// initializes the private variables (throws CryptographicException)
		private void Initialize(BigInteger p, BigInteger g, BigInteger x, int secretLen, bool checkInput) {
			if (!p.isProbablePrime() || g <= 0 || g >= p || (x != null && (x <= 0 || x > p - 2)))
				throw new CryptographicException();
			// default is to generate a number as large as the prime this
			// is usually overkill, but it's the most secure thing we can
			// do if the user doesn't specify a desired secret length ...
			if (secretLen == 0)
				secretLen = p.bitCount();
			m_P = p;
			m_G = g;
			if (x == null) {
				BigInteger pm1 = m_P - 1;
				for(m_X = BigInteger.genRandom(secretLen); m_X >= pm1 || m_X == 0; m_X = BigInteger.genRandom(secretLen)) {}
			} else {
				m_X = x;
			}
		}
			public static void PlusEq (BigInteger bi1, BigInteger bi2)
			{
				uint [] x, y;
				uint yMax, xMax, i = 0;
				bool flag = false;

				// x should be bigger
				if (bi1.length < bi2.length){
					flag = true;
					x = bi2.data;
					xMax = bi2.length;
					y = bi1.data;
					yMax = bi1.length;
				} else {
					x = bi1.data;
					xMax = bi1.length;
					y = bi2.data;
					yMax = bi2.length;
				}

				uint [] r = bi1.data;

				ulong sum = 0;

				// Add common parts of both numbers
				do {
					sum += ((ulong)x [i]) + ((ulong)y [i]);
					r [i] = (uint)sum;
					sum >>= 32;
				} while (++i < yMax);

				// Copy remainder of longer number while carry propagation is required
				bool carry = (sum != 0);

				if (carry){

					if (i < xMax) {
						do
							carry = ((r [i] = x [i] + 1) == 0);
						while (++i < xMax && carry);
					}

					if (carry) {
						r [i] = 1;
						bi1.length = ++i;
						return;
					}
				}

				// Copy the rest
				if (flag && i < xMax - 1) {
					do
						r [i] = x [i];
					while (++i < xMax);
				}

				bi1.length = xMax + 1;
				bi1.Normalize ();
			}
			public BigInteger EvenPow (BigInteger b, BigInteger exp)
			{
				BigInteger resultNum = new BigInteger ((BigInteger)1, mod.length << 1);
				BigInteger tempNum = new BigInteger (b % mod, mod.length << 1);  // ensures (tempNum * tempNum) < b^ (2k)

				uint totalBits = (uint)exp.bitCount ();

				uint [] wkspace = new uint [mod.length << 1];

				// perform squaring and multiply exponentiation
				for (uint pos = 0; pos < totalBits; pos++) {
					if (exp.testBit (pos)) {

						Array.Clear (wkspace, 0, wkspace.Length);
						Kernel.Multiply (resultNum.data, 0, resultNum.length, tempNum.data, 0, tempNum.length, wkspace, 0);
						resultNum.length += tempNum.length;
						uint [] t = wkspace;
						wkspace = resultNum.data;
						resultNum.data = t;

						BarrettReduction (resultNum);
					}

					Kernel.SquarePositive (tempNum, ref wkspace);
					BarrettReduction (tempNum);

					if (tempNum == 1) {
						return resultNum;
					}
				}

				return resultNum;
			}
			public BigInteger Difference (BigInteger a, BigInteger b)
			{
				Sign cmp = Kernel.Compare (a, b);
				BigInteger diff;

				switch (cmp) {
					case Sign.Zero:
						return 0;
					case Sign.Positive:
						diff = a - b; break;
					case Sign.Negative:
						diff = b - a; break;
					default:
						throw new Exception ();
				}

				if (diff >= mod) {
					if (diff.length >= mod.length << 1)
						diff %= mod;
					else
						BarrettReduction (diff);
				}
				if (cmp == Sign.Negative)
					diff = mod - diff;
				return diff;
			}
			public void BarrettReduction (BigInteger x)
			{
				BigInteger n = mod;
				uint k = n.length,
					kPlusOne = k+1,
					kMinusOne = k-1;

				// x < mod, so nothing to do.
				if (x.length < k) return;

				BigInteger q3;

				//
				// Validate pointers
				//
				if (x.data.Length < x.length) throw new IndexOutOfRangeException ("x out of range");

				// q1 = x / b^ (k-1)
				// q2 = q1 * constant
				// q3 = q2 / b^ (k+1), Needs to be accessed with an offset of kPlusOne

				// TODO: We should the method in HAC p 604 to do this (14.45)
				q3 = new BigInteger (Sign.Positive, x.length - kMinusOne + constant.length);
				Kernel.Multiply (x.data, kMinusOne, x.length - kMinusOne, constant.data, 0, constant.length, q3.data, 0);

				// r1 = x mod b^ (k+1)
				// i.e. keep the lowest (k+1) words

				uint lengthToCopy = (x.length > kPlusOne) ? kPlusOne : x.length;

				x.length = lengthToCopy;
				x.Normalize ();

				// r2 = (q3 * n) mod b^ (k+1)
				// partial multiplication of q3 and n

				BigInteger r2 = new BigInteger (Sign.Positive, kPlusOne);
				Kernel.MultiplyMod2p32pmod (q3.data, (int)kPlusOne, (int)q3.length - (int)kPlusOne, n.data, 0, (int)n.length, r2.data, 0, (int)kPlusOne);

				r2.Normalize ();

				if (r2 < x) {
					Kernel.MinusEq (x, r2);
				} else {
					BigInteger val = new BigInteger (Sign.Positive, kPlusOne + 1);
					val.data [kPlusOne] = 0x00000001;

					Kernel.MinusEq (val, r2);
					Kernel.PlusEq (x, val);
				}

				while (x >= n)
					Kernel.MinusEq (x, n);
			}
		/// <summary>
		/// Generates the smallest prime >= bi
		/// </summary>
		/// <param name="bi">A BigInteger</param>
		/// <returns>The smallest prime >= bi. More mathematically, if bi is prime: bi, else Prime [PrimePi [bi] + 1].</returns>
		public static BigInteger NextHightestPrime (BigInteger bi)
		{
			NextPrimeFinder npf = new NextPrimeFinder ();
			return npf.GenerateNewPrime (0, bi);
		}
		public BigInteger modInverse (BigInteger mod)
		{
			return Kernel.modInverse (this, mod);
		}
		public string ToString (uint radix, string charSet)
		{
			if (charSet.Length < radix)
				throw new ArgumentException ("charSet length less than radix", "charSet");
			if (radix == 1)
				throw new ArgumentException ("There is no such thing as radix one notation", "radix");

			if (this == 0) return "0";
			if (this == 1) return "1";

			string result = "";

			BigInteger a = new BigInteger (this);

			while (a != 0) {
				uint rem = Kernel.SingleByteDivideInPlace (a, radix);
				result = charSet [ (int)rem] + result;
			}

			return result;
		}
			public static BigInteger LeftShift (BigInteger bi, int n)
			{
				if (n == 0) return new BigInteger (bi, bi.length + 1);

				int w = n >> 5;
				n &= ((1 << 5) - 1);

				BigInteger ret = new BigInteger (Sign.Positive, bi.length + 1 + (uint)w);

				uint i = 0, l = bi.length;
				if (n != 0) {
					uint x, carry = 0;
					while (i < l) {
						x = bi.data [i];
						ret.data [i + w] = (x << n) | carry;
						carry = x >> (32 - n);
						i++;
					}
					ret.data [i + w] = carry;
				} else {
					while (i < l) {
						ret.data [i + w] = bi.data [i];
						i++;
					}
				}

				ret.Normalize ();
				return ret;
			}
			public static BigInteger [] multiByteDivide (BigInteger bi1, BigInteger bi2)
			{
				if (Kernel.Compare (bi1, bi2) == Sign.Negative)
					return new BigInteger [2] { 0, new BigInteger (bi1) };

				bi1.Normalize (); bi2.Normalize ();

				if (bi2.length == 1)
					return DwordDivMod (bi1, bi2.data [0]);

				uint remainderLen = bi1.length + 1;
				int divisorLen = (int)bi2.length + 1;

				uint mask = 0x80000000;
				uint val = bi2.data [bi2.length - 1];
				int shift = 0;
				int resultPos = (int)bi1.length - (int)bi2.length;

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

				BigInteger quot = new BigInteger (Sign.Positive, bi1.length - bi2.length + 1);
				BigInteger rem = (bi1 << shift);

				uint [] remainder = rem.data;

				bi2 = bi2 << shift;

				int j = (int)(remainderLen - bi2.length);
				int pos = (int)remainderLen - 1;

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

				while (j > 0) {
					ulong dividend = ((ulong)remainder [pos] << 32) + (ulong)remainder [pos-1];

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

					do {

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

							if (r_hat < 0x100000000)
								continue;
						}
						break;
					} while (true);

					//
					// At this point, q_hat is either exact, or one too large
					// (more likely to be exact) so, we attempt to multiply the
					// divisor by q_hat, if we get a borrow, we just subtract
					// one from q_hat and add the divisor back.
					//

					uint t;
					uint dPos = 0;
					int nPos = pos - divisorLen + 1;
					ulong mc = 0;
					uint uint_q_hat = (uint)q_hat;
					do {
						mc += (ulong)bi2.data [dPos] * (ulong)uint_q_hat;
						t = remainder [nPos];
						remainder [nPos] -= (uint)mc;
						mc >>= 32;
						if (remainder [nPos] > t) mc++;
						dPos++; nPos++;
					} while (dPos < divisorLen);

					nPos = pos - divisorLen + 1;
					dPos = 0;

					// Overestimate
					if (mc != 0) {
						uint_q_hat--;
						ulong sum = 0;

						do {
							sum = ((ulong)remainder [nPos]) + ((ulong)bi2.data [dPos]) + sum;
							remainder [nPos] = (uint)sum;
							sum >>= 32;
							dPos++; nPos++;
						} while (dPos < divisorLen);

					}

					quot.data [resultPos--] = (uint)uint_q_hat;

					pos--;
					j--;
				}

				quot.Normalize ();
				rem.Normalize ();
				BigInteger [] ret = new BigInteger [2] { quot, rem };

				if (shift != 0)
					ret [1] >>= shift;

				return ret;
			}
			public static BigInteger [] DwordDivMod (BigInteger n, uint d)
			{
				BigInteger ret = new BigInteger (Sign.Positive , n.length);

				ulong r = 0;
				uint i = n.length;

				while (i-- > 0) {
					r <<= 32;
					r |= n.data [i];
					ret.data [i] = (uint)(r / d);
					r %= d;
				}
				ret.Normalize ();

				BigInteger rem = (uint)r;

				return new BigInteger [] {ret, rem};
			}
			public static uint DwordMod (BigInteger n, uint d)
			{
				ulong r = 0;
				uint i = n.length;

				while (i-- > 0) {
					r <<= 32;
					r |= n.data [i];
					r %= d;
				}

				return (uint)r;
			}
		public static BigInteger Parse(string number) {
			if (number == null)
				throw new ArgumentNullException(number);
			int i = 0, len = number.Length;
			char c;
			bool digits_seen = false;
			BigInteger val = new BigInteger(0);
			if (number[i] == '+') {
				i++;
			} else if(number[i] == '-') {
				throw new FormatException("Only positive integers are allowed.");
			}
			for(; i < len; i++) {
				c = number[i];
				if (c == '\0') {
					i = len;
					continue;
				}
				if (c >= '0' && c <= '9'){
					val = val * 10 + (c - '0');
					digits_seen = true;
				} else {
					if (Char.IsWhiteSpace(c)){
						for (i++; i < len; i++){
							if (!Char.IsWhiteSpace (number[i]))
								throw new FormatException();
						}
						break;
					} else
						throw new FormatException();
				}
			}
			if (!digits_seen)
				throw new FormatException();
			return val;
		}
			/// <summary>
			/// Performs n / d and n % d in one operation.
			/// </summary>
			/// <param name="n">A BigInteger, upon exit this will hold n / d</param>
			/// <param name="d">The divisor</param>
			/// <returns>n % d</returns>
			public static uint SingleByteDivideInPlace (BigInteger n, uint d)
			{
				ulong r = 0;
				uint i = n.length;

				while (i-- > 0) {
					r <<= 32;
					r |= n.data [i];
					n.data [i] = (uint)(r / d);
					r %= d;
				}
				n.Normalize ();

				return (uint)r;
			}
			/// <summary>
			/// Compares two BigInteger
			/// </summary>
			/// <param name="bi1">A BigInteger</param>
			/// <param name="bi2">A BigInteger</param>
			/// <returns>The sign of bi1 - bi2</returns>
			public static Sign Compare (BigInteger bi1, BigInteger bi2)
			{
				//
				// Step 1. Compare the lengths
				//
				uint l1 = bi1.length, l2 = bi2.length;

				while (l1 > 0 && bi1.data [l1-1] == 0) l1--;
				while (l2 > 0 && bi2.data [l2-1] == 0) l2--;

				if (l1 == 0 && l2 == 0) return Sign.Zero;

				// bi1 len < bi2 len
				if (l1 < l2) return Sign.Negative;
				// bi1 len > bi2 len
				else if (l1 > l2) return Sign.Positive;

				//
				// Step 2. Compare the bits
				//

				uint pos = l1 - 1;

				while (pos != 0 && bi1.data [pos] == bi2.data [pos]) pos--;
				
				if (bi1.data [pos] < bi2.data [pos])
					return Sign.Negative;
				else if (bi1.data [pos] > bi2.data [pos])
					return Sign.Positive;
				else
					return Sign.Zero;
			}
		/// <summary>
		/// Generates a new, random BigInteger of the specified length.
		/// </summary>
		/// <param name="bits">The number of bits for the new number.</param>
		/// <param name="rng">A random number generator to use to obtain the bits.</param>
		/// <returns>A random number of the specified length.</returns>
		public static BigInteger genRandom (int bits, RandomNumberGenerator rng)
		{
			int dwords = bits >> 5;
			int remBits = bits & 0x1F;

			if (remBits != 0)
				dwords++;

			BigInteger ret = new BigInteger (Sign.Positive, (uint)dwords + 1);
			byte [] random = new byte [dwords << 2];

			rng.GetBytes (random);
			Buffer.BlockCopy (random, 0, ret.data, 0, (int)dwords << 2);

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

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

			ret.Normalize ();
			return ret;
		}
		public Sign Compare (BigInteger bi)
		{
			return Kernel.Compare (this, bi);
		}
			public static BigInteger RightShift (BigInteger bi, int n)
			{
				if (n == 0) return new BigInteger (bi);

				int w = n >> 5;
				int s = n & ((1 << 5) - 1);

				BigInteger ret = new BigInteger (Sign.Positive, bi.length - (uint)w + 1);
				uint l = (uint)ret.data.Length - 1;

				if (s != 0) {

					uint x, carry = 0;

					while (l-- > 0) {
						x = bi.data [l + w];
						ret.data [l] = (x >> n) | carry;
						carry = x << (32 - n);
					}
				} else {
					while (l-- > 0)
						ret.data [l] = bi.data [l + w];

				}
				ret.Normalize ();
				return ret;
			}
		public BigInteger gcd (BigInteger bi)
		{
			return Kernel.gcd (this, bi);
		}
			public static BigInteger MultiplyByDword (BigInteger n, uint f)
			{
				BigInteger ret = new BigInteger (Sign.Positive, n.length + 1);

				uint i = 0;
				ulong c = 0;

				do {
					c += (ulong)n.data [i] * (ulong)f;
					ret.data [i] = (uint)c;
					c >>= 32;
				} while (++i < n.length);
				ret.data [i] = (uint)c;
				ret.Normalize ();
				return ret;

			}
		public BigInteger modPow (BigInteger exp, BigInteger n)
		{
			ModulusRing mr = new ModulusRing (n);
			return mr.Pow (this, exp);
		}
			public static unsafe void SquarePositive (BigInteger bi, ref uint [] wkSpace)
			{
				uint [] t = wkSpace;
				wkSpace = bi.data;
				uint [] d = bi.data;
				uint dl = bi.length;
				bi.data = t;

				fixed (uint* dd = d, tt = t) {

					uint* ttE = tt + t.Length;
					// Clear the dest
					for (uint* ttt = tt; ttt < ttE; ttt++)
						*ttt = 0;

					uint* dP = dd, tP = tt;

					for (uint i = 0; i < dl; i++, dP++) {
						if (*dP == 0)
							continue;

						ulong mcarry = 0;
						uint bi1val = *dP;

						uint* dP2 = dP + 1, tP2 = tP + 2*i + 1;

						for (uint j = i + 1; j < dl; j++, tP2++, dP2++) {
							// k = i + j
							mcarry += ((ulong)bi1val * (ulong)*dP2) + *tP2;

							*tP2 = (uint)mcarry;
							mcarry >>= 32;
						}

						if (mcarry != 0)
							*tP2 = (uint)mcarry;
					}

					// Double t. Inlined for speed.

					tP = tt;

					uint x, carry = 0;
					while (tP < ttE) {
						x = *tP;
						*tP = (x << 1) | carry;
						carry = x >> (32 - 1);
						tP++;
					}
					if (carry != 0) *tP = carry;

					// Add in the diagnals

					dP = dd;
					tP = tt;
					for (uint* dE = dP + dl; (dP < dE); dP++, tP++) {
						ulong val = (ulong)*dP * (ulong)*dP + *tP;
						*tP = (uint)val;
						val >>= 32;
						*(++tP) += (uint)val;
						if (*tP < (uint)val) {
							uint* tP3 = tP;
							// Account for the first carry
							(*++tP3)++;

							// Keep adding until no carry
							while ((*tP3++) == 0x0)
								(*tP3)++;
						}

					}

					bi.length <<= 1;

					// Normalize length
					while (tt [bi.length-1] == 0 && bi.length > 1) bi.length--;

				}
			}
			public ModulusRing (BigInteger mod)
			{
				this.mod = mod;

				// calculate constant = b^ (2k) / m
				uint i = mod.length << 1;

				constant = new BigInteger (Sign.Positive, i + 1);
				constant.data [i] = 0x00000001;

				constant = constant / mod;
			}
			public static BigInteger gcd (BigInteger a, BigInteger b)
			{
				BigInteger x = a;
				BigInteger y = b;

				BigInteger g = y;

				while (x.length > 1) {
					g = x;
					x = y % x;
					y = g;

				}
				if (x == 0) return g;

				// TODO: should we have something here if we can convert to long?

				//
				// Now we can just do it with single precision. I am using the binary gcd method,
				// as it should be faster.
				//

				uint yy = x.data [0];
				uint xx = y % yy;

				int t = 0;

				while (((xx | yy) & 1) == 0) {
					xx >>= 1; yy >>= 1; t++;
				}
				while (xx != 0) {
					while ((xx & 1) == 0) xx >>= 1;
					while ((yy & 1) == 0) yy >>= 1;
					if (xx >= yy)
						xx = (xx - yy) >> 1;
					else
						yy = (yy - xx) >> 1;
				}

				return yy << t;
			}
			public BigInteger Multiply (BigInteger a, BigInteger b)
			{
				if (a == 0 || b == 0) return 0;

				if (a.length >= mod.length << 1)
					a %= mod;

				if (b.length >= mod.length << 1)
					b %= mod;

				if (a.length >= mod.length)
					BarrettReduction (a);

				if (b.length >= mod.length)
					BarrettReduction (b);

				BigInteger ret = new BigInteger (a * b);
				BarrettReduction (ret);

				return ret;
			}
			public static uint modInverse (BigInteger bi, uint modulus)
			{
				uint a = modulus, b = bi % modulus;
				uint p0 = 0, p1 = 1;

				while (b != 0) {
					if (b == 1)
						return p1;
					p0 += (a / b) * p1;
					a %= b;

					if (a == 0)
						break;
					if (a == 1)
						return modulus-p0;

					p1 += (b / a) * p0;
					b %= a;

				}
				return 0;
			}
			public BigInteger Pow (BigInteger b, BigInteger exp)
			{
				if ((mod.data [0] & 1) == 1) return OddPow (b, exp);
				else return EvenPow (b, exp);
			}
			public static BigInteger modInverse (BigInteger bi, BigInteger modulus)
			{
				if (modulus.length == 1) return modInverse (bi, modulus.data [0]);

				BigInteger [] p = { 0, 1 };
				BigInteger [] q = new BigInteger [2];    // quotients
				BigInteger [] r = { 0, 0 };             // remainders

				int step = 0;

				BigInteger a = modulus;
				BigInteger b = bi;

				ModulusRing mr = new ModulusRing (modulus);

				while (b != 0) {

					if (step > 1) {

						BigInteger pval = mr.Difference (p [0], p [1] * q [0]);
						p [0] = p [1]; p [1] = pval;
					}

					BigInteger [] divret = multiByteDivide (a, b);

					q [0] = q [1]; q [1] = divret [0];
					r [0] = r [1]; r [1] = divret [1];
					a = b;
					b = divret [1];

					step++;
				}

				if (r [0] != 1)
					throw (new ArithmeticException ("No inverse!"));

				return mr.Difference (p [0], p [1] * q [0]);

			}
			private BigInteger OddPow (BigInteger b, BigInteger exp)
			{
				BigInteger resultNum = new BigInteger (Montgomery.ToMont (1, mod), mod.length << 1);
				BigInteger tempNum = new BigInteger (Montgomery.ToMont (b, mod), mod.length << 1);  // ensures (tempNum * tempNum) < b^ (2k)
				uint mPrime = Montgomery.Inverse (mod.data [0]);
				uint totalBits = (uint)exp.bitCount ();

				uint [] wkspace = new uint [mod.length << 1];

				// perform squaring and multiply exponentiation
				for (uint pos = 0; pos < totalBits; pos++) {
					if (exp.testBit (pos)) {

						Array.Clear (wkspace, 0, wkspace.Length);
						Kernel.Multiply (resultNum.data, 0, resultNum.length, tempNum.data, 0, tempNum.length, wkspace, 0);
						resultNum.length += tempNum.length;
						uint [] t = wkspace;
						wkspace = resultNum.data;
						resultNum.data = t;

						Montgomery.Reduce (resultNum, mod, mPrime);
					}

					Kernel.SquarePositive (tempNum, ref wkspace);
					Montgomery.Reduce (tempNum, mod, mPrime);
				}

				Montgomery.Reduce (resultNum, mod, mPrime);
				return resultNum;
			}
		public static BigInteger operator * (BigInteger bi1, BigInteger bi2)
		{
			if (bi1 == 0 || bi2 == 0) return 0;

			//
			// Validate pointers
			//
			if (bi1.data.Length < bi1.length) throw new IndexOutOfRangeException ("bi1 out of range");
			if (bi2.data.Length < bi2.length) throw new IndexOutOfRangeException ("bi2 out of range");

			BigInteger ret = new BigInteger (Sign.Positive, bi1.length + bi2.length);

			Kernel.Multiply (bi1.data, 0, bi1.length, bi2.data, 0, bi2.length, ret.data, 0);

			ret.Normalize ();
			return ret;
		}