/**
  * Create an Erlang integer from the given value.
  *
  * @param val
  *                the long value to use.
  */
 public OtpErlangLong(BigInteger v)
 {
     if (v == null)
     {
         throw new NullReferenceException();
     }
     if (v.bitCount() < 64)
     {
         val = v.LongValue();
     }
     else
     {
         bigVal = v;
     }
 }
        //***********************************************************************
        // 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;
        }
Ejemplo n.º 3
0
 public void write_big_integer(BigInteger v)
 {
     if (v.bitCount() < 64)
     {
         this.write_long(v.LongValue(), true);
         return;
     }
     int signum = (v > (BigInteger)0) ? 1 : (v < (BigInteger)0) ? -1 : 0;
     if (signum < 0)
     {
         v = -v;
     }
     byte[] magnitude = v.getBytes();
     int n = magnitude.Length;
     // Reverse the array to make it little endian.
     for (int i = 0, j = n; i < j--; i++)
     {
         // Swap [i] with [j]
         byte b = magnitude[i];
         magnitude[i] = magnitude[j];
         magnitude[j] = b;
     }
     if ((n & 0xFF) == n)
     {
         write1(OtpExternal.smallBigTag);
         write1(n); // length
     }
     else
     {
         write1(OtpExternal.largeBigTag);
         write4BE(n); // length
     }
     write1(signum < 0 ? 1 : 0); // sign
     // Write the array
     writeN(magnitude);
 }
        //***********************************************************************
        // 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;
        }
        //***********************************************************************
        // 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;
        }
        //***********************************************************************
        // 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;
        }
        //***********************************************************************
        // 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;
        }
        //***********************************************************************
        // 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;
        }
        //***********************************************************************
        // 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;
        }
        //***********************************************************************
        // 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;
        }
        //***********************************************************************
        // 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;
        }
        //***********************************************************************
        // 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];
        }
        //***********************************************************************
        // 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));
        }
        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;
        }
        public static void Main(string[] args)
        {
                // Known problem -> these two pseudoprimes passes my implementation of
                // primality test but failed in JDK's isProbablePrime test.

                byte[] pseudoPrime1 = { (byte)0x00,
                        (byte)0x85, (byte)0x84, (byte)0x64, (byte)0xFD, (byte)0x70, (byte)0x6A,
                        (byte)0x9F, (byte)0xF0, (byte)0x94, (byte)0x0C, (byte)0x3E, (byte)0x2C,
                        (byte)0x74, (byte)0x34, (byte)0x05, (byte)0xC9, (byte)0x55, (byte)0xB3,
                        (byte)0x85, (byte)0x32, (byte)0x98, (byte)0x71, (byte)0xF9, (byte)0x41,
                        (byte)0x21, (byte)0x5F, (byte)0x02, (byte)0x9E, (byte)0xEA, (byte)0x56,
                        (byte)0x8D, (byte)0x8C, (byte)0x44, (byte)0xCC, (byte)0xEE, (byte)0xEE,
                        (byte)0x3D, (byte)0x2C, (byte)0x9D, (byte)0x2C, (byte)0x12, (byte)0x41,
                        (byte)0x1E, (byte)0xF1, (byte)0xC5, (byte)0x32, (byte)0xC3, (byte)0xAA,
                        (byte)0x31, (byte)0x4A, (byte)0x52, (byte)0xD8, (byte)0xE8, (byte)0xAF,
                        (byte)0x42, (byte)0xF4, (byte)0x72, (byte)0xA1, (byte)0x2A, (byte)0x0D,
                        (byte)0x97, (byte)0xB1, (byte)0x31, (byte)0xB3,
                };

		// warning CS0219: The variable `pseudoPrime2' is assigned but its value is never used
                //byte[] pseudoPrime2 = { (byte)0x00,
                //        (byte)0x99, (byte)0x98, (byte)0xCA, (byte)0xB8, (byte)0x5E, (byte)0xD7,
                //        (byte)0xE5, (byte)0xDC, (byte)0x28, (byte)0x5C, (byte)0x6F, (byte)0x0E,
                //        (byte)0x15, (byte)0x09, (byte)0x59, (byte)0x6E, (byte)0x84, (byte)0xF3,
                //        (byte)0x81, (byte)0xCD, (byte)0xDE, (byte)0x42, (byte)0xDC, (byte)0x93,
                //        (byte)0xC2, (byte)0x7A, (byte)0x62, (byte)0xAC, (byte)0x6C, (byte)0xAF,
                //        (byte)0xDE, (byte)0x74, (byte)0xE3, (byte)0xCB, (byte)0x60, (byte)0x20,
                //        (byte)0x38, (byte)0x9C, (byte)0x21, (byte)0xC3, (byte)0xDC, (byte)0xC8,
                //        (byte)0xA2, (byte)0x4D, (byte)0xC6, (byte)0x2A, (byte)0x35, (byte)0x7F,
                //        (byte)0xF3, (byte)0xA9, (byte)0xE8, (byte)0x1D, (byte)0x7B, (byte)0x2C,
                //        (byte)0x78, (byte)0xFA, (byte)0xB8, (byte)0x02, (byte)0x55, (byte)0x80,
                //        (byte)0x9B, (byte)0xC2, (byte)0xA5, (byte)0xCB,
                //};

                Console.WriteLine("List of primes < 2000\n---------------------");
                int limit = 100, count = 0;
                for(int i = 0; i < 2000; i++)
                {
                        if(i >= limit)
                        {
                                Console.WriteLine();
                                limit += 100;
                        }

                        BigInteger p = new BigInteger(-i);

                        if(p.isProbablePrime())
                        {
                                Console.Write(i + ", ");
                                count++;
                        }
                }
                Console.WriteLine("\nCount = " + count);


                BigInteger bi1 = new BigInteger(pseudoPrime1);
                Console.WriteLine("\n\nPrimality testing for\n" + bi1.ToString() + "\n");
                Console.WriteLine("SolovayStrassenTest(5) = " + bi1.SolovayStrassenTest(5));
                Console.WriteLine("RabinMillerTest(5) = " + bi1.RabinMillerTest(5));
                Console.WriteLine("FermatLittleTest(5) = " + bi1.FermatLittleTest(5));
                Console.WriteLine("isProbablePrime() = " + bi1.isProbablePrime());

                Console.Write("\nGenerating 512-bits random pseudoprime. . .");
                Random rand = new Random();
                BigInteger prime = BigInteger.genPseudoPrime(512, 5, rand);
                Console.WriteLine("\n" + prime);

                //int dwStart = System.Environment.TickCount;
                //BigInteger.MulDivTest(100000);
                //BigInteger.RSATest(10);
                //BigInteger.RSATest2(10);
                //Console.WriteLine(System.Environment.TickCount - dwStart);

        }
        //***********************************************************************
        // 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);
        }
        //***********************************************************************
        // 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;
        }
        //***********************************************************************
        // 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;
        }
        //***********************************************************************
        // 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;
        }
        //***********************************************************************
        // 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;
        }
        //***********************************************************************
        // 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;
        }
        //***********************************************************************
        // Tests the correct implementation of the /, %, * and + operators
        //***********************************************************************

        public static void MulDivTest(int rounds)
        {
                Random rand = new Random();
	        byte[] val = new byte[64];
	        byte[] val2 = new byte[64];

	        for(int count = 0; count < rounds; count++)
	        {
	                // generate 2 numbers of random length
		        int t1 = 0;
		        while(t1 == 0)
			        t1 = (int)(rand.NextDouble() * 65);

		        int t2 = 0;
		        while(t2 == 0)
			        t2 = (int)(rand.NextDouble() * 65);

		        bool done = false;
		        while(!done)
		        {
			        for(int i = 0; i < 64; i++)
			        {
				        if(i < t1)
					        val[i] = (byte)(rand.NextDouble() * 256);
				        else
					        val[i] = 0;

				        if(val[i] != 0)
					        done = true;
			        }
		        }

		        done = false;
		        while(!done)
		        {
			        for(int i = 0; i < 64; i++)
			        {
				        if(i < t2)
					        val2[i] = (byte)(rand.NextDouble() * 256);
				        else
					        val2[i] = 0;

				        if(val2[i] != 0)
					        done = true;
			        }
		        }

		        while(val[0] == 0)
		                val[0] = (byte)(rand.NextDouble() * 256);
		        while(val2[0] == 0)
		                val2[0] = (byte)(rand.NextDouble() * 256);

                        Console.WriteLine(count);
		        BigInteger bn1 = new BigInteger(val, t1);
		        BigInteger bn2 = new BigInteger(val2, t2);


                        // Determine the quotient and remainder by dividing
                        // the first number by the second.

		        BigInteger bn3 = bn1 / bn2;
		        BigInteger bn4 = bn1 % bn2;

		        // Recalculate the number
		        BigInteger bn5 = (bn3 * bn2) + bn4;

                        // Make sure they're the same
        		if(bn5 != bn1)
	        	{
		        	Console.WriteLine("Error at " + count);
                                Console.WriteLine(bn1 + "\n");
			        Console.WriteLine(bn2 + "\n");
			        Console.WriteLine(bn3 + "\n");
                                Console.WriteLine(bn4 + "\n");
			        Console.WriteLine(bn5 + "\n");
			        return;
		        }
	        }
        }
        //***********************************************************************
        // 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;
        }
        //***********************************************************************
        // Tests the correct implementation of the modulo exponential function
        // using RSA encryption and decryption (using pre-computed encryption and
        // decryption keys).
        //***********************************************************************

        public static void RSATest(int rounds)
        {
                Random rand = new Random(1);
	        byte[] val = new byte[64];

	        // private and public key
                BigInteger bi_e = new BigInteger("a932b948feed4fb2b692609bd22164fc9edb59fae7880cc1eaff7b3c9626b7e5b241c27a974833b2622ebe09beb451917663d47232488f23a117fc97720f1e7", 16);
                BigInteger bi_d = new BigInteger("4adf2f7a89da93248509347d2ae506d683dd3a16357e859a980c4f77a4e2f7a01fae289f13a851df6e9db5adaa60bfd2b162bbbe31f7c8f828261a6839311929d2cef4f864dde65e556ce43c89bbbf9f1ac5511315847ce9cc8dc92470a747b8792d6a83b0092d2e5ebaf852c85cacf34278efa99160f2f8aa7ee7214de07b7", 16);
                BigInteger bi_n = new BigInteger("e8e77781f36a7b3188d711c2190b560f205a52391b3479cdb99fa010745cbeba5f2adc08e1de6bf38398a0487c4a73610d94ec36f17f3f46ad75e17bc1adfec99839589f45f95ccc94cb2a5c500b477eb3323d8cfab0c8458c96f0147a45d27e45a4d11d54d77684f65d48f15fafcc1ba208e71e921b9bd9017c16a5231af7f", 16);

                Console.WriteLine("e =\n" + bi_e.ToString(10));
                Console.WriteLine("\nd =\n" + bi_d.ToString(10));
                Console.WriteLine("\nn =\n" + bi_n.ToString(10) + "\n");

	        for(int count = 0; count < rounds; count++)
	        {
	                // generate data of random length
		        int t1 = 0;
		        while(t1 == 0)
			        t1 = (int)(rand.NextDouble() * 65);

		        bool done = false;
		        while(!done)
		        {
			        for(int i = 0; i < 64; i++)
			        {
				        if(i < t1)
					        val[i] = (byte)(rand.NextDouble() * 256);
				        else
					        val[i] = 0;

				        if(val[i] != 0)
					        done = true;
			        }
		        }

		        while(val[0] == 0)
		                val[0] = (byte)(rand.NextDouble() * 256);

                        Console.Write("Round = " + count);

                        // encrypt and decrypt data
		        BigInteger bi_data = new BigInteger(val, t1);
                        BigInteger bi_encrypted = bi_data.modPow(bi_e, bi_n);
                        BigInteger bi_decrypted = bi_encrypted.modPow(bi_d, bi_n);

                        // compare
                        if(bi_decrypted != bi_data)
	        	{
		        	Console.WriteLine("\nError at round " + count);
                                Console.WriteLine(bi_data + "\n");
			        return;
		        }
		        Console.WriteLine(" <PASSED>.");
	        }

        }
        //***********************************************************************
        // 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;
        }
        //***********************************************************************
        // Tests the correct implementation of the modulo exponential and
        // inverse modulo functions using RSA encryption and decryption.  The two
        // pseudoprimes p and q are fixed, but the two RSA keys are generated
        // for each round of testing.
        //***********************************************************************

        public static void RSATest2(int rounds)
        {
                Random rand = new Random();
	        byte[] val = new byte[64];

                byte[] pseudoPrime1 = {
                        (byte)0x85, (byte)0x84, (byte)0x64, (byte)0xFD, (byte)0x70, (byte)0x6A,
                        (byte)0x9F, (byte)0xF0, (byte)0x94, (byte)0x0C, (byte)0x3E, (byte)0x2C,
                        (byte)0x74, (byte)0x34, (byte)0x05, (byte)0xC9, (byte)0x55, (byte)0xB3,
                        (byte)0x85, (byte)0x32, (byte)0x98, (byte)0x71, (byte)0xF9, (byte)0x41,
                        (byte)0x21, (byte)0x5F, (byte)0x02, (byte)0x9E, (byte)0xEA, (byte)0x56,
                        (byte)0x8D, (byte)0x8C, (byte)0x44, (byte)0xCC, (byte)0xEE, (byte)0xEE,
                        (byte)0x3D, (byte)0x2C, (byte)0x9D, (byte)0x2C, (byte)0x12, (byte)0x41,
                        (byte)0x1E, (byte)0xF1, (byte)0xC5, (byte)0x32, (byte)0xC3, (byte)0xAA,
                        (byte)0x31, (byte)0x4A, (byte)0x52, (byte)0xD8, (byte)0xE8, (byte)0xAF,
                        (byte)0x42, (byte)0xF4, (byte)0x72, (byte)0xA1, (byte)0x2A, (byte)0x0D,
                        (byte)0x97, (byte)0xB1, (byte)0x31, (byte)0xB3,
                };

                byte[] pseudoPrime2 = {
                        (byte)0x99, (byte)0x98, (byte)0xCA, (byte)0xB8, (byte)0x5E, (byte)0xD7,
                        (byte)0xE5, (byte)0xDC, (byte)0x28, (byte)0x5C, (byte)0x6F, (byte)0x0E,
                        (byte)0x15, (byte)0x09, (byte)0x59, (byte)0x6E, (byte)0x84, (byte)0xF3,
                        (byte)0x81, (byte)0xCD, (byte)0xDE, (byte)0x42, (byte)0xDC, (byte)0x93,
                        (byte)0xC2, (byte)0x7A, (byte)0x62, (byte)0xAC, (byte)0x6C, (byte)0xAF,
                        (byte)0xDE, (byte)0x74, (byte)0xE3, (byte)0xCB, (byte)0x60, (byte)0x20,
                        (byte)0x38, (byte)0x9C, (byte)0x21, (byte)0xC3, (byte)0xDC, (byte)0xC8,
                        (byte)0xA2, (byte)0x4D, (byte)0xC6, (byte)0x2A, (byte)0x35, (byte)0x7F,
                        (byte)0xF3, (byte)0xA9, (byte)0xE8, (byte)0x1D, (byte)0x7B, (byte)0x2C,
                        (byte)0x78, (byte)0xFA, (byte)0xB8, (byte)0x02, (byte)0x55, (byte)0x80,
                        (byte)0x9B, (byte)0xC2, (byte)0xA5, (byte)0xCB,
                };


                BigInteger bi_p = new BigInteger(pseudoPrime1);
                BigInteger bi_q = new BigInteger(pseudoPrime2);
                BigInteger bi_pq = (bi_p-1)*(bi_q-1);
                BigInteger bi_n = bi_p * bi_q;

	        for(int count = 0; count < rounds; count++)
	        {
	                // generate private and public key
                        BigInteger bi_e = bi_pq.genCoPrime(512, rand);
                        BigInteger bi_d = bi_e.modInverse(bi_pq);

                        Console.WriteLine("\ne =\n" + bi_e.ToString(10));
                        Console.WriteLine("\nd =\n" + bi_d.ToString(10));
                        Console.WriteLine("\nn =\n" + bi_n.ToString(10) + "\n");

	                // generate data of random length
		        int t1 = 0;
		        while(t1 == 0)
			        t1 = (int)(rand.NextDouble() * 65);

		        bool done = false;
		        while(!done)
		        {
			        for(int i = 0; i < 64; i++)
			        {
				        if(i < t1)
					        val[i] = (byte)(rand.NextDouble() * 256);
				        else
					        val[i] = 0;

				        if(val[i] != 0)
					        done = true;
			        }
		        }

		        while(val[0] == 0)
		                val[0] = (byte)(rand.NextDouble() * 256);

                        Console.Write("Round = " + count);

                        // encrypt and decrypt data
		        BigInteger bi_data = new BigInteger(val, t1);
                        BigInteger bi_encrypted = bi_data.modPow(bi_e, bi_n);
                        BigInteger bi_decrypted = bi_encrypted.modPow(bi_d, bi_n);

                        // compare
                        if(bi_decrypted != bi_data)
	        	{
		        	Console.WriteLine("\nError at round " + count);
                                Console.WriteLine(bi_data + "\n");
			        return;
		        }
		        Console.WriteLine(" <PASSED>.");
	        }

        }
 private bool BigIntegerIsNull(BigInteger val)
 {
     return ((Object)val) == null;
 }
        //***********************************************************************
        // Tests the correct implementation of sqrt() method.
        //***********************************************************************

        public static void SqrtTest(int rounds)
        {
                Random rand = new Random();
	        for(int count = 0; count < rounds; count++)
	        {
	                // generate data of random length
		        int t1 = 0;
		        while(t1 == 0)
			        t1 = (int)(rand.NextDouble() * 1024);

                        Console.Write("Round = " + count);

                        BigInteger a = new BigInteger();
                        a.genRandomBits(t1, rand);

                        BigInteger b = a.sqrt();
                        BigInteger c = (b+1)*(b+1);

                        // check that b is the largest integer such that b*b <= a
                        if(c <= a)
	        	{
		        	Console.WriteLine("\nError at round " + count);
                                Console.WriteLine(a + "\n");
			        return;
		        }
		        Console.WriteLine(" <PASSED>.");
	        }
        }
        /**
         * Create an Erlang integer from a stream containing an integer encoded in
         * Erlang external format.
         *
         * @param buf
         *                the stream containing the encoded value.
         *
         * @exception OtpErlangDecodeException
         *                    if the buffer does not contain a valid external
         *                    representation of an Erlang integer.
         */
        public OtpErlangLong(OtpInputStream buf)
        {
            byte[] b = buf.read_integer_byte_array();

            try
            {
                val = OtpInputStream.byte_array_to_long(b, false);
            }
            catch (OtpErlangDecodeException)
            {
                bigVal = new BigInteger(b);
            }
        }
        //***********************************************************************
        // 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;
        }