Ejemplo n.º 1
0
		//***********************************************************************
		// Overloading of bitwise XOR operator
		//***********************************************************************
		/// <summary>
		/// Overloading of bitwise XOR operator
		/// </summary>
		/// <param name="bi1"></param>
		/// <param name="bi2"></param>
		/// <returns></returns>
		public static BigInteger operator ^(BigInteger bi1, BigInteger bi2)
		{
			BigInteger result = new BigInteger();

			int len = (bi1.dataLength > bi2.dataLength) ? bi1.dataLength : bi2.dataLength;

			for (int i = 0; i < len; i++)
			{
				uint sum = (uint)(bi1.data[i] ^ bi2.data[i]);
				result.data[i] = sum;
			}

			result.dataLength = maxLength;

			while (result.dataLength > 1 && result.data[result.dataLength - 1] == 0)
				result.dataLength--;

			return result;
		}
Ejemplo n.º 2
0
		//***********************************************************************
		// Overloading of unary << operators
		//***********************************************************************
		/// <summary>
		/// Overloading of unary operators
		/// </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 = shiftLeft(result.data, shiftVal);

			return result;
		}
Ejemplo n.º 3
0
		//***********************************************************************
		// Overloading of the NOT operator (1's complement)
		//***********************************************************************
		/// <summary>
		/// Overloading of the NOT operator (1's complement)
		/// </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] = (uint)(~(bi1.data[i]));

			result.dataLength = maxLength;

			while (result.dataLength > 1 && result.data[result.dataLength - 1] == 0)
				result.dataLength--;

			return result;
		}
Ejemplo n.º 4
0
		//***********************************************************************
		// Overloading of the unary ++ operator
		//***********************************************************************
		/// <summary>
		/// Overloading of the unary ++ operator
		/// </summary>
		/// <param name="bi1"></param>
		/// <returns></returns>
		public static BigInteger operator ++(BigInteger bi1)
		{
			BigInteger result = new BigInteger(bi1);

			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 ((bi1.data[lastPos] & 0x80000000) == 0 &&
			   (result.data[lastPos] & 0x80000000) != (bi1.data[lastPos] & 0x80000000))
			{
				throw (new ArithmeticException("Overflow in ++."));
			}
			return result;
		}
Ejemplo n.º 5
0
		//***********************************************************************
		// Overloading of the unary -- operator
		//***********************************************************************
		/// <summary>
		/// Overloading of the unary -- operator
		/// </summary>
		/// <param name="bi1"></param>
		/// <returns></returns>
		public static BigInteger operator --(BigInteger bi1)
		{
			BigInteger result = new BigInteger(bi1);

			long val;
			bool carryIn = true;
			int index = 0;

			while (carryIn && index < maxLength)
			{
				val = (long)(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 ((bi1.data[lastPos] & 0x80000000) != 0 &&
			   (result.data[lastPos] & 0x80000000) != (bi1.data[lastPos] & 0x80000000))
			{
				throw (new ArithmeticException("Underflow in --."));
			}

			return result;
		}
Ejemplo n.º 6
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
		//
		//***********************************************************************

		internal 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;
		}
Ejemplo n.º 7
0
		//***********************************************************************
		// 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)
		{
			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;
		}
Ejemplo n.º 8
0
		//***********************************************************************
		// Fast calculation of modular reduction using Barrett's reduction.
		// Requires left < b^(2k), where b is the base.  In this case, base is
		// 2^32 (uint).
		//
		// Reference [4]
		//***********************************************************************

		private BigInteger BarrettReduction(BigInteger x, BigInteger n, BigInteger constant)
		{
			int k = n.dataLength,
				kPlusOne = k + 1,
				kMinusOne = k - 1;

			BigInteger q1 = new BigInteger();

			// q1 = left / 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 = left 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 = ((ulong)q3.data[i] * (ulong)n.data[j]) +
								 (ulong)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;
		}
Ejemplo n.º 9
0
		//***********************************************************************
		// Returns gcd(this, bi)
		//***********************************************************************

		internal BigInteger gcd(BigInteger bi)
		{
			BigInteger x;
			BigInteger y;

			if ((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;
		}
Ejemplo n.º 10
0
		//***********************************************************************
		// Constructor (Default value provided by BigInteger)
		//***********************************************************************
		/// <summary>
		/// Constructor (Default value provided by BigInteger)
		/// </summary>
		/// <param name="bi"></param>
		public BigInteger(BigInteger bi)
		{
			data = new uint[maxLength];

			dataLength = bi.dataLength;

			for (int i = 0; i < dataLength; i++)
				data[i] = bi.data[i];
		}
Ejemplo n.º 11
0
		//***********************************************************************
		// Modulo Exponentiation
		//***********************************************************************

		internal BigInteger modPow(BigInteger exp, BigInteger n)
		{
			if ((exp.data[maxLength - 1] & 0x80000000) != 0)
				throw (new ArithmeticException("Exponent must be positive."));

			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;

				for (int index = 0; index < 32; index++)
				{
					if ((exp.data[pos] & mask) != 0)
						resultNum = BarrettReduction(resultNum * tempNum, n, constant);

					mask <<= 1;

					tempNum = 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;
		}
Ejemplo n.º 12
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"
		//
		//***********************************************************************

		internal string ToString(int radix)
		{
			if (radix < 2 || radix > 36)
				throw (new ArgumentException("Radix must be >= 2 and <= 36"));

			string charSet = "ABCDEFGHIJKLMNOPQRSTUVWXYZ";
			string result = "";

			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;
		}
Ejemplo n.º 13
0
		//***********************************************************************
		// Returns min(this, bi)
		//***********************************************************************

		internal BigInteger min(BigInteger bi)
		{
			if (this < bi)
				return (new BigInteger(this));
			else
				return (new BigInteger(bi));

		}
Ejemplo n.º 14
0
		//***********************************************************************
		// Returns max(this, bi)
		//***********************************************************************

		internal BigInteger max(BigInteger bi)
		{
			if (this > bi)
				return (new BigInteger(this));
			else
				return (new BigInteger(bi));
		}
Ejemplo n.º 15
0
		//***********************************************************************
		// Generates a random number with the specified number of bits such
		// that gcd(number, this) = 1
		//***********************************************************************

		internal BigInteger genCoPrime(int bits, Random rand)
		{
			bool done = false;
			BigInteger result = new BigInteger();

			while (!done)
			{
				result.genRandomBits(bits, rand);

				// gcd test
				BigInteger g = result.gcd(this);
				if (g.dataLength == 1 && g.data[0] == 1)
					done = true;
			}

			return result;
		}
Ejemplo n.º 16
0
		//***********************************************************************
		// Probabilistic prime test based on Fermat's little theorem
		//
		// for any a < p (p does not divide a) if
		//      a^(p-1) mod p != 1 then p is not prime.
		//
		// Otherwise, p is probably prime (pseudoprime to the chosen base).
		//
		// Returns
		// -------
		// True if "this" is a pseudoprime to randomly chosen
		// bases.  The number of chosen bases is given by the "confidence"
		// parameter.
		//
		// False if "this" is definitely NOT prime.
		//
		// Note - this method is fast but fails for Carmichael numbers except
		// when the randomly chosen base is a factor of the number.
		//
		//***********************************************************************

		internal 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))
				{
					return false;
				}
			}

			return true;
		}
Ejemplo n.º 17
0
		//***********************************************************************
		// Returns the modulo inverse of this.  Throws ArithmeticException if
		// the inverse does not exist.  (i.e. gcd(this, modulus) != 1)
		//***********************************************************************

		internal 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);

				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;
		}
Ejemplo n.º 18
0
		//***********************************************************************
		// 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.
		//
		//***********************************************************************

		internal 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);

				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;
		}
Ejemplo n.º 19
0
		//***********************************************************************
		// 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
		//***********************************************************************

		internal 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);
		}
Ejemplo n.º 20
0
		//***********************************************************************
		// 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>
		/// Constructor (Default value provided by a string of digits of the specified base)
		/// </summary>
		/// <param name="value"></param>
		/// <param name="radix"></param>
		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 = (int)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."));
			}

			data = new uint[maxLength];
			for (int i = 0; i < result.dataLength; i++)
				data[i] = result.data[i];

			dataLength = result.dataLength;
		}
Ejemplo n.º 21
0
		//***********************************************************************
		// Overloading of addition operator
		//***********************************************************************
		/// <summary>
		/// Overloading of addition operator
		/// </summary>
		/// <param name="bi1"></param>
		/// <param name="bi2"></param>
		/// <returns></returns>
		public static BigInteger operator +(BigInteger bi1, BigInteger bi2)
		{
			BigInteger result = new BigInteger();

			result.dataLength = (bi1.dataLength > bi2.dataLength) ? bi1.dataLength : bi2.dataLength;

			long carry = 0;
			for (int i = 0; i < result.dataLength; i++)
			{
				long sum = (long)bi1.data[i] + (long)bi2.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 ((bi1.data[lastPos] & 0x80000000) == (bi2.data[lastPos] & 0x80000000) &&
			   (result.data[lastPos] & 0x80000000) != (bi1.data[lastPos] & 0x80000000))
			{
				throw (new ArithmeticException());
			}

			return result;
		}
Ejemplo n.º 22
0
		//***********************************************************************
		// 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.
		//
		//***********************************************************************

		internal 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);

				// if they are different then it is not prime
				if (expResult != jacob)
					return false;
			}

			return true;
		}
Ejemplo n.º 23
0
		//***********************************************************************
		// Overloading of subtraction operator
		//***********************************************************************
		/// <summary>
		/// Overloading of subtraction operator
		/// </summary>
		/// <param name="bi1"></param>
		/// <param name="bi2"></param>
		/// <returns></returns>
		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 = (long)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;
		}
Ejemplo n.º 24
0
		internal 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 = BigInteger.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;

			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;

					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 * BigInteger.Jacobi(Q, thisVal)) % thisVal;
					if ((temp.data[maxLength - 1] & 0x80000000) != 0)
						temp += thisVal;

					if (lucas[2] != temp)
						isPrime = false;
				}
			}

			return isPrime;
		}
Ejemplo n.º 25
0
		//***********************************************************************
		// Overloading of multiplication operator
		//***********************************************************************
		/// <summary>
		/// Overloading of multiplication operator
		/// </summary>
		/// <param name="bi1"></param>
		/// <param name="bi2"></param>
		/// <returns></returns>
		public static BigInteger operator *(BigInteger bi1, BigInteger bi2)
		{
			int lastPos = maxLength - 1;
			bool bi1Neg = false, bi2Neg = false;

			// take the absolute value of the inputs
			try
			{
				if ((bi1.data[lastPos] & 0x80000000) != 0)     // bi1 negative
				{
					bi1Neg = true; bi1 = -bi1;
				}
				if ((bi2.data[lastPos] & 0x80000000) != 0)     // bi2 negative
				{
					bi2Neg = true; bi2 = -bi2;
				}
			}
			catch (Exception) { }

			BigInteger result = new BigInteger();

			// multiply the absolute values
			try
			{
				for (int i = 0; i < bi1.dataLength; i++)
				{
					if (bi1.data[i] == 0) continue;

					ulong mcarry = 0;
					for (int j = 0, k = i; j < bi2.dataLength; j++, k++)
					{
						// k = i + j
						ulong val = ((ulong)bi1.data[i] * (ulong)bi2.data[j]) +
									 (ulong)result.data[k] + mcarry;

						result.data[k] = (uint)(val & 0xFFFFFFFF);
						mcarry = (val >> 32);
					}

					if (mcarry != 0)
						result.data[i + bi2.dataLength] = (uint)mcarry;
				}
			}
			catch (Exception)
			{
				throw (new ArithmeticException("Multiplication overflow."));
			}


			result.dataLength = bi1.dataLength + bi2.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 (bi1Neg != bi2Neg && 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 (bi1Neg != bi2Neg)
				return -result;

			return result;
		}
Ejemplo n.º 26
0
		//***********************************************************************
		// Computes the Jacobi Symbol for a and b.
		// Algorithm adapted from [3] and [4] with some optimizations
		//***********************************************************************

		internal 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));
		}
Ejemplo n.º 27
0
		//***********************************************************************
		// Overloading of unary >> operators
		//***********************************************************************
		/// <summary>
		/// Overloading of unary >> operators
		/// </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;
		}
Ejemplo n.º 28
0
		//***********************************************************************
		// Generates a positive BigInteger that is probably prime.
		// Overloaded to use the isProbablePrime method with no confidence value
		//***********************************************************************

		internal static BigInteger genPseudoPrime(int bits, 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();
			}
			return result;
		}
Ejemplo n.º 29
0
		//***********************************************************************
		// Overloading of the NEGATE operator (2's complement)
		//***********************************************************************
		/// <summary>
		/// Overloading of the NEGATE operator (2's complement)
		/// </summary>
		/// <param name="bi1"></param>
		/// <returns></returns>
		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] = (uint)(~(bi1.data[i]));

			// add one to result of 1's complement
			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 ((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;
		}
Ejemplo n.º 30
0
		//***********************************************************************
		// Overloading of modulus operator
		//***********************************************************************
		/// <summary>
		/// Overloading of modulus operator
		/// </summary>
		/// <param name="bi1"></param>
		/// <param name="bi2"></param>
		/// <returns></returns>
		public static BigInteger operator %(BigInteger bi1, BigInteger bi2)
		{
			BigInteger quotient = new BigInteger();
			BigInteger remainder = new BigInteger(bi1);

			int lastPos = maxLength - 1;
			bool dividendNeg = false;

			if ((bi1.data[lastPos] & 0x80000000) != 0)     // bi1 negative
			{
				bi1 = -bi1;
				dividendNeg = true;
			}
			if ((bi2.data[lastPos] & 0x80000000) != 0)     // bi2 negative
				bi2 = -bi2;

			if (bi1 < bi2)
			{
				return remainder;
			}

			else
			{
				if (bi2.dataLength == 1)
					singleByteDivide(bi1, bi2, quotient, remainder);
				else
					multiByteDivide(bi1, bi2, quotient, remainder);

				if (dividendNeg)
					return -remainder;

				return remainder;
			}
		}