|
using System;
|
using System.Collections.Generic;
|
using System.Linq;
|
using System.Text;
|
|
namespace Shared.Encryption
|
{
|
public class BigInteger
|
{
|
// maximum length of the BigInteger in uint (4 bytes)
|
// change this to suit the required level of precision.
|
|
private const int maxLength = 70;
|
|
// primes smaller than 2000 to test the generated prime number
|
|
public static readonly int[] primesBelow2000 = {
|
2, 3, 5, 7, 11, 13, 17, 19, 23, 29, 31, 37, 41, 43, 47, 53, 59, 61, 67, 71, 73, 79, 83, 89, 97,
|
101, 103, 107, 109, 113, 127, 131, 137, 139, 149, 151, 157, 163, 167, 173, 179, 181, 191, 193, 197, 199,
|
211, 223, 227, 229, 233, 239, 241, 251, 257, 263, 269, 271, 277, 281, 283, 293,
|
307, 311, 313, 317, 331, 337, 347, 349, 353, 359, 367, 373, 379, 383, 389, 397,
|
401, 409, 419, 421, 431, 433, 439, 443, 449, 457, 461, 463, 467, 479, 487, 491, 499,
|
503, 509, 521, 523, 541, 547, 557, 563, 569, 571, 577, 587, 593, 599,
|
601, 607, 613, 617, 619, 631, 641, 643, 647, 653, 659, 661, 673, 677, 683, 691,
|
701, 709, 719, 727, 733, 739, 743, 751, 757, 761, 769, 773, 787, 797,
|
809, 811, 821, 823, 827, 829, 839, 853, 857, 859, 863, 877, 881, 883, 887,
|
907, 911, 919, 929, 937, 941, 947, 953, 967, 971, 977, 983, 991, 997,
|
1009, 1013, 1019, 1021, 1031, 1033, 1039, 1049, 1051, 1061, 1063, 1069, 1087, 1091, 1093, 1097,
|
1103, 1109, 1117, 1123, 1129, 1151, 1153, 1163, 1171, 1181, 1187, 1193,
|
1201, 1213, 1217, 1223, 1229, 1231, 1237, 1249, 1259, 1277, 1279, 1283, 1289, 1291, 1297,
|
1301, 1303, 1307, 1319, 1321, 1327, 1361, 1367, 1373, 1381, 1399,
|
1409, 1423, 1427, 1429, 1433, 1439, 1447, 1451, 1453, 1459, 1471, 1481, 1483, 1487, 1489, 1493, 1499,
|
1511, 1523, 1531, 1543, 1549, 1553, 1559, 1567, 1571, 1579, 1583, 1597,
|
1601, 1607, 1609, 1613, 1619, 1621, 1627, 1637, 1657, 1663, 1667, 1669, 1693, 1697, 1699,
|
1709, 1721, 1723, 1733, 1741, 1747, 1753, 1759, 1777, 1783, 1787, 1789,
|
1801, 1811, 1823, 1831, 1847, 1861, 1867, 1871, 1873, 1877, 1879, 1889,
|
1901, 1907, 1913, 1931, 1933, 1949, 1951, 1973, 1979, 1987, 1993, 1997, 1999 };
|
|
|
private uint[] data = null; // stores bytes from the Big Integer
|
public int dataLength; // number of actual chars used
|
|
|
//***********************************************************************
|
// Constructor (Default value for BigInteger is 0
|
//***********************************************************************
|
|
public BigInteger()
|
{
|
data = new uint[maxLength];
|
dataLength = 1;
|
}
|
|
|
//***********************************************************************
|
// Constructor (Default value provided by long)
|
//***********************************************************************
|
|
public BigInteger(long value)
|
{
|
data = new uint[maxLength];
|
long tempVal = value;
|
|
// copy bytes from long to BigInteger without any assumption of
|
// the length of the long datatype
|
|
dataLength = 0;
|
while (value != 0 && dataLength < maxLength)
|
{
|
data[dataLength] = (uint)(value & 0xFFFFFFFF);
|
value >>= 32;
|
dataLength++;
|
}
|
|
if (tempVal > 0) // overflow check for +ve value
|
{
|
if (value != 0 || (data[maxLength - 1] & 0x80000000) != 0)
|
throw (new ArithmeticException("Positive overflow in constructor."));
|
}
|
else if (tempVal < 0) // underflow check for -ve value
|
{
|
if (value != -1 || (data[dataLength - 1] & 0x80000000) == 0)
|
throw (new ArithmeticException("Negative underflow in constructor."));
|
}
|
|
if (dataLength == 0)
|
dataLength = 1;
|
}
|
|
|
//***********************************************************************
|
// Constructor (Default value provided by ulong)
|
//***********************************************************************
|
|
public BigInteger(ulong value)
|
{
|
data = new uint[maxLength];
|
|
// copy bytes from ulong to BigInteger without any assumption of
|
// the length of the ulong datatype
|
|
dataLength = 0;
|
while (value != 0 && dataLength < maxLength)
|
{
|
data[dataLength] = (uint)(value & 0xFFFFFFFF);
|
value >>= 32;
|
dataLength++;
|
}
|
|
if (value != 0 || (data[maxLength - 1] & 0x80000000) != 0)
|
throw (new ArithmeticException("Positive overflow in constructor."));
|
|
if (dataLength == 0)
|
dataLength = 1;
|
}
|
|
|
|
//***********************************************************************
|
// Constructor (Default value provided by BigInteger)
|
//***********************************************************************
|
|
public BigInteger(BigInteger bi)
|
{
|
data = new uint[maxLength];
|
|
dataLength = bi.dataLength;
|
|
for (int i = 0; i < dataLength; i++)
|
data[i] = bi.data[i];
|
}
|
|
|
//***********************************************************************
|
// Constructor (Default value provided by a string of digits of the
|
// specified base)
|
//
|
// Example (base 10)
|
// -----------------
|
// To initialize "a" with the default value of 1234 in base 10
|
// BigInteger a = new BigInteger("1234", 10)
|
//
|
// To initialize "a" with the default value of -1234
|
// BigInteger a = new BigInteger("-1234", 10)
|
//
|
// Example (base 16)
|
// -----------------
|
// To initialize "a" with the default value of 0x1D4F in base 16
|
// BigInteger a = new BigInteger("1D4F", 16)
|
//
|
// To initialize "a" with the default value of -0x1D4F
|
// BigInteger a = new BigInteger("-1D4F", 16)
|
//
|
// Note that string values are specified in the <sign><magnitude>
|
// format.
|
//
|
//***********************************************************************
|
|
public BigInteger(string value, int radix)
|
{
|
BigInteger multiplier = new BigInteger(1);
|
BigInteger result = new BigInteger();
|
value = (value.ToUpper()).Trim();
|
int limit = 0;
|
|
if (value[0] == '-')
|
limit = 1;
|
|
for (int i = value.Length - 1; i >= limit; i--)
|
{
|
int posVal = (int)value[i];
|
|
if (posVal >= '0' && posVal <= '9')
|
posVal -= '0';
|
else if (posVal >= 'A' && posVal <= 'Z')
|
posVal = (posVal - 'A') + 10;
|
else
|
posVal = 9999999; // arbitrary large
|
|
|
if (posVal >= radix)
|
throw (new ArithmeticException("Invalid string in constructor."));
|
else
|
{
|
if (value[0] == '-')
|
posVal = -posVal;
|
|
result = result + (multiplier * posVal);
|
|
if ((i - 1) >= limit)
|
multiplier = multiplier * radix;
|
}
|
}
|
|
if (value[0] == '-') // negative values
|
{
|
if ((result.data[maxLength - 1] & 0x80000000) == 0)
|
throw (new ArithmeticException("Negative underflow in constructor."));
|
}
|
else // positive values
|
{
|
if ((result.data[maxLength - 1] & 0x80000000) != 0)
|
throw (new ArithmeticException("Positive overflow in constructor."));
|
}
|
|
data = new uint[maxLength];
|
for (int i = 0; i < result.dataLength; i++)
|
data[i] = result.data[i];
|
|
dataLength = result.dataLength;
|
}
|
|
|
//***********************************************************************
|
// Constructor (Default value provided by an array of bytes)
|
//
|
// The lowest index of the input byte array (i.e [0]) should contain the
|
// most significant byte of the number, and the highest index should
|
// contain the least significant byte.
|
//
|
// E.g.
|
// To initialize "a" with the default value of 0x1D4F in base 16
|
// byte[] temp = { 0x1D, 0x4F };
|
// BigInteger a = new BigInteger(temp)
|
//
|
// Note that this method of initialization does not allow the
|
// sign to be specified.
|
//
|
//***********************************************************************
|
|
public BigInteger(byte[] inData)
|
{
|
dataLength = inData.Length >> 2;
|
|
int leftOver = inData.Length & 0x3;
|
if (leftOver != 0) // length not multiples of 4
|
dataLength++;
|
|
|
if (dataLength > maxLength)
|
throw (new ArithmeticException("Byte overflow in constructor."));
|
|
data = new uint[maxLength];
|
|
for (int i = inData.Length - 1, j = 0; i >= 3; i -= 4, j++)
|
{
|
data[j] = (uint)((inData[i - 3] << 24) + (inData[i - 2] << 16) +
|
(inData[i - 1] << 8) + inData[i]);
|
}
|
|
if (leftOver == 1)
|
data[dataLength - 1] = (uint)inData[0];
|
else if (leftOver == 2)
|
data[dataLength - 1] = (uint)((inData[0] << 8) + inData[1]);
|
else if (leftOver == 3)
|
data[dataLength - 1] = (uint)((inData[0] << 16) + (inData[1] << 8) + inData[2]);
|
|
|
while (dataLength > 1 && data[dataLength - 1] == 0)
|
dataLength--;
|
|
//Shared.HDLUtils.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--;
|
|
//Shared.HDLUtils.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--;
|
|
//Shared.HDLUtils.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;
|
|
//Shared.HDLUtils.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--;
|
|
//Shared.HDLUtils.WriteLine("bufLen = " + bufLen + " buffer.Length = " + buffer.Length);
|
|
for (int count = shiftVal; count > 0;)
|
{
|
if (count < shiftAmount)
|
{
|
shiftAmount = count;
|
invShift = 32 - shiftAmount;
|
}
|
|
//Shared.HDLUtils.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;
|
}
|
|
//Shared.HDLUtils.WriteLine("shift = {0}", shift);
|
//Shared.HDLUtils.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;
|
|
/*
|
Shared.HDLUtils.WriteLine("bi1 Len = {0}, bi2 Len = {1}", bi1.dataLength, bi2.dataLength);
|
Shared.HDLUtils.WriteLine("dividend = " + bi1 + "\ndivisor = " + bi2);
|
for(int q = remainderLen - 1; q >= 0; q--)
|
Console.Write("{0:x2}", remainder[q]);
|
Shared.HDLUtils.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];
|
//Shared.HDLUtils.WriteLine("dividend = {0}", dividend);
|
|
ulong q_hat = dividend / firstDivisorByte;
|
ulong r_hat = dividend % firstDivisorByte;
|
|
//Shared.HDLUtils.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;
|
|
//Shared.HDLUtils.WriteLine("ss before = " + ss);
|
while (ss > kk)
|
{
|
q_hat--;
|
ss -= bi2;
|
//Shared.HDLUtils.WriteLine(ss);
|
}
|
BigInteger yy = kk - ss;
|
|
//Shared.HDLUtils.WriteLine("ss = " + ss);
|
//Shared.HDLUtils.WriteLine("kk = " + kk);
|
//Shared.HDLUtils.WriteLine("yy = " + yy);
|
|
for (int h = 0; h < divisorLen; h++)
|
remainder[pos - h] = yy.data[bi2.dataLength - h];
|
|
/*
|
Shared.HDLUtils.WriteLine("dividend = ");
|
for(int q = remainderLen - 1; q >= 0; q--)
|
Console.Write("{0:x2}", remainder[q]);
|
Shared.HDLUtils.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];
|
|
//Shared.HDLUtils.WriteLine("divisor = " + divisor + " dividend = " + dividend);
|
//Shared.HDLUtils.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)
|
{
|
//Shared.HDLUtils.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);
|
//Shared.HDLUtils.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;
|
//Shared.HDLUtils.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))
|
{
|
//Shared.HDLUtils.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);
|
|
/*
|
Shared.HDLUtils.WriteLine("a = " + a.ToString(10));
|
Shared.HDLUtils.WriteLine("b = " + b.ToString(10));
|
Shared.HDLUtils.WriteLine("t = " + t.ToString(10));
|
Shared.HDLUtils.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);
|
|
//Shared.HDLUtils.WriteLine("a = " + a.ToString(10) + " b = " + thisVal.ToString(10));
|
//Shared.HDLUtils.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;
|
}
|
|
//Shared.HDLUtils.WriteLine(D);
|
D = (Math.Abs(D) + 2) * sign;
|
sign = -sign;
|
}
|
dCount++;
|
}
|
|
long Q = (1 - D) >> 2;
|
|
/*
|
Shared.HDLUtils.WriteLine("D = " + D);
|
Shared.HDLUtils.WriteLine("Q = " + Q);
|
Shared.HDLUtils.WriteLine("(n,D) = " + thisVal.gcd(D));
|
Shared.HDLUtils.WriteLine("(n,Q) = " + thisVal.gcd(Q));
|
Shared.HDLUtils.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)
|
{
|
/*
|
Shared.HDLUtils.WriteLine("Not prime! Divisible by {0}\n",
|
primesBelow2000[p]);
|
*/
|
return false;
|
}
|
}
|
|
if (thisVal.RabinMillerTest(confidence))
|
return true;
|
else
|
{
|
//Shared.HDLUtils.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)
|
{
|
//Shared.HDLUtils.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);
|
//Shared.HDLUtils.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);
|
|
/*
|
Shared.HDLUtils.WriteLine(quotient.dataLength);
|
Shared.HDLUtils.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];
|
|
//Shared.HDLUtils.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;
|
|
//Shared.HDLUtils.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
|
{
|
//Shared.HDLUtils.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);
|
|
Shared.HDLUtils.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)
|
{
|
Shared.HDLUtils.WriteLine("Error at " + count);
|
Shared.HDLUtils.WriteLine(bn1 + "\n");
|
Shared.HDLUtils.WriteLine(bn2 + "\n");
|
Shared.HDLUtils.WriteLine(bn3 + "\n");
|
Shared.HDLUtils.WriteLine(bn4 + "\n");
|
Shared.HDLUtils.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);
|
|
Shared.HDLUtils.WriteLine("e =\n" + bi_e.ToString(10));
|
Shared.HDLUtils.WriteLine("\nd =\n" + bi_d.ToString(10));
|
Shared.HDLUtils.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)
|
{
|
Shared.HDLUtils.WriteLine("\nError at round " + count);
|
Shared.HDLUtils.WriteLine(bi_data + "\n");
|
return;
|
}
|
Shared.HDLUtils.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);
|
|
Shared.HDLUtils.WriteLine("\ne =\n" + bi_e.ToString(10));
|
Shared.HDLUtils.WriteLine("\nd =\n" + bi_d.ToString(10));
|
Shared.HDLUtils.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)
|
{
|
Shared.HDLUtils.WriteLine("\nError at round " + count);
|
Shared.HDLUtils.WriteLine(bi_data + "\n");
|
return;
|
}
|
Shared.HDLUtils.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)
|
{
|
Shared.HDLUtils.WriteLine("\nError at round " + count);
|
Shared.HDLUtils.WriteLine(a + "\n");
|
return;
|
}
|
Shared.HDLUtils.WriteLine(" <PASSED>.");
|
}
|
}
|
|
|
|
|
|
}
|
}
|
|
|
