public static void RunBigInteger() { BigInteger value = new BigInteger("1234DEADBEEF", 16); const string charSet = "ABCDEFGHIJKLMNOP"; string valueInLetters = value.ToString(charSet); Console.WriteLine("Value converted to base 16 using letters:\n{0}\n", valueInLetters); BigInteger valueFromLetters = new BigInteger(valueInLetters, charSet); Console.WriteLine("Value parsed from base 16 using letters:\n{0}\n", valueFromLetters.ToString(16)); }
public void PropertiesShouldBeLoadedCorrectly(string SectionName, string expectedModulus, string expectedExponent) { BigInteger mod = new BigInteger(expectedModulus, 16); BigInteger exp = new BigInteger(expectedExponent, 16); RsaSmallPublicKeySection config = (RsaSmallPublicKeySection)ConfigurationManager.GetSection(SectionName); Assert.IsNotNull(config, "Config is null."); Assert.IsNotNull(config.Value,"Value is null."); Assert.IsNotNull(config.Value.Modulus, "Modulus is null."); Assert.IsNotNull(config.Value.Exponent, "Exponent is null."); Assert.AreEqual(mod, config.Value.Modulus, "Modulus not loaded correctly."); Assert.AreEqual(exp, config.Value.Exponent, "Exponent not loaded correctly."); }
//*********************************************************************** // 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> /// 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. /// </summary> /// <param name="confidence">The confidence.</param> /// <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. /// </returns> public bool SolovayStrassenTest(int confidence) { unchecked { BigInteger thisVal; if ((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.GenerateRandomBits(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; } }
private Exception privateParse(string code, out BigInteger result) { result = null; if (code == null) return new ArgumentNullException("code"); string cleanCode = code.Trim(); cleanCode = cleanCode.ToUpperInvariant(); if (!Regex.IsMatch(cleanCode, validCodeRegex)) return new FormatException("Invalid code format"); string tranCode = translator.Translate(cleanCode); result = rsa.Decrypt(new BigInteger(tranCode, radix)); return null; }
/// <summary> /// Implements the operator *. /// </summary> /// <param name="operand">The first operand.</param> /// <param name="secondOperand">The second operand.</param> /// <returns>The result of the operator.</returns> public static BigInteger operator *(BigInteger operand, BigInteger secondOperand) { unchecked { if ((object)operand == null) throw new ArgumentNullException("operand"); if ((object)secondOperand == null) throw new ArgumentNullException("secondOperand"); int lastPos = maxLength - 1; bool operandNeg = false, secondOperandNeg = false; // take the absolute value of the inputs //try //{ if ((operand.data[lastPos] & 0x80000000) != 0) // operand negative { operandNeg = true; operand = -operand; } if ((secondOperand.data[lastPos] & 0x80000000) != 0) // secondOperand negative { secondOperandNeg = true; secondOperand = -secondOperand; } //} //catch (ApplicationException) //{ //} BigInteger result = new BigInteger(); // multiply the absolute values for (int i = 0; i < operand.dataLength; i++) { if (operand.data[i] == 0) continue; ulong mcarry = 0; for (int j = 0, k = i; j < secondOperand.dataLength; j++, k++) { // k = i + j if (k > maxLength) { throw new ArithmeticException("Multiplication overflow."); } ulong val = (operand.data[i] * (ulong)secondOperand.data[j]) + result.data[k] + mcarry; result.data[k] = (uint)(val & 0xFFFFFFFF); mcarry = (val >> 32); } if (mcarry != 0) result.data[i + secondOperand.dataLength] = (uint)mcarry; } result.dataLength = operand.dataLength + secondOperand.dataLength; if (result.dataLength > maxLength) result.dataLength = maxLength; while (result.dataLength > 1 && result.data[result.dataLength - 1] == 0) result.dataLength--; // overflow check (result is -ve) if ((result.data[lastPos] & 0x80000000) != 0) { if (operandNeg != secondOperandNeg && result.data[lastPos] == 0x80000000) // different sign { // handle the special case where multiplication produces // a max negative number in 2's complement. if (result.dataLength == 1) return result; else { bool isMaxNeg = true; for (int i = 0; i < result.dataLength - 1 && isMaxNeg; i++) { if (result.data[i] != 0) isMaxNeg = false; } if (isMaxNeg) return result; } } throw (new ArithmeticException("Multiplication overflow.")); } // if input has different signs, then result is -ve if (operandNeg != secondOperandNeg) return -result; return result; } }
/// <summary> /// Implements the operator --. /// </summary> /// <param name="operand">The operand.</param> /// <returns>The result of the operator.</returns> public static BigInteger operator --(BigInteger operand) { unchecked { if ((object)operand == null) throw new ArgumentNullException("operand"); BigInteger result = new BigInteger(operand); long val; bool carryIn = true; int index = 0; while (carryIn && index < maxLength) { val = (result.data[index]); val--; result.data[index] = (uint)(val & 0xFFFFFFFF); if (val >= 0) carryIn = false; index++; } if (index > result.dataLength) result.dataLength = index; while (result.dataLength > 1 && result.data[result.dataLength - 1] == 0) result.dataLength--; // overflow check int lastPos = maxLength - 1; // overflow if initial value was -ve but -- caused a sign // change to positive. if ((operand.data[lastPos] & 0x80000000) != 0 && (result.data[lastPos] & 0x80000000) != (operand.data[lastPos] & 0x80000000)) throw (new ArithmeticException("Underflow in --.")); return result; } }
//*********************************************************************** // Overloading of subtraction operator //*********************************************************************** /// <summary> /// Subtracts the specified operand from this. /// </summary> /// <param name="operand">The operand.</param> /// <returns></returns> public BigInteger Subtract(BigInteger operand) { return this - operand; }
/// <summary> /// Implements the operator +. /// </summary> /// <param name="operand">The first operand.</param> /// <param name="secondOperand">The second operand.</param> /// <returns>The result of the operator.</returns> public static BigInteger operator +(BigInteger operand, BigInteger secondOperand) { unchecked { if ((object)operand == null) return null; if ((object)secondOperand == null) return null; BigInteger result = new BigInteger(); result.dataLength = (operand.dataLength > secondOperand.dataLength) ? operand.dataLength : secondOperand.dataLength; long carry = 0; for (int i = 0; i < result.dataLength; i++) { long sum = operand.data[i] + (long)secondOperand.data[i] + carry; carry = sum >> 32; result.data[i] = (uint)(sum & 0xFFFFFFFF); } if (carry != 0 && result.dataLength < maxLength) { result.data[result.dataLength] = (uint)(carry); result.dataLength++; } while (result.dataLength > 1 && result.data[result.dataLength - 1] == 0) result.dataLength--; // overflow check int lastPos = maxLength - 1; if ((operand.data[lastPos] & 0x80000000) == (secondOperand.data[lastPos] & 0x80000000) && (result.data[lastPos] & 0x80000000) != (operand.data[lastPos] & 0x80000000)) throw (new ArithmeticException()); return result; } }
//*********************************************************************** // 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> /// Returns a value that is equivalent to the integer square root /// of the BigInteger. /// </summary> /// <returns></returns> public BigInteger Sqrt() { unchecked { uint numBits = (uint)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 modulo inverse of this. Throws ArithmeticException if // the inverse does not exist. (i.e. gcd(this, modulus) != 1) //*********************************************************************** /// <summary> /// Returns the modulo inverse of this. Throws ArithmeticException if /// the inverse does not exist. (i.e. gcd(this, modulus) != 1) /// </summary> /// <param name="modulus">The modulus.</param> /// <returns></returns> public BigInteger ModInverse(BigInteger modulus) { unchecked { 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; } }
//*********************************************************************** // Generates a random number with the specified number of bits such // that gcd(number, this) = 1 //*********************************************************************** /// <summary> /// Generates a random number with the specified number of bits such /// that gcd(number, this) = 1 /// </summary> /// <param name="bits">The bits.</param> /// <param name="rand">The rand.</param> /// <returns></returns> public BigInteger GenerateCoprime(int bits, Random rand) { unchecked { bool done = false; BigInteger result = new BigInteger(); while (!done) { result.GenerateRandomBits(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; } }
//*********************************************************************** // Generates a positive BigInteger that is probably prime. //*********************************************************************** /// <summary> /// Generates a positive BigInteger that is probably prime. /// </summary> /// <param name="bits">The bits.</param> /// <param name="confidence">The confidence.</param> /// <param name="rand">The random number generator.</param> /// <returns></returns> public static BigInteger GeneratePseudoPrime(int bits, int confidence, Random rand) { unchecked { BigInteger result = new BigInteger(); bool done = false; while (!done) { result.GenerateRandomBits(bits, rand); result.data[0] |= 0x01; // make it odd // prime test done = result.IsProbablePrime(confidence); } return result; } }
//*********************************************************************** // Computes the Jacobi Symbol for a and b. // Algorithm adapted from [3] and [4] with some optimizations //*********************************************************************** /// <summary> /// Computes the Jacobi Symbol for a and b. /// </summary> /// <param name="a">a.</param> /// <param name="b">b.</param> /// <returns></returns> public static int Jacobi(BigInteger a, BigInteger b) { unchecked { if ((object)a == null) throw new ArgumentNullException("a"); if ((object)b == null) throw new ArgumentNullException("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)); } }
//*********************************************************************** // Constructor (Default value provided by BigInteger) //*********************************************************************** /// <summary> /// Initializes a new instance of the <see cref="BigInteger"/> class. /// </summary> /// <param name="bi">The bi.</param> public BigInteger(BigInteger bi) { unchecked { if ((object)bi == null) throw new ArgumentNullException("bi"); data = new uint[maxLength]; dataLength = bi.dataLength; for (int i = 0; i < dataLength; i++) data[i] = bi.data[i]; } }
private static bool LucasStrongTestHelper(BigInteger thisVal) { unchecked { // 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] = 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] = 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; } }
internal static void Main(string[] args) { // Known problem -> these two pseudoprimes passes my implementation of // primality test but failed in JDK's isProbablePrime test. byte[] pseudoPrime1 = { 0x00, 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 = { // 0x00, // 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, // }; Console.WriteLine("List of primes < 2000\n---------------------"); int limit = 100, count = 0; for (int i = 0; i < 2000; i++) { if (i >= limit) { Console.WriteLine(); limit += 100; } BigInteger p = new BigInteger(-i); if (p.IsProbablePrime()) { Console.Write(i + ", "); count++; } } Console.WriteLine("\nCount = " + count); BigInteger operand = new BigInteger(pseudoPrime1); Console.WriteLine("\n\nPrimality testing for\n" + operand + "\n"); Console.WriteLine("SolovayStrassenTest(5) = " + operand.SolovayStrassenTest(5)); Console.WriteLine("RabinMillerTest(5) = " + operand.RabinMillerTest(5)); Console.WriteLine("FermatLittleTest(5) = " + operand.FermatLittleTest(5)); Console.WriteLine("isProbablePrime() = " + operand.IsProbablePrime()); Console.Write("\nGenerating 512-bits random pseudoprime. . ."); Random rand = new Random(); BigInteger prime = GeneratePseudoPrime(512, 5, rand); Console.WriteLine("\n" + prime); //int dwStart = System.Environment.TickCount; //BigInteger.MulDivTest(100000); //BigInteger.RSATest(10); //BigInteger.RSATest2(10); //Console.WriteLine(System.Environment.TickCount - dwStart); }
//*********************************************************************** // Overloading of addition operator //*********************************************************************** /// <summary> /// Adds the specified operands. /// </summary> /// <param name="operand">The operand.</param> /// <returns></returns> public BigInteger Add(BigInteger operand) { return this + operand; }
//*********************************************************************** // 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> /// Initializes a new instance of the <see cref="BigInteger"/> class. /// </summary> /// <param name="value">The value.</param> /// <param name="charSet">The character set used to represent the value.</param> public BigInteger(string value, string charSet) { int radix = charSet.Length; unchecked { if (value == null) throw new ArgumentNullException("value"); BigInteger multiplier = new BigInteger(1); BigInteger result = new BigInteger(); value = (value.ToUpperInvariant()).Trim(); int limit = 0; if (value[0] == '-') limit = 1; for (int i = value.Length - 1; i >= limit; i--) { int posVal = charSet.IndexOf(value[i]); if (posVal < 0) 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.")); } data = new uint[maxLength]; for (int i = 0; i < result.dataLength; i++) data[i] = result.data[i]; dataLength = result.dataLength; } }
/// <summary> /// Implements the operator ++. /// </summary> /// <param name="operand">The operand.</param> /// <returns>The result of the operator.</returns> public static BigInteger operator ++(BigInteger operand) { unchecked { if ((object)operand == null) throw new ArgumentNullException("operand"); BigInteger result = new BigInteger(operand); long val, carry = 1; int index = 0; while (carry != 0 && index < maxLength) { val = (long)(result.data[index]); val++; result.data[index] = (uint)(val & 0xFFFFFFFF); carry = val >> 32; index++; } if (index > result.dataLength) result.dataLength = index; else { while (result.dataLength > 1 && result.data[result.dataLength - 1] == 0) result.dataLength--; } // overflow check int lastPos = maxLength - 1; // overflow if initial value was +ve but ++ caused a sign // change to negative. if ((operand.data[lastPos] & 0x80000000) == 0 && (result.data[lastPos] & 0x80000000) != (operand.data[lastPos] & 0x80000000)) throw (new ArithmeticException("Overflow in ++.")); 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 //*********************************************************************** /// <summary> /// Returns the k_th number in the Lucas Sequence reduced modulo n. /// </summary> /// <param name="P">P.</param> /// <param name="Q">Q.</param> /// <param name="k">k.</param> /// <param name="n">n.</param> /// <returns></returns> public static BigInteger[] LucasSequence(BigInteger P, BigInteger Q, BigInteger k, BigInteger n) { unchecked { if ((object)P == null) throw new ArgumentNullException("P"); if ((object)Q == null) throw new ArgumentNullException("Q"); if ((object)k == null) throw new ArgumentNullException("k"); if ((object)n == null) throw new ArgumentNullException("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> /// Implements the operator -. /// </summary> /// <param name="operand">The first operand.</param> /// <param name="secondOperand">The second operand.</param> /// <returns>The result of the operator.</returns> public static BigInteger operator -(BigInteger operand, BigInteger secondOperand) { unchecked { if ((object)operand == null) throw new ArgumentNullException("operand"); if ((object)secondOperand == null) throw new ArgumentNullException("secondOperand"); BigInteger result = new BigInteger(); result.dataLength = (operand.dataLength > secondOperand.dataLength) ? operand.dataLength : secondOperand.dataLength; long carryIn = 0; for (int i = 0; i < result.dataLength; i++) { long diff; diff = operand.data[i] - (long)secondOperand.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 ((operand.data[lastPos] & 0x80000000) != (secondOperand.data[lastPos] & 0x80000000) && (result.data[lastPos] & 0x80000000) != (operand.data[lastPos] & 0x80000000)) throw (new ArithmeticException()); return result; } }
//*********************************************************************** // 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 //*********************************************************************** private static BigInteger[] LucasSequenceHelper(BigInteger P, BigInteger Q, BigInteger k, BigInteger n, BigInteger constant, int s) { unchecked { 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 = BarrettReduction(v1 * v1, n, constant); v1 = (v1 - ((Q_k * Q) << 1)) % n; if (flag) flag = false; else Q_k = 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 = BarrettReduction(v * v, n, constant); v = (v - (Q_k << 1)) % n; if (flag) { Q_k = Q % n; flag = false; } else Q_k = 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 = 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 = BarrettReduction(Q_k * Q_k, n, constant); } result[0] = u1; result[1] = v; result[2] = Q_k; return result; } }
//*********************************************************************** // Overloading of multiplication operator //*********************************************************************** /// <summary> /// Multiplies this instance with the specified operand. /// </summary> /// <param name="operand">The operand.</param> /// <returns>The result of the operator.</returns> public BigInteger Multiply(BigInteger operand) { return this*operand; }
//*********************************************************************** // Tests the correct implementation of the /, %, * and + operators //*********************************************************************** internal 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; } } }
//*********************************************************************** // Tests the correct implementation of the modulo exponential function // using RSA encryption and decryption (using pre-computed encryption and // decryption keys). //*********************************************************************** internal 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>."); } }
//*********************************************************************** // 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. //*********************************************************************** internal 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.GenerateCoprime(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> /// Tries to parse the code and returns true if successful. The parsed value is passed in result. /// </summary> /// <param name="code">The code.</param> /// <param name="result">The result.</param> /// <returns></returns> public bool TryParse(string code, out BigInteger result) { Exception e = privateParse(code, out result); return e == null; }
//*********************************************************************** // Tests the correct implementation of sqrt() method. //*********************************************************************** internal 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.GenerateRandomBits(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>."); } }
/// <summary> /// Initializes a new instance of the <see cref="RsaSmallFullKey"/> class. /// </summary> /// <param name="modulus">The modulus.</param> /// <param name="privateExponent">The private exponent.</param> /// <param name="publicExponent">The public exponent.</param> public RsaSmallFullKey(BigInteger modulus, BigInteger privateExponent, BigInteger publicExponent) { privateKey = new RsaSmallPrivateKey(modulus, privateExponent); publicKey = new RsaSmallPublicKey(modulus, publicExponent); }
//*********************************************************************** // Probabilistic prime test based on Rabin-Miller's // // for any p > 0 with p - 1 = 2^s * t // // p is probably prime (strong pseudoprime) if for any a < p, // 1) a^t mod p = 1 or // 2) a^((2^j)*t) mod p = p-1 for some 0 <= j <= s-1 // // Otherwise, p is composite. // // Returns // ------- // True if "this" is a strong pseudoprime to randomly chosen // bases. The number of chosen bases is given by the "confidence" // parameter. // // False if "this" is definitely NOT prime. // //*********************************************************************** /// <summary> /// Probabilistic prime test based on Rabin-Miller's /// /// for any p > 0 with p - 1 = 2^s * t /// /// p is probably prime (strong pseudoprime) if for any a < p, /// 1) a^t mod p = 1 or /// 2) a^((2^j)*t) mod p = p-1 for some 0 <= j <= s-1 /// /// Otherwise, p is composite. /// </summary> /// <param name="confidence">The confidence.</param> /// <returns> /// True if "this" is a strong pseudoprime to randomly chosen /// bases. The number of chosen bases is given by the "confidence" /// parameter. /// /// False if "this" is definitely NOT prime. /// </returns> public bool RabinMillerTest(int confidence) { unchecked { BigInteger thisVal; if ((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.GenerateRandomBits(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; } }