/lib/otp.net/Otp/Erlang/BigInteger.cs

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//************************************************************************************
// BigInteger Class Version 1.03
//
// Copyright (c) 2002 Chew Keong TAN
// All rights reserved.
//
// Permission is hereby granted, free of charge, to any person obtaining a
// copy of this software and associated documentation files (the
// "Software"), to deal in the Software without restriction, including
// without limitation the rights to use, copy, modify, merge, publish,
// distribute, and/or sell copies of the Software, and to permit persons
// to whom the Software is furnished to do so, provided that the above
// copyright notice(s) and this permission notice appear in all copies of
// the Software and that both the above copyright notice(s) and this
// permission notice appear in supporting documentation.
//
// THE SOFTWARE IS PROVIDED "AS IS", WITHOUT WARRANTY OF ANY KIND, EXPRESS
// OR IMPLIED, INCLUDING BUT NOT LIMITED TO THE WARRANTIES OF
// MERCHANTABILITY, FITNESS FOR A PARTICULAR PURPOSE AND NONINFRINGEMENT
// OF THIRD PARTY RIGHTS. IN NO EVENT SHALL THE COPYRIGHT HOLDER OR
// HOLDERS INCLUDED IN THIS NOTICE BE LIABLE FOR ANY CLAIM, OR ANY SPECIAL
// INDIRECT OR CONSEQUENTIAL DAMAGES, OR ANY DAMAGES WHATSOEVER RESULTING
// FROM LOSS OF USE, DATA OR PROFITS, WHETHER IN AN ACTION OF CONTRACT,
// NEGLIGENCE OR OTHER TORTIOUS ACTION, ARISING OUT OF OR IN CONNECTION
// WITH THE USE OR PERFORMANCE OF THIS SOFTWARE.
//
//
// Disclaimer
// ----------
// Although reasonable care has been taken to ensure the correctness of this
// implementation, this code should never be used in any application without
// proper verification and testing.  I disclaim all liability and responsibility
// to any person or entity with respect to any loss or damage caused, or alleged
// to be caused, directly or indirectly, by the use of this BigInteger class.
//
// Comments, bugs and suggestions to
// (http://www.codeproject.com/csharp/biginteger.asp)
//
//
// Overloaded Operators +, -, *, /, %, >>, <<, ==, !=, >, <, >=, <=, &, |, ^, ++, --, ~
//
// Features
// --------
// 1) Arithmetic operations involving large signed integers (2's complement).
// 2) Primality test using Fermat little theorm, Rabin Miller's method,
//    Solovay Strassen's method and Lucas strong pseudoprime.
// 3) Modulo exponential with Barrett's reduction.
// 4) Inverse modulo.
// 5) Pseudo prime generation.
// 6) Co-prime generation.
//
//
// Known Problem
// -------------
// This pseudoprime 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,
//       };
//
//
// Change Log
// ----------
// 1) September 23, 2002 (Version 1.03)
//    - Fixed operator- to give correct data length.
//    - Added Lucas sequence generation.
//    - Added Strong Lucas Primality test.
//    - Added integer square root method.
//    - Added setBit/unsetBit methods.
//    - New isProbablePrime() method which do not require the
//      confident parameter.
//
// 2) August 29, 2002 (Version 1.02)
//    - Fixed bug in the exponentiation of negative numbers.
//    - Faster modular exponentiation using Barrett reduction.
//    - Added getBytes() method.
//    - Fixed bug in ToHexString method.
//    - Added overloading of ^ operator.
//    - Faster computation of Jacobi symbol.
//
// 3) August 19, 2002 (Version 1.01)
//    - Big integer is stored and manipulated as unsigned integers (4 bytes) instead of
//      individual bytes this gives significant performance improvement.
//    - Updated Fermat's Little Theorem test to use a^(p-1) mod p = 1
//    - Added isProbablePrime method.
//    - Updated documentation.
//
// 4) August 9, 2002 (Version 1.0)
//    - Initial Release.
//
//
// References
// [1] D. E. Knuth, "Seminumerical Algorithms", The Art of Computer Programming Vol. 2,
//     3rd Edition, Addison-Wesley, 1998.
//
// [2] K. H. Rosen, "Elementary Number Theory and Its Applications", 3rd Ed,
//     Addison-Wesley, 1993.
//
// [3] B. Schneier, "Applied Cryptography", 2nd Ed, John Wiley & Sons, 1996.
//
// [4] A. Menezes, P. van Oorschot, and S. Vanstone, "Handbook of Applied Cryptography",
//     CRC Press, 1996, www.cacr.math.uwaterloo.ca/hac
//
// [5] A. Bosselaers, R. Govaerts, and J. Vandewalle, "Comparison of Three Modular
//     Reduction Functions," Proc. CRYPTO'93, pp.175-186.
//
// [6] R. Baillie and S. S. Wagstaff Jr, "Lucas Pseudoprimes", Mathematics of Computation,
//     Vol. 35, No. 152, Oct 1980, pp. 1391-1417.
//
// [7] H. C. Williams, "Édouard Lucas and Primality Testing", Canadian Mathematical
//     Society Series of Monographs and Advance Texts, vol. 22, John Wiley & Sons, New York,
//     NY, 1998.
//
// [8] P. Ribenboim, "The new book of prime number records", 3rd edition, Springer-Verlag,
//     New York, NY, 1995.
//
// [9] M. Joye and J.-J. Quisquater, "Efficient computation of full Lucas sequences",
//     Electronics Letters, 32(6), 1996, pp 537-538.
//
//************************************************************************************

using System;


public class BigInteger
{
        // maximum length of the BigInteger in uint (4 bytes)
        // change this to suit the required level of precision.

        private const int maxLength = 70;

        // primes smaller than 2000 to test the generated prime number

        public static readonly int[] primesBelow2000 = {
        2, 3, 5, 7, 11, 13, 17, 19, 23, 29, 31, 37, 41, 43, 47, 53, 59, 61, 67, 71, 73, 79, 83, 89, 97,
        101, 103, 107, 109, 113, 127, 131, 137, 139, 149, 151, 157, 163, 167, 173, 179, 181, 191, 193, 197, 199,
	211, 223, 227, 229, 233, 239, 241, 251, 257, 263, 269, 271, 277, 281, 283, 293,
	307, 311, 313, 317, 331, 337, 347, 349, 353, 359, 367, 373, 379, 383, 389, 397,
	401, 409, 419, 421, 431, 433, 439, 443, 449, 457, 461, 463, 467, 479, 487, 491, 499,
	503, 509, 521, 523, 541, 547, 557, 563, 569, 571, 577, 587, 593, 599,
	601, 607, 613, 617, 619, 631, 641, 643, 647, 653, 659, 661, 673, 677, 683, 691,
	701, 709, 719, 727, 733, 739, 743, 751, 757, 761, 769, 773, 787, 797,
	809, 811, 821, 823, 827, 829, 839, 853, 857, 859, 863, 877, 881, 883, 887,
	907, 911, 919, 929, 937, 941, 947, 953, 967, 971, 977, 983, 991, 997,
	1009, 1013, 1019, 1021, 1031, 1033, 1039, 1049, 1051, 1061, 1063, 1069, 1087, 1091, 1093, 1097,
	1103, 1109, 1117, 1123, 1129, 1151, 1153, 1163, 1171, 1181, 1187, 1193,
	1201, 1213, 1217, 1223, 1229, 1231, 1237, 1249, 1259, 1277, 1279, 1283, 1289, 1291, 1297,
	1301, 1303, 1307, 1319, 1321, 1327, 1361, 1367, 1373, 1381, 1399,
	1409, 1423, 1427, 1429, 1433, 1439, 1447, 1451, 1453, 1459, 1471, 1481, 1483, 1487, 1489, 1493, 1499,
	1511, 1523, 1531, 1543, 1549, 1553, 1559, 1567, 1571, 1579, 1583, 1597,
	1601, 1607, 1609, 1613, 1619, 1621, 1627, 1637, 1657, 1663, 1667, 1669, 1693, 1697, 1699,
	1709, 1721, 1723, 1733, 1741, 1747, 1753, 1759, 1777, 1783, 1787, 1789,
	1801, 1811, 1823, 1831, 1847, 1861, 1867, 1871, 1873, 1877, 1879, 1889,
	1901, 1907, 1913, 1931, 1933, 1949, 1951, 1973, 1979, 1987, 1993, 1997, 1999 };


        private uint[] data = null;             // stores bytes from the Big Integer
        public int dataLength;                 // number of actual chars used


        //***********************************************************************
        // Constructor (Default value for BigInteger is 0
        //***********************************************************************

        public BigInteger()
        {
                data = new uint[maxLength];
                dataLength = 1;
        }


        //***********************************************************************
        // Constructor (Default value provided by long)
        //***********************************************************************

        public BigInteger(long value)
        {
                data = new uint[maxLength];
                long tempVal = value;

                // copy bytes from long to BigInteger without any assumption of
                // the length of the long datatype

                dataLength = 0;
                while(value != 0 && dataLength < maxLength)
                {
                        data[dataLength] = (uint)(value & 0xFFFFFFFF);
                        value >>= 32;
                        dataLength++;
                }

                if(tempVal > 0)         // overflow check for +ve value
                {
                        if(value != 0 || (data[maxLength-1] & 0x80000000) != 0)
                                throw(new ArithmeticException("Positive overflow in constructor."));
                }
                else if(tempVal < 0)    // underflow check for -ve value
                {
                        if(value != -1 || (data[dataLength-1] & 0x80000000) == 0)
                                throw(new ArithmeticException("Negative underflow in constructor."));
                }

                if(dataLength == 0)
                        dataLength = 1;
        }


        //***********************************************************************
        // Constructor (Default value provided by ulong)
        //***********************************************************************

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

                // copy bytes from ulong to BigInteger without any assumption of
                // the length of the ulong datatype

                dataLength = 0;
                while(value != 0 && dataLength < maxLength)
                {
                        data[dataLength] = (uint)(value & 0xFFFFFFFF);
                        value >>= 32;
                        dataLength++;
                }

                if(value != 0 || (data[maxLength-1] & 0x80000000) != 0)
                        throw(new ArithmeticException("Positive overflow in constructor."));

                if(dataLength == 0)
                        dataLength = 1;
        }



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


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


        //***********************************************************************
        // Constructor (Default value provided by an array of bytes)
        //
        // The lowest index of the input byte array (i.e [0]) should contain the
        // most significant byte of the number, and the highest index should
        // contain the least significant byte.
        //
        // E.g.
        // To initialize "a" with the default value of 0x1D4F in base 16
        //      byte[] temp = { 0x1D, 0x4F };
        //      BigInteger a = new BigInteger(temp)
        //
        // Note that this method of initialization does not allow the
        // sign to be specified.
        //
        //***********************************************************************

        public BigInteger(byte[] inData)
        {
                dataLength = inData.Length >> 2;

                int leftOver = inData.Length & 0x3;
                if(leftOver != 0)         // length not multiples of 4
                        dataLength++;


                if(dataLength > maxLength)
                      throw(new ArithmeticException("Byte overflow in constructor."));

                data = new uint[maxLength];

                for(int i = inData.Length - 1, j = 0; i >= 3; i -= 4, j++)
                {
                        data[j] = (uint)((inData[i-3] << 24) + (inData[i-2] << 16) +
                                         (inData[i-1] <<  8) + inData[i]);
                }

                if(leftOver == 1)
                        data[dataLength-1] = (uint)inData[0];
                else if(leftOver == 2)
                        data[dataLength-1] = (uint)((inData[0] << 8) + inData[1]);
                else if(leftOver == 3)
                        data[dataLength-1] = (uint)((inData[0] << 16) + (inData[1] << 8) + inData[2]);


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

                //Console.WriteLine("Len = " + dataLength);
        }


        //***********************************************************************
        // Constructor (Default value provided by an array of bytes of the
        // specified length.)
        //***********************************************************************

        public BigInteger(byte[] inData, int inLen)
        {
                dataLength = inLen >> 2;

                int leftOver = inLen & 0x3;
                if(leftOver != 0)         // length not multiples of 4
                        dataLength++;

                if(dataLength > maxLength || inLen > inData.Length)
                      throw(new ArithmeticException("Byte overflow in constructor."));


                data = new uint[maxLength];

                for(int i = inLen - 1, j = 0; i >= 3; i -= 4, j++)
                {
                        data[j] = (uint)((inData[i-3] << 24) + (inData[i-2] << 16) +
                                         (inData[i-1] <<  8) + inData[i]);
                }

                if(leftOver == 1)
                        data[dataLength-1] = (uint)inData[0];
                else if(leftOver == 2)
                        data[dataLength-1] = (uint)((inData[0] << 8) + inData[1]);
                else if(leftOver == 3)
                        data[dataLength-1] = (uint)((inData[0] << 16) + (inData[1] << 8) + inData[2]);


                if(dataLength == 0)
                        dataLength = 1;

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

                //Console.WriteLine("Len = " + dataLength);
        }


        //***********************************************************************
        // Constructor (Default value provided by an array of unsigned integers)
        //*********************************************************************

        public BigInteger(uint[] inData)
        {
                dataLength = inData.Length;

                if(dataLength > maxLength)
                      throw(new ArithmeticException("Byte overflow in constructor."));

                data = new uint[maxLength];

                for(int i = dataLength - 1, j = 0; i >= 0; i--, j++)
                        data[j] = inData[i];

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

                //Console.WriteLine("Len = " + dataLength);
        }


        //***********************************************************************
        // Overloading of the typecast operator.
        // For BigInteger bi = 10;
        //***********************************************************************

        public static implicit operator BigInteger(long value)
        {
                return (new BigInteger(value));
        }

        public static implicit operator BigInteger(ulong value)
        {
                return (new BigInteger(value));
        }

        public static implicit operator BigInteger(int value)
        {
                return (new BigInteger((long)value));
        }

        public static implicit operator BigInteger(uint value)
        {
                return (new BigInteger((ulong)value));
        }


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


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


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


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



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


        // least significant bits at lower part of buffer

        private static int shiftLeft(uint[] buffer, int shiftVal)
        {
                int shiftAmount = 32;
                int bufLen = buffer.Length;

                while(bufLen > 1 && buffer[bufLen-1] == 0)
                        bufLen--;

                for(int count = shiftVal; count > 0;)
                {
                        if(count < shiftAmount)
                                shiftAmount = count;

                        //Console.WriteLine("shiftAmount = {0}", shiftAmount);

                        ulong carry = 0;
                        for(int i = 0; i < bufLen; i++)
                        {
                                ulong val = ((ulong)buffer[i]) << shiftAmount;
                                val |= carry;

                                buffer[i] = (uint)(val & 0xFFFFFFFF);
                                carry = val >> 32;
                        }

                        if(carry != 0)
                        {
                                if(bufLen + 1 <= buffer.Length)
                                {
                                        buffer[bufLen] = (uint)carry;
                                        bufLen++;
                                }
                        }
                        count -= shiftAmount;
                }
                return bufLen;
        }


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


        private static int shiftRight(uint[] buffer, int shiftVal)
        {
                int shiftAmount = 32;
                int invShift = 0;
                int bufLen = buffer.Length;

                while(bufLen > 1 && buffer[bufLen-1] == 0)
                        bufLen--;

                //Console.WriteLine("bufLen = " + bufLen + " buffer.Length = " + buffer.Length);

                for(int count = shiftVal; count > 0;)
                {
                        if(count < shiftAmount)
                        {
                                shiftAmount = count;
                                invShift = 32 - shiftAmount;
                        }

                        //Console.WriteLine("shiftAmount = {0}", shiftAmount);

                        ulong carry = 0;
                        for(int i = bufLen - 1; i >= 0; i--)
                        {
                                ulong val = ((ulong)buffer[i]) >> shiftAmount;
                                val |= carry;

                                carry = ((ulong)buffer[i]) << invShift;
                                buffer[i] = (uint)(val);
                        }

                        count -= shiftAmount;
                }

                while(bufLen > 1 && buffer[bufLen-1] == 0)
                        bufLen--;

                return bufLen;
        }


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


        //***********************************************************************
        // Overloading of equality operator
        //***********************************************************************

        public static bool operator ==(BigInteger bi1, BigInteger bi2)
        {
                return bi1.Equals(bi2);
        }


        public static bool operator !=(BigInteger bi1, BigInteger bi2)
        {
                return !(bi1.Equals(bi2));
        }


        public override bool Equals(object o)
        {
                BigInteger bi = (BigInteger)o;

                if(this.dataLength != bi.dataLength)
                        return false;

                for(int i = 0; i < this.dataLength; i++)
                {
                        if(this.data[i] != bi.data[i])
                                return false;
                }
                return true;
        }


        public override int GetHashCode()
        {
                return this.ToString().GetHashCode();
        }


        //***********************************************************************
        // Overloading of inequality operator
        //***********************************************************************

        public static bool operator >(BigInteger bi1, BigInteger bi2)
        {
        	int pos = maxLength - 1;

        	// bi1 is negative, bi2 is positive
        	if((bi1.data[pos] & 0x80000000) != 0 && (bi2.data[pos] & 0x80000000) == 0)
        	        return false;

                // bi1 is positive, bi2 is negative
        	else if((bi1.data[pos] & 0x80000000) == 0 && (bi2.data[pos] & 0x80000000) != 0)
        	        return true;

                // same sign
        	int len = (bi1.dataLength > bi2.dataLength) ? bi1.dataLength : bi2.dataLength;
	        for(pos = len - 1; pos >= 0 && bi1.data[pos] == bi2.data[pos]; pos--);

	        if(pos >= 0)
	        {
		        if(bi1.data[pos] > bi2.data[pos])
			        return true;
		        return false;
	        }
	        return false;
        }


        public static bool operator <(BigInteger bi1, BigInteger bi2)
        {
        	int pos = maxLength - 1;

        	// bi1 is negative, bi2 is positive
        	if((bi1.data[pos] & 0x80000000) != 0 && (bi2.data[pos] & 0x80000000) == 0)
        	        return true;

                // bi1 is positive, bi2 is negative
        	else if((bi1.data[pos] & 0x80000000) == 0 && (bi2.data[pos] & 0x80000000) != 0)
        	        return false;

        	// same sign
        	int len = (bi1.dataLength > bi2.dataLength) ? bi1.dataLength : bi2.dataLength;
	        for(pos = len - 1; pos >= 0 && bi1.data[pos] == bi2.data[pos]; pos--);

	        if(pos >= 0)
	        {
		        if(bi1.data[pos] < bi2.data[pos])
			        return true;
		        return false;
	        }
	        return false;
        }


        public static bool operator >=(BigInteger bi1, BigInteger bi2)
        {
                return (bi1 == bi2 || bi1 > bi2);
        }


        public static bool operator <=(BigInteger bi1, BigInteger bi2)
        {
                return (bi1 == bi2 || bi1 < bi2);
        }


        //***********************************************************************
        // Private function that supports the division of two numbers with
        // a divisor that has more than 1 digit.
        //
        // Algorithm taken from [1]
        //***********************************************************************

        private static void multiByteDivide(BigInteger bi1, BigInteger bi2,
                                            BigInteger outQuotient, BigInteger outRemainder)
        {
                uint[] result = new uint[maxLength];

                int remainderLen = bi1.dataLength + 1;
                uint[] remainder = new uint[remainderLen];

                uint mask = 0x80000000;
                uint val = bi2.data[bi2.dataLength - 1];
                int shift = 0, resultPos = 0;

                while(mask != 0 && (val & mask) == 0)
                {
                        shift++; mask >>= 1;
                }

                //Console.WriteLine("shift = {0}", shift);
                //Console.WriteLine("Before bi1 Len = {0}, bi2 Len = {1}", bi1.dataLength, bi2.dataLength);

                for(int i = 0; i < bi1.dataLength; i++)
                        remainder[i] = bi1.data[i];
                shiftLeft(remainder, shift);
                bi2 = bi2 << shift;

                /*
                Console.WriteLine("bi1 Len = {0}, bi2 Len = {1}", bi1.dataLength, bi2.dataLength);
                Console.WriteLine("dividend = " + bi1 + "\ndivisor = " + bi2);
                for(int q = remainderLen - 1; q >= 0; q--)
                        Console.Write("{0:x2}", remainder[q]);
                Console.WriteLine();
                */

                int j = remainderLen - bi2.dataLength;
                int pos = remainderLen - 1;

                ulong firstDivisorByte = bi2.data[bi2.dataLength-1];
                ulong secondDivisorByte = bi2.data[bi2.dataLength-2];

                int divisorLen = bi2.dataLength + 1;
                uint[] dividendPart = new uint[divisorLen];

                while(j > 0)
                {
                        ulong dividend = ((ulong)remainder[pos] << 32) + (ulong)remainder[pos-1];
                        //Console.WriteLine("dividend = {0}", dividend);

                        ulong q_hat = dividend / firstDivisorByte;
                        ulong r_hat = dividend % firstDivisorByte;

                        //Console.WriteLine("q_hat = {0:X}, r_hat = {1:X}", q_hat, r_hat);

                        bool done = false;
                        while(!done)
                        {
                                done = true;

                                if(q_hat == 0x100000000 ||
                                   (q_hat * secondDivisorByte) > ((r_hat << 32) + remainder[pos-2]))
                                {
                                        q_hat--;
                                        r_hat += firstDivisorByte;

                                        if(r_hat < 0x100000000)
                                                done = false;
                                }
                        }

                        for(int h = 0; h < divisorLen; h++)
                                dividendPart[h] = remainder[pos-h];

                        BigInteger kk = new BigInteger(dividendPart);
                        BigInteger ss = bi2 * (long)q_hat;

                        //Console.WriteLine("ss before = " + ss);
                        while(ss > kk)
                        {
                                q_hat--;
                                ss -= bi2;
                                //Console.WriteLine(ss);
                        }
                        BigInteger yy = kk - ss;

                        //Console.WriteLine("ss = " + ss);
                        //Console.WriteLine("kk = " + kk);
                        //Console.WriteLine("yy = " + yy);

                        for(int h = 0; h < divisorLen; h++)
                                remainder[pos-h] = yy.data[bi2.dataLength-h];

                        /*
                        Console.WriteLine("dividend = ");
                        for(int q = remainderLen - 1; q >= 0; q--)
                                Console.Write("{0:x2}", remainder[q]);
                        Console.WriteLine("\n************ q_hat = {0:X}\n", q_hat);
                        */

                        result[resultPos++] = (uint)q_hat;

                        pos--;
                        j--;
                }

                outQuotient.dataLength = resultPos;
                int y = 0;
                for(int x = outQuotient.dataLength - 1; x >= 0; x--, y++)
                        outQuotient.data[y] = result[x];
                for(; y < maxLength; y++)
                        outQuotient.data[y] = 0;

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

                if(outQuotient.dataLength == 0)
                        outQuotient.dataLength = 1;

                outRemainder.dataLength = shiftRight(remainder, shift);

                for(y = 0; y < outRemainder.dataLength; y++)
                        outRemainder.data[y] = remainder[y];
                for(; y < maxLength; y++)
                        outRemainder.data[y] = 0;
        }


        //***********************************************************************
        // Private function that supports the division of two numbers with
        // a divisor that has only 1 digit.
        //***********************************************************************

        private static void singleByteDivide(BigInteger bi1, BigInteger bi2,
                                             BigInteger outQuotient, BigInteger outRemainder)
        {
                uint[] result = new uint[maxLength];
                int resultPos = 0;

                // copy dividend to reminder
                for(int i = 0; i < maxLength; i++)
                        outRemainder.data[i] = bi1.data[i];
                outRemainder.dataLength = bi1.dataLength;

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

                ulong divisor = (ulong)bi2.data[0];
                int pos = outRemainder.dataLength - 1;
                ulong dividend = (ulong)outRemainder.data[pos];

                //Console.WriteLine("divisor = " + divisor + " dividend = " + dividend);
                //Console.WriteLine("divisor = " + bi2 + "\ndividend = " + bi1);

                if(dividend >= divisor)
                {
                        ulong quotient = dividend / divisor;
                        result[resultPos++] = (uint)quotient;

                        outRemainder.data[pos] = (uint)(dividend % divisor);
                }
                pos--;

                while(pos >= 0)
                {
                        //Console.WriteLine(pos);

                        dividend = ((ulong)outRemainder.data[pos+1] << 32) + (ulong)outRemainder.data[pos];
                        ulong quotient = dividend / divisor;
                        result[resultPos++] = (uint)quotient;

                        outRemainder.data[pos+1] = 0;
                        outRemainder.data[pos--] = (uint)(dividend % divisor);
                        //Console.WriteLine(">>>> " + bi1);
                }

                outQuotient.dataLength = resultPos;
                int j = 0;
                for(int i = outQuotient.dataLength - 1; i >= 0; i--, j++)
                        outQuotient.data[j] = result[i];
                for(; j < maxLength; j++)
                        outQuotient.data[j] = 0;

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

                if(outQuotient.dataLength == 0)
                        outQuotient.dataLength = 1;

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


        //***********************************************************************
        // Overloading of division operator
        //***********************************************************************

        public static BigInteger operator /(BigInteger bi1, BigInteger bi2)
        {
                BigInteger quotient = new BigInteger();
                BigInteger remainder = new BigInteger();

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

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

                if(bi1 < bi2)
                {
                        return quotient;
                }

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

                        if(dividendNeg != divisorNeg)
                                return -quotient;

                        return quotient;
                }
        }


        //***********************************************************************
        // Overloading of modulus operator
        //***********************************************************************

        public static BigInteger operator %(BigInteger bi1, BigInteger bi2)
      …

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