예제 #1
0
 /**
  *
  */
 private void generateKeyPair()
 {
     BigInteger p, g, x, y;
     int qLength = strength - 1;
     p = parameter.P;
     g = parameter.G;
     //
     // berechne den private key
     //
     x = new BigInteger();
     x.genRandomBits(qLength, random);
     //
     // berechne den public key.
     //
     y = g.modPow(x, p);
     this.publicKey = new PublicKey(y, parameter);
     this.privateKey = new PrivateKey(x, parameter);
 }
예제 #2
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"
	//
	//***********************************************************************

	public 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;
	}
예제 #3
0
 /**
  *
  * <param name="x"/>
  * <param name="params"/>
  */
 public PrivateKey(BigInteger x, Parameter parameter)
     : base(true, parameter)
 {
     this.x = x;
 }
예제 #4
0
	//***********************************************************************
	// Overloading of the NEGATE operator (2's complement)
	//***********************************************************************

	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;
	}
예제 #5
0
	//***********************************************************************
	// Overloading of unary >> operators
	//***********************************************************************

	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;
	}
예제 #6
0
	//***********************************************************************
	// Overloading of multiplication operator
	//***********************************************************************

	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;
	}
예제 #7
0
	//***********************************************************************
	// Overloading of subtraction operator
	//***********************************************************************

	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;
	}
예제 #8
0
	//***********************************************************************
	// Overloading of addition operator
	//***********************************************************************

	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;
	}
예제 #9
0
	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 = 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;

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

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

		return isPrime;
	}
예제 #10
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.
	//
	//***********************************************************************

	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;
	}
예제 #11
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.
	//
	//***********************************************************************

	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;
	}
예제 #12
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.
	//
	//***********************************************************************

	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;
	}
예제 #13
0
	//***********************************************************************
	// Returns gcd(this, bi)
	//***********************************************************************

	public 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;
	}
예제 #14
0
	//***********************************************************************
	// Fast calculation of modular reduction using Barrett's reduction.
	// Requires x < 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 = 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 = ((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;
	}
예제 #15
0
	//***********************************************************************
	// Modulo Exponentiation
	//***********************************************************************

	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 = 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;
	}
예제 #16
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	//***********************************************************************
	// 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;
	}
예제 #17
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	//***********************************************************************
	// 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.
	//
	//***********************************************************************

	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;
	}
예제 #18
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	//***********************************************************************
	// Computes the Jacobi Symbol for a and b.
	// Algorithm adapted from [3] and [4] with some optimizations
	//***********************************************************************

	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));
	}
예제 #19
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	//***********************************************************************
	// Overloading of the unary ++ operator
	//***********************************************************************

	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;
	}
예제 #20
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	//***********************************************************************
	// Constructor (Default value provided by BigInteger)
	//***********************************************************************

	public BigInteger(BigInteger bi)
	{
		data = new uint[maxLength];

		dataLength = bi.dataLength;

		for(int i = 0; i < dataLength; i++)
			data[i] = bi.data[i];
	}
예제 #21
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	//***********************************************************************
	// Overloading of the unary -- operator
	//***********************************************************************

	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;
	}
예제 #22
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	//***********************************************************************
	// Generates a positive BigInteger that is probably prime.
	//***********************************************************************

	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;
	}
예제 #23
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	//***********************************************************************
	// Overloading of unary << operators
	//***********************************************************************

	public static BigInteger operator <<(BigInteger bi1, int shiftVal)
	{
		BigInteger result = new BigInteger(bi1);
		result.dataLength = shiftLeft(result.data, shiftVal);

		return result;
	}
예제 #24
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	//***********************************************************************
	// Generates a random number with the specified number of bits such
	// that gcd(number, this) = 1
	//***********************************************************************

	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;
	}
예제 #25
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	//***********************************************************************
	// Overloading of the NOT operator (1's complement)
	//***********************************************************************

	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;
	}
예제 #26
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	//***********************************************************************
	// Returns the modulo inverse of this.  Throws ArithmeticException if
	// the inverse does not exist.  (i.e. gcd(this, modulus) != 1)
	//***********************************************************************

	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;
	}
예제 #27
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	//***********************************************************************
	// 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
	//
	//***********************************************************************

	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;
	}
예제 #28
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
	//***********************************************************************

	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);
	}
예제 #29
0
 /**
  *
  * <param name="y"/>
  * <param name="params"/>
  */
 public PublicKey(BigInteger y, Parameter parameter)
     : base(false, parameter)
 {
     this.y = y;
 }
예제 #30
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	//***********************************************************************
	// Returns min(this, bi)
	//***********************************************************************

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

	}