mono/mcs/class/corlib/System.Security.Cryptography/BigInteger.cs

//
// BigInteger.cs - Big Integer implementation
//
// Authors:
//	Chew Keong TAN
//	Sebastien Pouliot ([email protected])
//
// Copyright (c) 2002 Chew Keong TAN
// All rights reserved.
//
// Modifications from original
// - Removed all reference to Random class (not secure enough)
// - Moved all static Test function into BigIntegerTest.cs (for NUnit)
//

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

namespace System.Security.Cryptography {

internal class BigRandom {
	RandomNumberGenerator rng;

	public BigRandom () 
	{
		rng = RandomNumberGenerator.Create ();
	}

	public void Get (uint[] data) 
	{
		byte[] random = new byte [4 * data.Length];
		rng.GetBytes (random);
		int n = 0;
		for (int i=0; i < data.Length; i++) {
			data[i] = BitConverter.ToUInt32 (random, n);
			n+=4;
		}
	}

	public int GetInt (int maxValue) 
	{
		// calculate mask
		int mask = Int32.MaxValue;
		while ((mask & maxValue) == maxValue)
			mask >>= 1;
		// undo last iteration
		mask <<= 1;
		mask |= 0x01;
		byte[] data = new byte [4];
		int result = -1;
		while ((result < 0) || (result > maxValue)) {
			rng.GetBytes (data);
			result = (BitConverter.ToInt32 (data, 0) & mask);
		}
		return result;
	}

	public byte GetByte() 
	{
		byte[] data = new byte [1];
		rng.GetBytes (data);
		return data [0];
	}
}

internal 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;
	// FIXME: actually this limit us to approx. 2048 bits keypair for RSA
	private const int maxLength = 140;

	private BigRandom random;

	// 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);
	}

	private BigRandom rng {
		get {
			if (random == null)
				random = new BigRandom ();
			return random;
		}
	}

	// 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)
	{
		BigInteger quotient = new BigInteger();
		BigInteger remainder = new BigInteger(bi1);

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

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

		if(bi1 < bi2) {
			return remainder;
		}

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

			if(dividendNeg)
				return -remainder;

			return remainder;
		}
	}

	// Overloading of bitwise AND operator
	public static BigInteger operator & (BigInteger bi1, BigInteger bi2) 
	{
		BigInteger result = new BigInteger();

		int len = (bi1.dataLength > bi2.dataLength) ? bi1.dataLength : bi2.dataLength;

		for(int i = 0; i < len; i++) {
			uint sum = (uint)(bi1.data[i] & bi2.data[i]);
			result.data[i] = sum;
		}

		result.dataLength = maxLength;

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

		return result;
	}

	// Overloading of bitwise OR operator
	public static BigInteger operator | (BigInteger bi1, BigInteger bi2) 
	{
		BigInteger result = new BigInteger();

		int len = (bi1.dataLength > bi2.dataLength) ? bi1.dataLength : bi2.dataLength;

		for(int i = 0; i < len; i++) {
			uint sum = (uint)(bi1.data[i] | bi2.data[i]);
			result.data[i] = sum;
		}

		result.dataLength = maxLength;

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

		return result;
	}

	// Overloading of bitwise XOR operator
	public static BigInteger operator ^ (BigInteger bi1, BigInteger bi2) 
	{
		BigInteger result = new BigInteger();

		int len = (bi1.dataLength > bi2.dataLength) ? bi1.dataLength : bi2.dataLength;

		for(int i = 0; i < len; i++) {
			uint sum = (uint)(bi1.data[i] ^ bi2.data[i]);
			result.data[i] = sum;
		}

		result.dataLength = maxLength;

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

		return result;
	}

	// Returns max(this, bi)
	public BigInteger max (BigInteger bi) 
	{
		if(this > bi)
			return (new BigInteger(this));
		else
			return (new BigInteger(bi));
	}

	// Returns min(this, bi)
	public BigInteger min (BigInteger bi) 
	{
		if (this < bi)
			return (new BigInteger (this));
		else
			return (new BigInteger (bi));
	}

	// Returns the absolute value
	public BigInteger abs () 
	{
		if((this.data[maxLength - 1] & 0x80000000) != 0)
			return (-this);
		else
			return (new BigInteger (this));
	}

	// Returns a string representing the BigInteger in base 10.
	public override string ToString () 
	{
		return ToString (10);
	}

	// Returns a string representing the BigInteger in sign-and-magnitude
	// format in the specified radix.
	//
	// Example
	// -------
	// If the value of BigInteger is -255 in base 10, then
	// ToString(16) returns "-FF"
	public string ToString (int radix) 
	{
		if(radix < 2 || radix > 36)
			throw (new ArgumentException("Radix must be >= 2 and <= 36"));

		string charSet = "ABCDEFGHIJKLMNOPQRSTUVWXYZ";
		string result = "";

		BigInteger a = this;

		bool negative = false;
		if((a.data[maxLength-1] & 0x80000000) != 0) {
			negative = true;
			try {
				a = -a;
			}
			catch(Exception) {}
		}

		BigInteger quotient = new BigInteger();
		BigInteger remainder = new BigInteger();
		BigInteger biRadix = new BigInteger(radix);

		if(a.dataLength == 1 && a.data[0] == 0)
			result = "0";
		else {
			while(a.dataLength > 1 || (a.dataLength == 1 && a.data[0] != 0)) {
				singleByteDivide(a, biRadix, quotient, remainder);

				if(remainder.data[0] < 10)
					result = remainder.data[0] + result;
				else
					result = charSet[(int)remainder.data[0] - 10] + result;

				a = quotient;
			}
			if(negative)
				result = "-" + result;
		}

		return result;
	}


	// Returns a hex string showing the contains of the BigInteger
	//
	// Examples
	// -------
	// 1) If the value of BigInteger is 255 in base 10, then
	//    ToHexString() returns "FF"
	//
	// 2) If the value of BigInteger is -255 in base 10, then
	//    ToHexString() returns ".....FFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFF01",
	//    which is the 2's complement representation of -255.
	public string ToHexString () 
	{
		string result = data[dataLength - 1].ToString("X");

		for(int i = dataLength - 2; i >= 0; i--) {
			result += data[i].ToString("X8");
		}

		return result;
	}

	// Modulo Exponentiation
	public BigInteger modPow(BigInteger exp, BigInteger n) 
	{
		if((exp.data[maxLength-1] & 0x80000000) != 0)
			throw (new ArithmeticException("Positive exponents only."));

		BigInteger resultNum = 1;
		BigInteger tempNum;
		bool thisNegative = false;

		if((this.data[maxLength-1] & 0x80000000) != 0) {   // negative this
			tempNum = -this % n;
			thisNegative = true;
		}
		else
			tempNum = this % n;  // ensures (tempNum * tempNum) < b^(2k)

		if((n.data[maxLength-1] & 0x80000000) != 0)   // negative n
			n = -n;

		// calculate constant = b^(2k) / m
		BigInteger constant = new BigInteger ();

		int i = n.dataLength << 1;
		constant.data[i] = 0x00000001;
		constant.dataLength = i + 1;

		constant = constant / n;
		int totalBits = exp.bitCount ();
		int count = 0;

		// perform squaring and multiply exponentiation
		for(int pos = 0; pos < exp.dataLength; pos++) {
			uint mask = 0x01;
			//Console.WriteLine("pos = " + pos);

			for(int index = 0; index < 32; index++) {
				if((exp.data[pos] & mask) != 0)
					resultNum = BarrettReduction(resultNum * tempNum, n, constant);

				mask <<= 1;

				tempNum = BarrettReduction(tempNum * tempNum, n, constant);


				if(tempNum.dataLength == 1 && tempNum.data[0] == 1) {
					if(thisNegative && (exp.data[0] & 0x1) != 0)    //odd exp
						return -resultNum;
					return resultNum;
				}
				count++;
				if(count == totalBits)
					break;
			}
		}

		if(thisNegative && (exp.data[0] & 0x1) != 0)    //odd exp
			return -resultNum;

		return resultNum;
	}

	// 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;
	}

	// 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;
	}

	// Populates "this" with the specified amount of random bits
	public void genRandomBits (int bits) 
	{
		genRandomBits (bits, new BigRandom ());
	}

	public void genRandomBits (int bits, BigRandom rng)
	{
		int dwords = bits >> 5;
		int remBits = bits & 0x1F;

		if (remBits != 0)
			dwords++;

		if (dwords > maxLength)
			throw (new ArithmeticException("Number of required bits > maxLength."));

		rng.Get (data);
		for (int i = dwords; i < maxLength; i++)
			data[i] = 0;

		if (remBits != 0) {
			uint mask = (uint)(0x01 << (remBits-1));
			data[dwords-1] |= mask;

			mask = (uint)(0xFFFFFFFF >> (32 - remBits));
			data[dwords-1] &= mask;
		}
		else
			data[dwords-1] |= 0x80000000;

		dataLength = dwords;

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

	// Returns the position of the most significant bit in the BigInteger.
	// Eg.  The result is 0, if the value of BigInteger is 0...0000 0000
	//      The result is 1, if the value of BigInteger is 0...0000 0001
	//      The result is 2, if the value of BigInteger is 0...0000 0010
	//      The result is 2, if the value of BigInteger is 0...0000 0011
	public int bitCount () 
	{
		while(dataLength > 1 && data[dataLength-1] == 0)
			dataLength--;

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

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

		return bits;
	}

	// Probabilistic prime test based on Fermat's little theorem
	//
	// for any a < p (p does not divide a) if
	//      a^(p-1) mod p != 1 then p is not prime.
	//
	// Otherwise, p is probably prime (pseudoprime to the chosen base).
	//
	// Returns
	// -------
	// True if "this" is a pseudoprime to randomly chosen
	// bases.  The number of chosen bases is given by the "confidence"
	// parameter.
	//
	// False if "this" is definitely NOT prime.
	//
	// Note - this method is fast but fails for Carmichael numbers except
	// when the randomly chosen base is a factor of the number.
	public bool FermatLittleTest (int confidence) 
	{
		BigInteger thisVal;
		if((this.data[maxLength-1] & 0x80000000) != 0)        // negative
			thisVal = -this;
		else
			thisVal = this;

		if(thisVal.dataLength == 1) {
			// test small numbers
			if(thisVal.data[0] == 0 || thisVal.data[0] == 1)
				return false;
			else if(thisVal.data[0] == 2 || thisVal.data[0] == 3)
				return true;
		}

		if((thisVal.data[0] & 0x1) == 0)     // even numbers
			return false;

		int bits = thisVal.bitCount();
		BigInteger a = new BigInteger();
		BigInteger p_sub1 = thisVal - (new BigInteger(1));

		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 = rng.GetInt (bits);

				a.genRandomBits (testBits);

				int byteLen = a.dataLength;

				// make sure "a" is not 0
				if(byteLen > 1 || (byteLen == 1 && a.data[0] != 1))
					done = true;
			}

			// check whether a factor exists (fix for version 1.03)
			BigInteger gcdTest = a.gcd(thisVal);
			if(gcdTest.dataLength == 1 && gcdTest.data[0] != 1)
				return false;

			// calculate a^(p-1) mod p
			BigInteger expResult = a.modPow(p_sub1, thisVal);

			int resultLen = expResult.dataLength;

			// is NOT prime is a^(p-1) mod p != 1

			if(resultLen > 1 || (resultLen == 1 && expResult.data[0] != 1)) {
				//Console.WriteLine("a = " + a.ToString());
				return false;
			}
		}

		return true;
	}

	// Probabilistic prime test based on Rabin-Miller's
	//
	// for any p > 0 with p - 1 = 2^s * t
	//
	// p is probably prime (strong pseudoprime) if for any a < p,
	// 1) a^t mod p = 1 or
	// 2) a^((2^j)*t) mod p = p-1 for some 0 <= j <= s-1
	//
	// Otherwise, p is composite.
	//
	// Returns
	// -------
	// True if "this" is a strong pseudoprime to randomly chosen
	// bases.  The number of chosen bases is given by the "confidence"
	// parameter.
	//
	// False if "this" is definitely NOT prime.
	public bool RabinMillerTest(int confidence) 
	{
		BigInteger thisVal;
		if((this.data[maxLength-1] & 0x80000000) != 0)        // negative
			thisVal = -this;
		else
			thisVal = this;

		if(thisVal.dataLength == 1) {
			// test small numbers
			if(thisVal.data[0] == 0 || thisVal.data[0] == 1)
				return false;
			else if(thisVal.data[0] == 2 || thisVal.data[0] == 3)
				return true;
		}

		if((thisVal.data[0] & 0x1) == 0)     // even numbers
			return false;

		// calculate values of s and t
		BigInteger p_sub1 = thisVal - (new BigInteger(1));
		int s = 0;

		for(int index = 0; index < p_sub1.dataLength; index++) {
			uint mask = 0x01;

			for(int i = 0; i < 32; i++) {
				if((p_sub1.data[index] & mask) != 0) {
					index = p_sub1.dataLength;      // to break the outer loop
					break;
				}
				mask <<= 1;
				s++;
			}
		}

		BigInteger t = p_sub1 >> s;

		int bits = thisVal.bitCount();
		BigInteger a = new BigInteger();

		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 = rng.GetInt (bits);

				a.genRandomBits (testBits);

				int byteLen = a.dataLength;

				// make sure "a" is not 0
				if(byteLen > 1 || (byteLen == 1 && a.data[0] != 1))
					done = true;
			}

			// check whether a factor exists (fix for version 1.03)
			BigInteger gcdTest = a.gcd(thisVal);
			if(gcdTest.dataLength == 1 && gcdTest.data[0] != 1)
				return false;

			BigInteger b = a.modPow(t, thisVal);

			/*
			Console.WriteLine("a = " + a.ToString(10));
			Console.WriteLine("b = " + b.ToString(10));
			Console.WriteLine("t = " + t.ToString(10));
			Console.WriteLine("s = " + s);
			*/

			bool result = false;

			if(b.dataLength == 1 && b.data[0] == 1)         // a^t mod p = 1
				result = true;

			for(int j = 0; result == false && j < s; j++) {
				if(b == p_sub1) {         // a^((2^j)*t) mod p = p-1 for some 0 <= j <= s-1
					result = true;
					break;
				}

				b = (b * b) % thisVal;
			}

			if(result == false)
				return false;
		}
		return true;
	}

	// Probabilistic prime test based on Solovay-Strassen (Euler Criterion)
	//
	// p is probably prime if for any a < p (a is not multiple of p),
	// a^((p-1)/2) mod p = J(a, p)
	//
	// where J is the Jacobi symbol.
	//
	// Otherwise, p is composite.
	//
	// Returns
	// -------
	// True if "this" is a Euler pseudoprime to randomly chosen
	// bases.  The number of chosen bases is given by the "confidence"
	// parameter.
	//
	// False if "this" is definitely NOT prime.
	public bool SolovayStrassenTest(int confidence) 
	{
		BigInteger thisVal;
		if((this.data[maxLength-1] & 0x80000000) != 0)        // negative
			thisVal = -this;
		else
			thisVal = this;

		if(thisVal.dataLength == 1) {
			// test small numbers
			if(thisVal.data[0] == 0 || thisVal.data[0] == 1)
				return false;
			else if(thisVal.data[0] == 2 || thisVal.data[0] == 3)
				return true;
		}

		if((thisVal.data[0] & 0x1) == 0)     // even numbers
			return false;

		int bits = thisVal.bitCount();
		BigInteger a = new BigInteger();
		BigInteger p_sub1 = thisVal - 1;
		BigInteger p_sub1_shift = p_sub1 >> 1;

		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 = rng.GetInt (bits);

				a.genRandomBits (testBits);

				int byteLen = a.dataLength;

				// make sure "a" is not 0
				if(byteLen > 1 || (byteLen == 1 && a.data[0] != 1))
					done = true;
			}

			// check whether a factor exists (fix for version 1.03)
			BigInteger gcdTest = a.gcd(thisVal);
			if(gcdTest.dataLength == 1 && gcdTest.data[0] != 1)
				return false;

			// calculate a^((p-1)/2) mod p

			BigInteger expResult = a.modPow(p_sub1_shift, thisVal);
			if(expResult == p_sub1)
				expResult = -1;

			// calculate Jacobi symbol
			BigInteger jacob = Jacobi(a, thisVal);

			//Console.WriteLine("a = " + a.ToString(10) + " b = " + thisVal.ToString(10));
			//Console.WriteLine("expResult = " + expResult.ToString(10) + " Jacob = " + jacob.ToString(10));

			// if they are different then it is not prime
			if(expResult != jacob)
				return false;
		}

		return true;
	}

	// Implementation of the Lucas Strong Pseudo Prime test.
	//
	// Let n be an odd number with gcd(n,D) = 1, and n - J(D, n) = 2^s * d
	// with d odd and s >= 0.
	//
	// If Ud mod n = 0 or V2^r*d mod n = 0 for some 0 <= r < s, then n
	// is a strong Lucas pseudoprime with parameters (P, Q).  We select
	// P and Q based on Selfridge.
	//
	// Returns True if number is a strong Lucus pseudo prime.
	// Otherwise, returns False indicating that number is composite.
	public bool LucasStrongTest() 
	{
		BigInteger thisVal;
		if((this.data[maxLength-1] & 0x80000000) != 0)        // negative
			thisVal = -this;
		else
			thisVal = this;

		if(thisVal.dataLength == 1) {
			// test small numbers
			if(thisVal.data[0] == 0 || thisVal.data[0] == 1)
				return false;
			else if(thisVal.data[0] == 2 || thisVal.data[0] == 3)
				return true;
		}

		if((thisVal.data[0] & 0x1) == 0)     // even numbers
			return false;

		return LucasStrongTestHelper(thisVal);
	}

	private bool LucasStrongTestHelper(BigInteger thisVal) 
	{
		// Do the test (selects D based on Selfridge)
		// Let D be the first element of the sequence
		// 5, -7, 9, -11, 13, ... for which J(D,n) = -1
		// Let P = 1, Q = (1-D) / 4

		long D = 5, sign = -1, dCount = 0;
		bool done = false;

		while(!done) {
			int Jresult = BigInteger.Jacobi(D, thisVal);

			if(Jresult == -1)
				done = true;    // J(D, this) = 1
			else {
				if(Jresult == 0 && System.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 = (System.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;
	}

	// Determines whether a number is probably prime, using the Rabin-Miller's
	// test.  Before applying the test, the number is tested for divisibility
	// by primes < 2000
	//
	// Returns true if number is probably prime.
	public bool isProbablePrime(int confidence) 
	{
		BigInteger thisVal;
		if((this.data[maxLength-1] & 0x80000000) != 0)        // negative
			thisVal = -this;
		else
			thisVal = this;


		// test for divisibility by primes < 2000
		for(int p = 0; p < primesBelow2000.Length; p++) {
			BigInteger divisor = primesBelow2000[p];

			if(divisor >= thisVal)
				break;

			BigInteger resultNum = thisVal % divisor;
			if(resultNum.IntValue() == 0) {
				/*
				Console.WriteLine("Not prime!  Divisible by {0}\n",
							primesBelow2000[p]);
				*/
				return false;
			}
		}

		if(thisVal.RabinMillerTest(confidence))
			return true;
		else {
			//Console.WriteLine("Not prime!  Failed primality test\n");
			return false;
		}
	}

	// Determines whether this BigInteger is probably prime using a
	// combination of base 2 strong pseudoprime test and Lucas strong
	// pseudoprime test.
	//
	// The sequence of the primality test is as follows,
	//
	// 1) Trial divisions are carried out using prime numbers below 2000.
	//    if any of the primes divides this BigInteger, then it is not prime.
	//
	// 2) Perform base 2 strong pseudoprime test.  If this BigInteger is a
	//    base 2 strong pseudoprime, proceed on to the next step.
	//
	// 3) Perform strong Lucas pseudoprime test.
	//
	// Returns True if this BigInteger is both a base 2 strong pseudoprime
	// and a strong Lucas pseudoprime.
	//
	// For a detailed discussion of this primality test, see [6].
	public bool isProbablePrime() 
	{
		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;

		// test for divisibility by primes < 2000
		for(int p = 0; p < primesBelow2000.Length; p++) {
			BigInteger divisor = primesBelow2000[p];

			if(divisor >= thisVal)
				break;

			BigInteger resultNum = thisVal % divisor;
			if(resultNum.IntValue() == 0) {
				//Console.WriteLine("Not prime!  Divisible by {0}\n",
				//                  primesBelow2000[p]);

				return false;
			}
		}

		// Perform BASE 2 Rabin-Miller Test

		// 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 = 2;

		// b = a^t mod p
		BigInteger b = a.modPow(t, thisVal);
		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 number is strong pseudoprime to base 2, then do a strong lucas test
		if(result)
			result = LucasStrongTestHelper(thisVal);

		return result;
	}

	// Returns the lowest 4 bytes of the BigInteger as an int.
	public int IntValue () 
	{
		return (int)data[0];
	}

	// Returns the lowest 8 bytes of the BigInteger as a long.
	public long LongValue () 
	{
		long val = 0;

		val = (long)data[0];
		try {
			// exception if maxLength = 1
			val |= (long)data[1] << 32;
		}
		catch(Exception) {
			if((data[0] & 0x80000000) != 0) // negative
				val = (int)data[0];
		}

		return val;
	}

	// 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) 
	{
		BigInteger result = new BigInteger ();
		bool done = false;

		while (!done) {
			result.genRandomBits (bits);
			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)
	{
		bool done = false;
		BigInteger result = new BigInteger ();

		while(!done) {
			result.genRandomBits (bits);
			//Console.WriteLine(result.ToString(16));

			// gcd test
			BigInteger g = result.gcd(this);
			if (g.dataLength == 1 && g.data[0] == 1)
				done = true;
		}

		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 the value of the BigInteger as a byte array.  The lowest
	// index contains the MSB.
	public byte[] getBytes() 
	{                
		int numBits = bitCount();                
		byte[] result = null;                
		if(numBits == 0) {                        
			result = new byte[1];                        
			result[0] = 0;                
		}                
		else {                        
			int numBytes = numBits >> 3;                        
			if((numBits & 0x7) != 0)                                
				numBytes++;                        
			result = new byte[numBytes];                        
			//Console.WriteLine(result.Length);                        
			int numBytesInWord = numBytes & 0x3;                        
			if(numBytesInWord == 0)                                
				numBytesInWord = 4;                        
			int pos = 0;                        
			for(int i = dataLength - 1; i >= 0; i--) {                                
				uint val = data[i];                                
				for(int j = numBytesInWord - 1; j >= 0; j--) {                                        
					result[pos+j] = (byte)(val & 0xFF);                                        
					val >>= 8;                                
				}                                
				pos += numBytesInWord;                                
				numBytesInWord = 4;                        
			}                
		}                
		return result;        
	}

	// Return true if the value of the specified bit is 1, false otherwise
	public bool testBit (uint bitNum) 
	{
		uint bytePos = bitNum >> 5;             // divide by 32
		byte bitPos = (byte)(bitNum & 0x1F);    // get the lowest 5 bits

		uint mask = (uint)1 << bitPos;
		return ((this.data[bytePos] | mask) == this.data[bytePos]);
	}

	// Sets the value of the specified bit to 1
	// The Least Significant Bit position is 0.
	public void setBit(uint bitNum) 
	{
		uint bytePos = bitNum >> 5;             // divide by 32
		byte bitPos = (byte)(bitNum & 0x1F);    // get the lowest 5 bits

		uint mask = (uint)1 << bitPos;
		this.data[bytePos] |= mask;

		if(bytePos >= this.dataLength)
			this.dataLength = (int)bytePos + 1;
	}

	// Sets the value of the specified bit to 0
	// The Least Significant Bit position is 0.
	public void unsetBit(uint bitNum) 
	{
		uint bytePos = bitNum >> 5;

		if(bytePos < this.dataLength) {
			byte bitPos = (byte)(bitNum & 0x1F);

			uint mask = (uint)1 << bitPos;
			uint mask2 = 0xFFFFFFFF ^ mask;

			this.data[bytePos] &= mask2;

			if(this.dataLength > 1 && this.data[this.dataLength - 1] == 0)
				this.dataLength--;
		}
	}

	// 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;
				s++;
			}
		}

		BigInteger t = k >> s;

		//Console.WriteLine("s = " + s + " t = " + t);
		return LucasSequenceHelper(P, Q, t, n, constant, s);
	}

	// 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;
	}
}

}