9999999; // arbitrary large } if (posVal >= radix) { throw new ArithmeticException("Invalid string in constructor."); } else { if (value[0] == '-') { posVal = -posVal; } result = result + (multiplier * posVal); if ((i - 1) >= limit) { multiplier = multiplier * radix; } } } if (value[0] == '-') { // negative values if ((result.data[maxLength - 1] & 0x80000000) == 0) { throw new ArithmeticException("Negative underflow in constructor."); } } else { // positive values if ((result.data[maxLength - 1] & 0x80000000) != 0) { throw new ArithmeticException("Positive overflow in constructor."); } } this.data = new uint[maxLength]; for (int i = 0; i < result.dataLength; i++) { this.data[i] = result.data[i]; } this.dataLength = result.dataLength; }
/// <summary> /// </summary> /// <param name="bi1"> /// </param> /// <param name="bi2"> /// </param> /// <param name="outQuotient"> /// </param> /// <param name="outRemainder"> /// </param> private static void singleByteDivide( BigInteger bi1, BigInteger bi2, BigInteger outQuotient, BigInteger outRemainder) { uint[] result = new uint[maxLength]; int resultPos = 0; // copy dividend to reminder for (int i = 0; i < maxLength; i++) { outRemainder.data[i] = bi1.data[i]; } outRemainder.dataLength = bi1.dataLength; while (outRemainder.dataLength > 1 && outRemainder.data[outRemainder.dataLength - 1] == 0) { outRemainder.dataLength--; } ulong divisor = bi2.data[0]; int pos = outRemainder.dataLength - 1; ulong dividend = outRemainder.data[pos]; // Console.WriteLine("divisor = " + divisor + " dividend = " + dividend); // Console.WriteLine("divisor = " + bi2 + "\ndividend = " + bi1); if (dividend >= divisor) { ulong quotient = dividend / divisor; result[resultPos++] = (uint)quotient; outRemainder.data[pos] = (uint)(dividend % divisor); } pos--; while (pos >= 0) { // Console.WriteLine(pos); dividend = ((ulong)outRemainder.data[pos + 1] << 32) + outRemainder.data[pos]; ulong quotient = dividend / divisor; result[resultPos++] = (uint)quotient; outRemainder.data[pos + 1] = 0; outRemainder.data[pos--] = (uint)(dividend % divisor); // Console.WriteLine(">>>> " + bi1); } outQuotient.dataLength = resultPos; int j = 0; for (int i = outQuotient.dataLength - 1; i >= 0; i--, j++) { outQuotient.data[j] = result[i]; } for (; j < maxLength; j++) { outQuotient.data[j] = 0; } while (outQuotient.dataLength > 1 && outQuotient.data[outQuotient.dataLength - 1] == 0) { outQuotient.dataLength--; } if (outQuotient.dataLength == 0) { outQuotient.dataLength = 1; } while (outRemainder.dataLength > 1 && outRemainder.data[outRemainder.dataLength - 1] == 0) { outRemainder.dataLength--; } }
// *********************************************************************** // Returns a hex string showing the contains of the BigInteger // Examples // ------- // 1) If the value of BigInteger is 255 in base 10, then // ToHexString() returns "FF" // 2) If the value of BigInteger is -255 in base 10, then // ToHexString() returns ".....FFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFF01", // which is the 2's complement representation of -255. // *********************************************************************** // *********************************************************************** // Returns gcd(this, bi) // *********************************************************************** /// <summary> /// </summary> /// <param name="bi"> /// </param> /// <returns> /// </returns> public BigInteger gcd(BigInteger bi) { BigInteger x; BigInteger y; if ((this.data[maxLength - 1] & 0x80000000) != 0) { // negative x = -this; } else { x = this; } if ((bi.data[maxLength - 1] & 0x80000000) != 0) { // negative y = -bi; } else { y = bi; } BigInteger g = y; while (x.dataLength > 1 || (x.dataLength == 1 && x.data[0] != 0)) { g = x; x = y % x; y = g; } return g; }
/// <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; }
/// <summary> /// </summary> /// <param name="bits"> /// </param> /// <param name="rand"> /// </param> /// <returns> /// </returns> public BigInteger genCoPrime(int bits, Random rand) { bool done = false; BigInteger result = new BigInteger(); while (!done) { result.genRandomBits(bits, rand); // Console.WriteLine(result.ToString(16)); // gcd test BigInteger g = result.gcd(this); if (g.dataLength == 1 && g.data[0] == 1) { done = true; } } return result; }
/// <summary> /// </summary> /// <param name="P"> /// </param> /// <param name="Q"> /// </param> /// <param name="k"> /// </param> /// <param name="n"> /// </param> /// <returns> /// </returns> public static BigInteger[] LucasSequence(BigInteger P, BigInteger Q, BigInteger k, BigInteger n) { if (k.dataLength == 1 && k.data[0] == 0) { BigInteger[] result = new BigInteger[3]; result[0] = 0; result[1] = 2 % n; result[2] = 1 % n; return result; } // calculate constant = b^(2k) / m // for Barrett Reduction BigInteger constant = new BigInteger(); int nLen = n.dataLength << 1; constant.data[nLen] = 0x00000001; constant.dataLength = nLen + 1; constant = constant / n; // calculate values of s and t int s = 0; for (int index = 0; index < k.dataLength; index++) { uint mask = 0x01; for (int i = 0; i < 32; i++) { if ((k.data[index] & mask) != 0) { index = k.dataLength; // to break the outer loop break; } mask <<= 1; s++; } } BigInteger t = k >> s; // Console.WriteLine("s = " + s + " t = " + t); return LucasSequenceHelper(P, Q, t, n, constant, s); }
/// <summary> /// </summary> /// <param name="bi"> /// </param> /// <returns> /// </returns> public BigInteger max(BigInteger bi) { if (this > bi) { return new BigInteger(this); } else { return new BigInteger(bi); } }
// *********************************************************************** // Tests the correct implementation of the modulo exponential function // using RSA encryption and decryption (using pre-computed encryption and // decryption keys). // *********************************************************************** /// <summary> /// </summary> /// <param name="rounds"> /// </param> public static void RSATest(int rounds) { Random rand = new Random(1); byte[] val = new byte[64]; // private and public key BigInteger bi_e = new BigInteger( "a932b948feed4fb2b692609bd22164fc9edb59fae7880cc1eaff7b3c9626b7e5b241c27a974833b2622ebe09beb451917663d47232488f23a117fc97720f1e7", 16); BigInteger bi_d = new BigInteger( "4adf2f7a89da93248509347d2ae506d683dd3a16357e859a980c4f77a4e2f7a01fae289f13a851df6e9db5adaa60bfd2b162bbbe31f7c8f828261a6839311929d2cef4f864dde65e556ce43c89bbbf9f1ac5511315847ce9cc8dc92470a747b8792d6a83b0092d2e5ebaf852c85cacf34278efa99160f2f8aa7ee7214de07b7", 16); BigInteger bi_n = new BigInteger( "e8e77781f36a7b3188d711c2190b560f205a52391b3479cdb99fa010745cbeba5f2adc08e1de6bf38398a0487c4a73610d94ec36f17f3f46ad75e17bc1adfec99839589f45f95ccc94cb2a5c500b477eb3323d8cfab0c8458c96f0147a45d27e45a4d11d54d77684f65d48f15fafcc1ba208e71e921b9bd9017c16a5231af7f", 16); Console.WriteLine("e =\n" + bi_e.ToString(10)); Console.WriteLine("\nd =\n" + bi_d.ToString(10)); Console.WriteLine("\nn =\n" + bi_n.ToString(10) + "\n"); for (int count = 0; count < rounds; count++) { // generate data of random length int t1 = 0; while (t1 == 0) { t1 = (int)(rand.NextDouble() * 65); } bool done = false; while (!done) { for (int i = 0; i < 64; i++) { if (i < t1) { val[i] = (byte)(rand.NextDouble() * 256); } else { val[i] = 0; } if (val[i] != 0) { done = true; } } } while (val[0] == 0) { val[0] = (byte)(rand.NextDouble() * 256); } Console.Write("Round = " + count); // encrypt and decrypt data BigInteger bi_data = new BigInteger(val, t1); BigInteger bi_encrypted = bi_data.modPow(bi_e, bi_n); BigInteger bi_decrypted = bi_encrypted.modPow(bi_d, bi_n); // compare if (bi_decrypted != bi_data) { Console.WriteLine("\nError at round " + count); Console.WriteLine(bi_data + "\n"); return; } Console.WriteLine(" <PASSED>."); } }
// *********************************************************************** // Returns min(this, bi) // *********************************************************************** /// <summary> /// </summary> /// <param name="bi"> /// </param> /// <returns> /// </returns> public BigInteger min(BigInteger bi) { if (this < bi) { return new BigInteger(this); } else { return new BigInteger(bi); } }
// *********************************************************************** // Tests the correct implementation of sqrt() method. // *********************************************************************** /// <summary> /// </summary> /// <param name="rounds"> /// </param> public static void SqrtTest(int rounds) { Random rand = new Random(); for (int count = 0; count < rounds; count++) { // generate data of random length int t1 = 0; while (t1 == 0) { t1 = (int)(rand.NextDouble() * 1024); } Console.Write("Round = " + count); BigInteger a = new BigInteger(); a.genRandomBits(t1, rand); BigInteger b = a.sqrt(); BigInteger c = (b + 1) * (b + 1); // check that b is the largest integer such that b*b <= a if (c <= a) { Console.WriteLine("\nError at round " + count); Console.WriteLine(a + "\n"); return; } Console.WriteLine(" <PASSED>."); } }
// *********************************************************************** // Returns the lowest 4 bytes of the BigInteger as an int. // *********************************************************************** // *********************************************************************** // Returns the modulo inverse of this. Throws ArithmeticException if // the inverse does not exist. (i.e. gcd(this, modulus) != 1) // *********************************************************************** /// <summary> /// </summary> /// <param name="modulus"> /// </param> /// <returns> /// </returns> /// <exception cref="ArithmeticException"> /// </exception> public BigInteger modInverse(BigInteger modulus) { BigInteger[] p = { 0, 1 }; BigInteger[] q = new BigInteger[2]; // quotients BigInteger[] r = { 0, 0 }; // remainders int step = 0; BigInteger a = modulus; BigInteger b = this; while (b.dataLength > 1 || (b.dataLength == 1 && b.data[0] != 0)) { BigInteger quotient = new BigInteger(); BigInteger remainder = new BigInteger(); if (step > 1) { BigInteger pval = (p[0] - (p[1] * q[0])) % modulus; p[0] = p[1]; p[1] = pval; } if (b.dataLength == 1) { singleByteDivide(a, b, quotient, remainder); } else { multiByteDivide(a, b, quotient, remainder); } /* Console.WriteLine(quotient.dataLength); Console.WriteLine("{0} = {1}({2}) + {3} p = {4}", a.ToString(10), b.ToString(10), quotient.ToString(10), remainder.ToString(10), p[1].ToString(10)); */ q[0] = q[1]; r[0] = r[1]; q[1] = quotient; r[1] = remainder; a = b; b = remainder; step++; } if (r[0].dataLength > 1 || (r[0].dataLength == 1 && r[0].data[0] != 1)) { throw new ArithmeticException("No inverse!"); } BigInteger result = (p[0] - (p[1] * q[0])) % modulus; if ((result.data[maxLength - 1] & 0x80000000) != 0) { result += modulus; // get the least positive modulus } return result; }
/// <summary> /// </summary> /// <param name="exp"> /// </param> /// <param name="n"> /// </param> /// <returns> /// </returns> /// <exception cref="ArithmeticException"> /// </exception> public BigInteger modPow(BigInteger exp, BigInteger n) { if ((exp.data[maxLength - 1] & 0x80000000) != 0) { throw new ArithmeticException("Positive exponents only."); } BigInteger resultNum = 1; BigInteger tempNum; bool thisNegative = false; if ((this.data[maxLength - 1] & 0x80000000) != 0) { // negative this tempNum = -this % n; thisNegative = true; } else { tempNum = this % n; // ensures (tempNum * tempNum) < b^(2k) } if ((n.data[maxLength - 1] & 0x80000000) != 0) { // negative n n = -n; } // calculate constant = b^(2k) / m BigInteger constant = new BigInteger(); int i = n.dataLength << 1; constant.data[i] = 0x00000001; constant.dataLength = i + 1; constant = constant / n; int totalBits = exp.bitCount(); int count = 0; // perform squaring and multiply exponentiation for (int pos = 0; pos < exp.dataLength; pos++) { uint mask = 0x01; // Console.WriteLine("pos = " + pos); for (int index = 0; index < 32; index++) { if ((exp.data[pos] & mask) != 0) { resultNum = this.BarrettReduction(resultNum * tempNum, n, constant); } mask <<= 1; tempNum = this.BarrettReduction(tempNum * tempNum, n, constant); if (tempNum.dataLength == 1 && tempNum.data[0] == 1) { if (thisNegative && (exp.data[0] & 0x1) != 0) { // odd exp return -resultNum; } return resultNum; } count++; if (count == totalBits) { break; } } } if (thisNegative && (exp.data[0] & 0x1) != 0) { // odd exp return -resultNum; } return resultNum; }
// least significant bits at lower part of buffer // *********************************************************************** // Overloading of the NOT operator (1's complement) // *********************************************************************** /// <summary> /// </summary> /// <param name="bi1"> /// </param> /// <returns> /// </returns> public static BigInteger operator ~(BigInteger bi1) { BigInteger result = new BigInteger(bi1); for (int i = 0; i < maxLength; i++) { result.data[i] = ~bi1.data[i]; } result.dataLength = maxLength; while (result.dataLength > 1 && result.data[result.dataLength - 1] == 0) { result.dataLength--; } return result; }