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djhohnstein/FastDoubling.cs

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FastDoubling - Mod RSA 129
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FastDoubling.cs
using System;
class GFG{
// Function calculate the N-th fibonacci
// number using fast doubling method
static void FastDoubling(BigInteger n, BigInteger []res)
{
BigInteger a, b, c, d;
BigInteger MOD = new BigInteger ("114381625757888867669235779976146612010218296721242362562561842935706935245733897830597123563958705058989075147599290026879543541",10);
// Base Condition
if (n == 0)
{
res[0] = 0;
res[1] = 1;
return;
}
FastDoubling((n / 2), res);
// Here a = F(n)
a = res[0];
// Here b = F(n+1)
b = res[1];
c = 2 * b - a;
if (c < 0)
c += MOD;
// As F(2n) = F(n)[2F(n+1) – F(n)]
// Here c = F(2n)
c = (a * c) % MOD;
// As F(2n + 1) = F(n)^2 + F(n+1)^2
// Here d = F(2n + 1)
d = (a * a + b * b) % MOD;
// Check if N is odd
// or even
if (n % 2 == 0)
{
res[0] = c;
res[1] = d;
}
else
{
res[0] = d;
res[1] = c + d;
}
}
// Driver code
public static void Main()
{
BigInteger N = new BigInteger("3490529510847650949147849619903898133417764638493387843990820577",10);
BigInteger []res = new BigInteger[2];
FastDoubling(N, res);
Console.Write(res[0]);
}
}
//************************************************************************************
// 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.
//
//************************************************************************************
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 = 2048;
// 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)
{
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, Random rand)
{
int dwords = bits >> 5;
int remBits = bits & 0x1F;
if(remBits != 0)
dwords++;
if(dwords > maxLength)
throw (new ArithmeticException("Number of required bits > maxLength."));
for(int i = 0; i < dwords; i++)
data[i] = (uint)(rand.NextDouble() * 0x100000000);
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));
Random rand = new Random();
for(int round = 0; round < confidence; round++)
{
bool done = false;
while(!done) // generate a < n
{
int testBits = 0;
// make sure "a" has at least 2 bits
while(testBits < 2)
testBits = (int)(rand.NextDouble() * bits);
a.genRandomBits(testBits, rand);
int byteLen = a.dataLength;
// make sure "a" is not 0
if(byteLen > 1 || (byteLen == 1 && a.data[0] != 1))
done = true;
}
// check whether a factor exists (fix for version 1.03)
BigInteger gcdTest = a.gcd(thisVal);
if(gcdTest.dataLength == 1 && gcdTest.data[0] != 1)
return false;
// calculate a^(p-1) mod p
BigInteger expResult = a.modPow(p_sub1, thisVal);
int resultLen = expResult.dataLength;
// is NOT prime is a^(p-1) mod p != 1
if(resultLen > 1 || (resultLen == 1 && expResult.data[0] != 1))
{
//Console.WriteLine("a = " + a.ToString());
return false;
}
}
return true;
}
//***********************************************************************
// Probabilistic prime test based on Rabin-Miller's
//
// for any p > 0 with p - 1 = 2^s * t
//
// p is probably prime (strong pseudoprime) if for any a < p,
// 1) a^t mod p = 1 or
// 2) a^((2^j)*t) mod p = p-1 for some 0 <= j <= s-1
//
// Otherwise, p is composite.
//
// Returns
// -------
// True if "this" is a strong pseudoprime to randomly chosen
// bases. The number of chosen bases is given by the "confidence"
// parameter.
//
// False if "this" is definitely NOT prime.
//
//***********************************************************************
public bool RabinMillerTest(int confidence)
{
BigInteger thisVal;
if((this.data[maxLength-1] & 0x80000000) != 0) // negative
thisVal = -this;
else
thisVal = this;
if(thisVal.dataLength == 1)
{
// test small numbers
if(thisVal.data[0] == 0 || thisVal.data[0] == 1)
return false;
else if(thisVal.data[0] == 2 || thisVal.data[0] == 3)
return true;
}
if((thisVal.data[0] & 0x1) == 0) // even numbers
return false;
// calculate values of s and t
BigInteger p_sub1 = thisVal - (new BigInteger(1));
int s = 0;
for(int index = 0; index < p_sub1.dataLength; index++)
{
uint mask = 0x01;
for(int i = 0; i < 32; i++)
{
if((p_sub1.data[index] & mask) != 0)
{
index = p_sub1.dataLength; // to break the outer loop
break;
}
mask <<= 1;
s++;
}
}
BigInteger t = p_sub1 >> s;
int bits = thisVal.bitCount();
BigInteger a = new BigInteger();
Random rand = new Random();
for(int round = 0; round < confidence; round++)
{
bool done = false;
while(!done) // generate a < n
{
int testBits = 0;
// make sure "a" has at least 2 bits
while(testBits < 2)
testBits = (int)(rand.NextDouble() * bits);
a.genRandomBits(testBits, rand);
int byteLen = a.dataLength;
// make sure "a" is not 0
if(byteLen > 1 || (byteLen == 1 && a.data[0] != 1))
done = true;
}
// check whether a factor exists (fix for version 1.03)
BigInteger gcdTest = a.gcd(thisVal);
if(gcdTest.dataLength == 1 && gcdTest.data[0] != 1)
return false;
BigInteger b = a.modPow(t, thisVal);
/*
Console.WriteLine("a = " + a.ToString(10));
Console.WriteLine("b = " + b.ToString(10));
Console.WriteLine("t = " + t.ToString(10));
Console.WriteLine("s = " + s);
*/
bool result = false;
if(b.dataLength == 1 && b.data[0] == 1) // a^t mod p = 1
result = true;
for(int j = 0; result == false && j < s; j++)
{
if(b == p_sub1) // a^((2^j)*t) mod p = p-1 for some 0 <= j <= s-1
{
result = true;
break;
}
b = (b * b) % thisVal;
}
if(result == false)
return false;
}
return true;
}
//***********************************************************************
// Probabilistic prime test based on Solovay-Strassen (Euler Criterion)
//
// p is probably prime if for any a < p (a is not multiple of p),
// a^((p-1)/2) mod p = J(a, p)
//
// where J is the Jacobi symbol.
//
// Otherwise, p is composite.
//
// Returns
// -------
// True if "this" is a Euler pseudoprime to randomly chosen
// bases. The number of chosen bases is given by the "confidence"
// parameter.
//
// False if "this" is definitely NOT prime.
//
//***********************************************************************
public bool SolovayStrassenTest(int confidence)
{
BigInteger thisVal;
if((this.data[maxLength-1] & 0x80000000) != 0) // negative
thisVal = -this;
else
thisVal = this;
if(thisVal.dataLength == 1)
{
// test small numbers
if(thisVal.data[0] == 0 || thisVal.data[0] == 1)
return false;
else if(thisVal.data[0] == 2 || thisVal.data[0] == 3)
return true;
}
if((thisVal.data[0] & 0x1) == 0) // even numbers
return false;
int bits = thisVal.bitCount();
BigInteger a = new BigInteger();
BigInteger p_sub1 = thisVal - 1;
BigInteger p_sub1_shift = p_sub1 >> 1;
Random rand = new Random();
for(int round = 0; round < confidence; round++)
{
bool done = false;
while(!done) // generate a < n
{
int testBits = 0;
// make sure "a" has at least 2 bits
while(testBits < 2)
testBits = (int)(rand.NextDouble() * bits);
a.genRandomBits(testBits, rand);
int byteLen = a.dataLength;
// make sure "a" is not 0
if(byteLen > 1 || (byteLen == 1 && a.data[0] != 1))
done = true;
}
// check whether a factor exists (fix for version 1.03)
BigInteger gcdTest = a.gcd(thisVal);
if(gcdTest.dataLength == 1 && gcdTest.data[0] != 1)
return false;
// calculate a^((p-1)/2) mod p
BigInteger expResult = a.modPow(p_sub1_shift, thisVal);
if(expResult == p_sub1)
expResult = -1;
// calculate Jacobi symbol
BigInteger jacob = Jacobi(a, thisVal);
//Console.WriteLine("a = " + a.ToString(10) + " b = " + thisVal.ToString(10));
//Console.WriteLine("expResult = " + expResult.ToString(10) + " Jacob = " + jacob.ToString(10));
// if they are different then it is not prime
if(expResult != jacob)
return false;
}
return true;
}
//***********************************************************************
// 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 && Math.Abs(D) < thisVal) // divisor found
return false;
if(dCount == 20)
{
// check for square
BigInteger root = thisVal.sqrt();
if(root * root == thisVal)
return false;
}
//Console.WriteLine(D);
D = (Math.Abs(D) + 2) * sign;
sign = -sign;
}
dCount++;
}
long Q = (1 - D) >> 2;
/*
Console.WriteLine("D = " + D);
Console.WriteLine("Q = " + Q);
Console.WriteLine("(n,D) = " + thisVal.gcd(D));
Console.WriteLine("(n,Q) = " + thisVal.gcd(Q));
Console.WriteLine("J(D|n) = " + BigInteger.Jacobi(D, thisVal));
*/
BigInteger p_add1 = thisVal + 1;
int s = 0;
for(int index = 0; index < p_add1.dataLength; index++)
{
uint mask = 0x01;
for(int i = 0; i < 32; i++)
{
if((p_add1.data[index] & mask) != 0)
{
index = p_add1.dataLength; // to break the outer loop
break;
}
mask <<= 1;
s++;
}
}
BigInteger t = p_add1 >> s;
// calculate constant = b^(2k) / m
// for Barrett Reduction
BigInteger constant = new BigInteger();
int nLen = thisVal.dataLength << 1;
constant.data[nLen] = 0x00000001;
constant.dataLength = nLen + 1;
constant = constant / thisVal;
BigInteger[] lucas = LucasSequenceHelper(1, Q, t, thisVal, constant, 0);
bool isPrime = false;
if((lucas[0].dataLength == 1 && lucas[0].data[0] == 0) ||
(lucas[1].dataLength == 1 && lucas[1].data[0] == 0))
{
// u(t) = 0 or V(t) = 0
isPrime = true;
}
for(int i = 1; i < s; i++)
{
if(!isPrime)
{
// doubling of index
lucas[1] = thisVal.BarrettReduction(lucas[1] * lucas[1], thisVal, constant);
lucas[1] = (lucas[1] - (lucas[2] << 1)) % thisVal;
//lucas[1] = ((lucas[1] * lucas[1]) - (lucas[2] << 1)) % thisVal;
if((lucas[1].dataLength == 1 && lucas[1].data[0] == 0))
isPrime = true;
}
lucas[2] = thisVal.BarrettReduction(lucas[2] * lucas[2], thisVal, constant); //Q^k
}
if(isPrime) // additional checks for composite numbers
{
// If n is prime and gcd(n, Q) == 1, then
// Q^((n+1)/2) = Q * Q^((n-1)/2) is congruent to (Q * J(Q, n)) mod n
BigInteger g = thisVal.gcd(Q);
if(g.dataLength == 1 && g.data[0] == 1) // gcd(this, Q) == 1
{
if((lucas[2].data[maxLength-1] & 0x80000000) != 0)
lucas[2] += thisVal;
BigInteger temp = (Q * BigInteger.Jacobi(Q, thisVal)) % thisVal;
if((temp.data[maxLength-1] & 0x80000000) != 0)
temp += thisVal;
if(lucas[2] != temp)
isPrime = false;
}
}
return isPrime;
}
//***********************************************************************
// 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, Random rand)
{
BigInteger result = new BigInteger();
bool done = false;
while(!done)
{
result.genRandomBits(bits, rand);
result.data[0] |= 0x01; // make it odd
// prime test
done = result.isProbablePrime(confidence);
}
return result;
}
//***********************************************************************
// Generates a random number with the specified number of bits such
// that gcd(number, this) = 1
//***********************************************************************
public BigInteger genCoPrime(int bits, Random rand)
{
bool done = false;
BigInteger result = new BigInteger();
while(!done)
{
result.genRandomBits(bits, rand);
//Console.WriteLine(result.ToString(16));
// gcd test
BigInteger g = result.gcd(this);
if(g.dataLength == 1 && g.data[0] == 1)
done = true;
}
return result;
}
//***********************************************************************
// 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();
int numBytes = numBits >> 3;
if((numBits & 0x7) != 0)
numBytes++;
byte[] result = new byte[numBytes];
//Console.WriteLine(result.Length);
int pos = 0;
uint tempVal, val = data[dataLength - 1];
if((tempVal = (val >> 24 & 0xFF)) != 0)
result[pos++] = (byte)tempVal;
if((tempVal = (val >> 16 & 0xFF)) != 0)
result[pos++] = (byte)tempVal;
if((tempVal = (val >> 8 & 0xFF)) != 0)
result[pos++] = (byte)tempVal;
if((tempVal = (val & 0xFF)) != 0)
result[pos++] = (byte)tempVal;
for(int i = dataLength - 2; i >= 0; i--, pos += 4)
{
val = data[i];
result[pos+3] = (byte)(val & 0xFF);
val >>= 8;
result[pos+2] = (byte)(val & 0xFF);
val >>= 8;
result[pos+1] = (byte)(val & 0xFF);
val >>= 8;
result[pos] = (byte)(val & 0xFF);
}
return result;
}
//***********************************************************************
// 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;
}
//***********************************************************************
// Tests the correct implementation of the /, %, * and + operators
//***********************************************************************
public static void MulDivTest(int rounds)
{
Random rand = new Random();
byte[] val = new byte[64];
byte[] val2 = new byte[64];
for(int count = 0; count < rounds; count++)
{
// generate 2 numbers of random length
int t1 = 0;
while(t1 == 0)
t1 = (int)(rand.NextDouble() * 65);
int t2 = 0;
while(t2 == 0)
t2 = (int)(rand.NextDouble() * 65);
bool done = false;
while(!done)
{
for(int i = 0; i < 64; i++)
{
if(i < t1)
val[i] = (byte)(rand.NextDouble() * 256);
else
val[i] = 0;
if(val[i] != 0)
done = true;
}
}
done = false;
while(!done)
{
for(int i = 0; i < 64; i++)
{
if(i < t2)
val2[i] = (byte)(rand.NextDouble() * 256);
else
val2[i] = 0;
if(val2[i] != 0)
done = true;
}
}
while(val[0] == 0)
val[0] = (byte)(rand.NextDouble() * 256);
while(val2[0] == 0)
val2[0] = (byte)(rand.NextDouble() * 256);
Console.WriteLine(count);
BigInteger bn1 = new BigInteger(val, t1);
BigInteger bn2 = new BigInteger(val2, t2);
// Determine the quotient and remainder by dividing
// the first number by the second.
BigInteger bn3 = bn1 / bn2;
BigInteger bn4 = bn1 % bn2;
// Recalculate the number
BigInteger bn5 = (bn3 * bn2) + bn4;
// Make sure they're the same
if(bn5 != bn1)
{
Console.WriteLine("Error at " + count);
Console.WriteLine(bn1 + "\n");
Console.WriteLine(bn2 + "\n");
Console.WriteLine(bn3 + "\n");
Console.WriteLine(bn4 + "\n");
Console.WriteLine(bn5 + "\n");
return;
}
}
}
//***********************************************************************
// Tests the correct implementation of the modulo exponential function
// using RSA encryption and decryption (using pre-computed encryption and
// decryption keys).
//***********************************************************************
public static void RSATest(int rounds)
{
Random rand = new Random(1);
byte[] val = new byte[64];
// private and public key
BigInteger bi_e = new BigInteger("a932b948feed4fb2b692609bd22164fc9edb59fae7880cc1eaff7b3c9626b7e5b241c27a974833b2622ebe09beb451917663d47232488f23a117fc97720f1e7", 16);
BigInteger bi_d = new BigInteger("4adf2f7a89da93248509347d2ae506d683dd3a16357e859a980c4f77a4e2f7a01fae289f13a851df6e9db5adaa60bfd2b162bbbe31f7c8f828261a6839311929d2cef4f864dde65e556ce43c89bbbf9f1ac5511315847ce9cc8dc92470a747b8792d6a83b0092d2e5ebaf852c85cacf34278efa99160f2f8aa7ee7214de07b7", 16);
BigInteger bi_n = new BigInteger("e8e77781f36a7b3188d711c2190b560f205a52391b3479cdb99fa010745cbeba5f2adc08e1de6bf38398a0487c4a73610d94ec36f17f3f46ad75e17bc1adfec99839589f45f95ccc94cb2a5c500b477eb3323d8cfab0c8458c96f0147a45d27e45a4d11d54d77684f65d48f15fafcc1ba208e71e921b9bd9017c16a5231af7f", 16);
Console.WriteLine("e =\n" + bi_e.ToString(10));
Console.WriteLine("\nd =\n" + bi_d.ToString(10));
Console.WriteLine("\nn =\n" + bi_n.ToString(10) + "\n");
for(int count = 0; count < rounds; count++)
{
// generate data of random length
int t1 = 0;
while(t1 == 0)
t1 = (int)(rand.NextDouble() * 65);
bool done = false;
while(!done)
{
for(int i = 0; i < 64; i++)
{
if(i < t1)
val[i] = (byte)(rand.NextDouble() * 256);
else
val[i] = 0;
if(val[i] != 0)
done = true;
}
}
while(val[0] == 0)
val[0] = (byte)(rand.NextDouble() * 256);
Console.Write("Round = " + count);
// encrypt and decrypt data
BigInteger bi_data = new BigInteger(val, t1);
BigInteger bi_encrypted = bi_data.modPow(bi_e, bi_n);
BigInteger bi_decrypted = bi_encrypted.modPow(bi_d, bi_n);
// compare
if(bi_decrypted != bi_data)
{
Console.WriteLine("\nError at round " + count);
Console.WriteLine(bi_data + "\n");
return;
}
Console.WriteLine(" <PASSED>.");
}
}
//***********************************************************************
// Tests the correct implementation of the modulo exponential and
// inverse modulo functions using RSA encryption and decryption. The two
// pseudoprimes p and q are fixed, but the two RSA keys are generated
// for each round of testing.
//***********************************************************************
public static void RSATest2(int rounds)
{
Random rand = new Random();
byte[] val = new byte[64];
byte[] pseudoPrime1 = {
(byte)0x85, (byte)0x84, (byte)0x64, (byte)0xFD, (byte)0x70, (byte)0x6A,
(byte)0x9F, (byte)0xF0, (byte)0x94, (byte)0x0C, (byte)0x3E, (byte)0x2C,
(byte)0x74, (byte)0x34, (byte)0x05, (byte)0xC9, (byte)0x55, (byte)0xB3,
(byte)0x85, (byte)0x32, (byte)0x98, (byte)0x71, (byte)0xF9, (byte)0x41,
(byte)0x21, (byte)0x5F, (byte)0x02, (byte)0x9E, (byte)0xEA, (byte)0x56,
(byte)0x8D, (byte)0x8C, (byte)0x44, (byte)0xCC, (byte)0xEE, (byte)0xEE,
(byte)0x3D, (byte)0x2C, (byte)0x9D, (byte)0x2C, (byte)0x12, (byte)0x41,
(byte)0x1E, (byte)0xF1, (byte)0xC5, (byte)0x32, (byte)0xC3, (byte)0xAA,
(byte)0x31, (byte)0x4A, (byte)0x52, (byte)0xD8, (byte)0xE8, (byte)0xAF,
(byte)0x42, (byte)0xF4, (byte)0x72, (byte)0xA1, (byte)0x2A, (byte)0x0D,
(byte)0x97, (byte)0xB1, (byte)0x31, (byte)0xB3,
};
byte[] pseudoPrime2 = {
(byte)0x99, (byte)0x98, (byte)0xCA, (byte)0xB8, (byte)0x5E, (byte)0xD7,
(byte)0xE5, (byte)0xDC, (byte)0x28, (byte)0x5C, (byte)0x6F, (byte)0x0E,
(byte)0x15, (byte)0x09, (byte)0x59, (byte)0x6E, (byte)0x84, (byte)0xF3,
(byte)0x81, (byte)0xCD, (byte)0xDE, (byte)0x42, (byte)0xDC, (byte)0x93,
(byte)0xC2, (byte)0x7A, (byte)0x62, (byte)0xAC, (byte)0x6C, (byte)0xAF,
(byte)0xDE, (byte)0x74, (byte)0xE3, (byte)0xCB, (byte)0x60, (byte)0x20,
(byte)0x38, (byte)0x9C, (byte)0x21, (byte)0xC3, (byte)0xDC, (byte)0xC8,
(byte)0xA2, (byte)0x4D, (byte)0xC6, (byte)0x2A, (byte)0x35, (byte)0x7F,
(byte)0xF3, (byte)0xA9, (byte)0xE8, (byte)0x1D, (byte)0x7B, (byte)0x2C,
(byte)0x78, (byte)0xFA, (byte)0xB8, (byte)0x02, (byte)0x55, (byte)0x80,
(byte)0x9B, (byte)0xC2, (byte)0xA5, (byte)0xCB,
};
BigInteger bi_p = new BigInteger(pseudoPrime1);
BigInteger bi_q = new BigInteger(pseudoPrime2);
BigInteger bi_pq = (bi_p-1)*(bi_q-1);
BigInteger bi_n = bi_p * bi_q;
for(int count = 0; count < rounds; count++)
{
// generate private and public key
BigInteger bi_e = bi_pq.genCoPrime(512, rand);
BigInteger bi_d = bi_e.modInverse(bi_pq);
Console.WriteLine("\ne =\n" + bi_e.ToString(10));
Console.WriteLine("\nd =\n" + bi_d.ToString(10));
Console.WriteLine("\nn =\n" + bi_n.ToString(10) + "\n");
// generate data of random length
int t1 = 0;
while(t1 == 0)
t1 = (int)(rand.NextDouble() * 65);
bool done = false;
while(!done)
{
for(int i = 0; i < 64; i++)
{
if(i < t1)
val[i] = (byte)(rand.NextDouble() * 256);
else
val[i] = 0;
if(val[i] != 0)
done = true;
}
}
while(val[0] == 0)
val[0] = (byte)(rand.NextDouble() * 256);
Console.Write("Round = " + count);
// encrypt and decrypt data
BigInteger bi_data = new BigInteger(val, t1);
BigInteger bi_encrypted = bi_data.modPow(bi_e, bi_n);
BigInteger bi_decrypted = bi_encrypted.modPow(bi_d, bi_n);
// compare
if(bi_decrypted != bi_data)
{
Console.WriteLine("\nError at round " + count);
Console.WriteLine(bi_data + "\n");
return;
}
Console.WriteLine(" <PASSED>.");
}
}
//***********************************************************************
// Tests the correct implementation of sqrt() method.
//***********************************************************************
public static void SqrtTest(int rounds)
{
Random rand = new Random();
for(int count = 0; count < rounds; count++)
{
// generate data of random length
int t1 = 0;
while(t1 == 0)
t1 = (int)(rand.NextDouble() * 1024);
Console.Write("Round = " + count);
BigInteger a = new BigInteger();
a.genRandomBits(t1, rand);
BigInteger b = a.sqrt();
BigInteger c = (b+1)*(b+1);
// check that b is the largest integer such that b*b <= a
if(c <= a)
{
Console.WriteLine("\nError at round " + count);
Console.WriteLine(a + "\n");
return;
}
Console.WriteLine(" <PASSED>.");
}
}
/*
public static void Main(string[] args)
{
// Known problem -> these two pseudoprimes passes my implementation of
// primality test but failed in JDK's isProbablePrime test.
byte[] pseudoPrime1 = { (byte)0x00,
(byte)0x85, (byte)0x84, (byte)0x64, (byte)0xFD, (byte)0x70, (byte)0x6A,
(byte)0x9F, (byte)0xF0, (byte)0x94, (byte)0x0C, (byte)0x3E, (byte)0x2C,
(byte)0x74, (byte)0x34, (byte)0x05, (byte)0xC9, (byte)0x55, (byte)0xB3,
(byte)0x85, (byte)0x32, (byte)0x98, (byte)0x71, (byte)0xF9, (byte)0x41,
(byte)0x21, (byte)0x5F, (byte)0x02, (byte)0x9E, (byte)0xEA, (byte)0x56,
(byte)0x8D, (byte)0x8C, (byte)0x44, (byte)0xCC, (byte)0xEE, (byte)0xEE,
(byte)0x3D, (byte)0x2C, (byte)0x9D, (byte)0x2C, (byte)0x12, (byte)0x41,
(byte)0x1E, (byte)0xF1, (byte)0xC5, (byte)0x32, (byte)0xC3, (byte)0xAA,
(byte)0x31, (byte)0x4A, (byte)0x52, (byte)0xD8, (byte)0xE8, (byte)0xAF,
(byte)0x42, (byte)0xF4, (byte)0x72, (byte)0xA1, (byte)0x2A, (byte)0x0D,
(byte)0x97, (byte)0xB1, (byte)0x31, (byte)0xB3,
};
byte[] pseudoPrime2 = { (byte)0x00,
(byte)0x99, (byte)0x98, (byte)0xCA, (byte)0xB8, (byte)0x5E, (byte)0xD7,
(byte)0xE5, (byte)0xDC, (byte)0x28, (byte)0x5C, (byte)0x6F, (byte)0x0E,
(byte)0x15, (byte)0x09, (byte)0x59, (byte)0x6E, (byte)0x84, (byte)0xF3,
(byte)0x81, (byte)0xCD, (byte)0xDE, (byte)0x42, (byte)0xDC, (byte)0x93,
(byte)0xC2, (byte)0x7A, (byte)0x62, (byte)0xAC, (byte)0x6C, (byte)0xAF,
(byte)0xDE, (byte)0x74, (byte)0xE3, (byte)0xCB, (byte)0x60, (byte)0x20,
(byte)0x38, (byte)0x9C, (byte)0x21, (byte)0xC3, (byte)0xDC, (byte)0xC8,
(byte)0xA2, (byte)0x4D, (byte)0xC6, (byte)0x2A, (byte)0x35, (byte)0x7F,
(byte)0xF3, (byte)0xA9, (byte)0xE8, (byte)0x1D, (byte)0x7B, (byte)0x2C,
(byte)0x78, (byte)0xFA, (byte)0xB8, (byte)0x02, (byte)0x55, (byte)0x80,
(byte)0x9B, (byte)0xC2, (byte)0xA5, (byte)0xCB,
};
Console.WriteLine("List of primes < 2000\n---------------------");
int limit = 100, count = 0;
for(int i = 0; i < 2000; i++)
{
if(i >= limit)
{
Console.WriteLine();
limit += 100;
}
BigInteger p = new BigInteger(-i);
if(p.isProbablePrime())
{
Console.Write(i + ", ");
count++;
}
}
Console.WriteLine("\nCount = " + count);
BigInteger bi1 = new BigInteger(pseudoPrime1);
Console.WriteLine("\n\nPrimality testing for\n" + bi1.ToString() + "\n");
Console.WriteLine("SolovayStrassenTest(5) = " + bi1.SolovayStrassenTest(5));
Console.WriteLine("RabinMillerTest(5) = " + bi1.RabinMillerTest(5));
Console.WriteLine("FermatLittleTest(5) = " + bi1.FermatLittleTest(5));
Console.WriteLine("isProbablePrime() = " + bi1.isProbablePrime());
Console.Write("\nGenerating 512-bits random pseudoprime. . .");
Random rand = new Random();
BigInteger prime = BigInteger.genPseudoPrime(512, 5, rand);
Console.WriteLine("\n" + prime);
//int dwStart = System.Environment.TickCount;
//BigInteger.MulDivTest(100000);
//BigInteger.RSATest(10);
//BigInteger.RSATest2(10);
//Console.WriteLine(System.Environment.TickCount - dwStart);
}
*/
}
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