biginteger.cs

JAVA实现的RSA公钥加密方法

		//
		//***********************************************************************

		public int bitCount()
		{
			while(dataLength > 1 && data[dataLength-1] == 0)
				dataLength--;

			uint value = data[dataLength - 1];
			uint mask = 0x80000000;
			int bits = 32;

			while(bits > 0 && (value & mask) == 0)
			{
				bits--;
				mask >>= 1;
			}
			bits += ((dataLength - 1) << 5);

			return bits;
		}


		//***********************************************************************
		// 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;
		}


		//***********************************************************************
		// 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;
		}


		//***********************************************************************
		// 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;
		}


		//***********************************************************************
		// Implementation of the Lucas Strong Pseudo Prime test.
		//
		// Let n be an odd number with gcd(n,D) = 1, and n - J(D, n) = 2^s * d
		// with d odd and s >= 0.
		//
		// If Ud mod n = 0 or V2^r*d mod n = 0 for some 0 <= r < s, then n
		// is a strong Lucas pseudoprime with parameters (P, Q).  We select
		// P and Q based on Selfridge.
		//
		// Returns True if number is a strong Lucus pseudo prime.
		// Otherwise, returns False indicating that number is composite.
		//***********************************************************************

		public bool LucasStrongTest()
		{
			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;

			return LucasStrongTestHelper(thisVal);
		}


		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)
				{