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

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

        //***********************************************************************
        // 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>.");
            }
        }

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

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

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

        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);
//.........这里部分代码省略.........

        //***********************************************************************
        // 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;
//.........这里部分代码省略.........

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

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

        //***********************************************************************
        // 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>.");
            }

        }

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

        //***********************************************************************
        // 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>.");
            }

        }

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

        public static void Main1(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,
            };

            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 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;
        }

        //***********************************************************************
        // 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;