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3271 lines
102 KiB
3271 lines
102 KiB
using System;
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using System.Security.Cryptography;
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// compile with: /doc:BigIntegerDocComment.xml
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/// <summary>
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/// This is a BigInteger class. Holds integer that is more than 64-bit (long).
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/// </summary>
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/// <remarks>
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/// This class contains overloaded arithmetic operators(+, -, *, /, %), bitwise operators(&, |) and other
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/// operations that can be done with normal integer. It also contains implementation of various prime test.
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/// This class also contains methods dealing with cryptography such as generating prime number, generating
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/// a coprime number.
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/// </remarks>
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public class BigInteger
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{
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// maximum length of the BigInteger in uint (4 bytes)
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// change this to suit the required level of precision.
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private const int maxLength = 70;
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// primes smaller than 2000 to test the generated prime number
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public static readonly int[] primesBelow2000 = {
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2, 3, 5, 7, 11, 13, 17, 19, 23, 29, 31, 37, 41, 43, 47, 53, 59, 61, 67, 71,
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73, 79, 83, 89, 97, 101, 103, 107, 109, 113, 127, 131, 137, 139, 149, 151, 157, 163, 167, 173,
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179, 181, 191, 193, 197, 199, 211, 223, 227, 229, 233, 239, 241, 251, 257, 263, 269, 271, 277, 281,
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283, 293, 307, 311, 313, 317, 331, 337, 347, 349, 353, 359, 367, 373, 379, 383, 389, 397, 401, 409,
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419, 421, 431, 433, 439, 443, 449, 457, 461, 463, 467, 479, 487, 491, 499, 503, 509, 521, 523, 541,
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547, 557, 563, 569, 571, 577, 587, 593, 599, 601, 607, 613, 617, 619, 631, 641, 643, 647, 653, 659,
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661, 673, 677, 683, 691, 701, 709, 719, 727, 733, 739, 743, 751, 757, 761, 769, 773, 787, 797, 809,
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811, 821, 823, 827, 829, 839, 853, 857, 859, 863, 877, 881, 883, 887, 907, 911, 919, 929, 937, 941,
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947, 953, 967, 971, 977, 983, 991, 997, 1009, 1013, 1019, 1021, 1031, 1033, 1039, 1049, 1051, 1061, 1063, 1069,
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1087, 1091, 1093, 1097, 1103, 1109, 1117, 1123, 1129, 1151, 1153, 1163, 1171, 1181, 1187, 1193, 1201, 1213, 1217, 1223,
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1229, 1231, 1237, 1249, 1259, 1277, 1279, 1283, 1289, 1291, 1297, 1301, 1303, 1307, 1319, 1321, 1327, 1361, 1367, 1373,
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1381, 1399, 1409, 1423, 1427, 1429, 1433, 1439, 1447, 1451, 1453, 1459, 1471, 1481, 1483, 1487, 1489, 1493, 1499, 1511,
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1523, 1531, 1543, 1549, 1553, 1559, 1567, 1571, 1579, 1583, 1597, 1601, 1607, 1609, 1613, 1619, 1621, 1627, 1637, 1657,
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1663, 1667, 1669, 1693, 1697, 1699, 1709, 1721, 1723, 1733, 1741, 1747, 1753, 1759, 1777, 1783, 1787, 1789, 1801, 1811,
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1823, 1831, 1847, 1861, 1867, 1871, 1873, 1877, 1879, 1889, 1901, 1907, 1913, 1931, 1933, 1949, 1951, 1973, 1979, 1987,
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1993, 1997, 1999 };
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private uint[] data = null; // stores bytes from the Big Integer
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public int dataLength; // number of actual chars used
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/// <summary>
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/// Default constructor for BigInteger of value 0
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/// </summary>
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public BigInteger()
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{
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data = new uint[maxLength];
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dataLength = 1;
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}
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/// <summary>
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/// Constructor (Default value provided by long)
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/// </summary>
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/// <param name="value">Turn the long value into BigInteger type</param>
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public BigInteger(long value)
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{
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data = new uint[maxLength];
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long tempVal = value;
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// copy bytes from long to BigInteger without any assumption of
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// the length of the long datatype
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dataLength = 0;
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while (value != 0 && dataLength < maxLength)
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{
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data[dataLength] = (uint)(value & 0xFFFFFFFF);
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value >>= 32;
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dataLength++;
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}
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if (tempVal > 0) // overflow check for +ve value
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{
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if (value != 0 || (data[maxLength - 1] & 0x80000000) != 0)
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throw (new ArithmeticException("Positive overflow in constructor."));
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}
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else if (tempVal < 0) // underflow check for -ve value
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{
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if (value != -1 || (data[dataLength - 1] & 0x80000000) == 0)
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throw (new ArithmeticException("Negative underflow in constructor."));
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}
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if (dataLength == 0)
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dataLength = 1;
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}
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/// <summary>
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/// Constructor (Default value provided by ulong)
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/// </summary>
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/// <param name="value">Turn the unsigned long value into BigInteger type</param>
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public BigInteger(ulong value)
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{
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data = new uint[maxLength];
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// copy bytes from ulong to BigInteger without any assumption of
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// the length of the ulong datatype
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dataLength = 0;
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while (value != 0 && dataLength < maxLength)
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{
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data[dataLength] = (uint)(value & 0xFFFFFFFF);
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value >>= 32;
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dataLength++;
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}
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if (value != 0 || (data[maxLength - 1] & 0x80000000) != 0)
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throw (new ArithmeticException("Positive overflow in constructor."));
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if (dataLength == 0)
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dataLength = 1;
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}
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/// <summary>
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/// Constructor (Default value provided by BigInteger)
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/// </summary>
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/// <param name="bi"></param>
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public BigInteger(BigInteger bi)
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{
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data = new uint[maxLength];
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dataLength = bi.dataLength;
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for (int i = 0; i < dataLength; i++)
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data[i] = bi.data[i];
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}
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/// <summary>
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/// Constructor (Default value provided by a string of digits of the specified base)
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/// </summary>
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/// <example>
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/// To initialize "a" with the default value of 1234 in base 10:
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/// BigInteger a = new BigInteger("1234", 10)
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/// To initialize "a" with the default value of -0x1D4F in base 16:
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/// BigInteger a = new BigInteger("-1D4F", 16)
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/// </example>
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///
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/// <param name="value">String value in the format of [sign][magnitude]</param>
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/// <param name="radix">The base of value</param>
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public BigInteger(string value, int radix)
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{
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BigInteger multiplier = new BigInteger(1);
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BigInteger result = new BigInteger();
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value = (value.ToUpper()).Trim();
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int limit = 0;
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if (value[0] == '-')
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limit = 1;
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for (int i = value.Length - 1; i >= limit; i--)
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{
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int posVal = (int)value[i];
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if (posVal >= '0' && posVal <= '9')
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posVal -= '0';
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else if (posVal >= 'A' && posVal <= 'Z')
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posVal = (posVal - 'A') + 10;
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else
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posVal = 9999999; // arbitrary large
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if (posVal >= radix)
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throw (new ArithmeticException("Invalid string in constructor."));
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else
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{
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if (value[0] == '-')
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posVal = -posVal;
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result = result + (multiplier * posVal);
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if ((i - 1) >= limit)
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multiplier = multiplier * radix;
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}
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}
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if (value[0] == '-') // negative values
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{
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if ((result.data[maxLength - 1] & 0x80000000) == 0)
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throw (new ArithmeticException("Negative underflow in constructor."));
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}
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else // positive values
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{
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if ((result.data[maxLength - 1] & 0x80000000) != 0)
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throw (new ArithmeticException("Positive overflow in constructor."));
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}
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data = new uint[maxLength];
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for (int i = 0; i < result.dataLength; i++)
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data[i] = result.data[i];
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dataLength = result.dataLength;
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}
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//***********************************************************************
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// Constructor (Default value provided by an array of bytes)
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//
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// The lowest index of the input byte array (i.e [0]) should contain the
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// most significant byte of the number, and the highest index should
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// contain the least significant byte.
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//
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// E.g.
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// To initialize "a" with the default value of 0x1D4F in base 16
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// byte[] temp = { 0x1D, 0x4F };
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// BigInteger a = new BigInteger(temp)
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//
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// Note that this method of initialization does not allow the
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// sign to be specified.
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//
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//***********************************************************************
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/*public BigInteger(byte[] inData)
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{
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dataLength = inData.Length >> 2;
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int leftOver = inData.Length & 0x3;
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if (leftOver != 0) // length not multiples of 4
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dataLength++;
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if (dataLength > maxLength)
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throw (new ArithmeticException("Byte overflow in constructor."));
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data = new uint[maxLength];
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for (int i = inData.Length - 1, j = 0; i >= 3; i -= 4, j++)
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{
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data[j] = ((uint)(inData[i - 3]) << 24) + ((uint)(inData[i - 2]) << 16) +
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((uint)(inData[i - 1] << 8)) + ((uint)(inData[i]));
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}
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if (leftOver == 1)
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data[dataLength - 1] = (uint)inData[0];
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else if (leftOver == 2)
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data[dataLength - 1] = (uint)((inData[0] << 8) + inData[1]);
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else if (leftOver == 3)
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data[dataLength - 1] = (uint)((inData[0] << 16) + (inData[1] << 8) + inData[2]);
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while (dataLength > 1 && data[dataLength - 1] == 0)
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dataLength--;
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}*/
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/// <summary>
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/// Constructor (Default value provided by an array of bytes of the specified length.)
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/// </summary>
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/// <param name="inData">A list of byte values</param>
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/// <param name="length">Default -1</param>
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/// <param name="offset">Default 0</param>
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public BigInteger(System.Collections.Generic.IList<byte> inData, int length = -1, int offset = 0)
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{
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var inLen = length == -1 ? inData.Count - offset : length;
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dataLength = inLen >> 2;
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int leftOver = inLen & 0x3;
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if (leftOver != 0) // length not multiples of 4
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dataLength++;
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if (dataLength > maxLength || inLen > inData.Count - offset)
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throw (new ArithmeticException("Byte overflow in constructor."));
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data = new uint[maxLength];
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for (int i = inLen - 1, j = 0; i >= 3; i -= 4, j++)
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{
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data[j] = (uint)((inData[offset + i - 3] << 24) + (inData[offset + i - 2] << 16) +
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(inData[offset + i - 1] << 8) + inData[offset + i]);
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}
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if (leftOver == 1)
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data[dataLength - 1] = (uint)inData[offset + 0];
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else if (leftOver == 2)
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data[dataLength - 1] = (uint)((inData[offset + 0] << 8) + inData[offset + 1]);
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else if (leftOver == 3)
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data[dataLength - 1] = (uint)((inData[offset + 0] << 16) + (inData[offset + 1] << 8) + inData[offset + 2]);
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if (dataLength == 0)
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dataLength = 1;
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while (dataLength > 1 && data[dataLength - 1] == 0)
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dataLength--;
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}
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/// <summary>
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/// Constructor (Default value provided by an array of unsigned integers)
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/// </summary>
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/// <param name="inData">Array of unsigned integer</param>
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public BigInteger(uint[] inData)
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{
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dataLength = inData.Length;
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if (dataLength > maxLength)
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throw (new ArithmeticException("Byte overflow in constructor."));
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data = new uint[maxLength];
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for (int i = dataLength - 1, j = 0; i >= 0; i--, j++)
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data[j] = inData[i];
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while (dataLength > 1 && data[dataLength - 1] == 0)
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dataLength--;
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}
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/// <summary>
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/// Cast a type long value to type BigInteger value
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/// </summary>
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/// <param name="value">A long value</param>
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public static implicit operator BigInteger(long value)
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{
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return (new BigInteger(value));
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}
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/// <summary>
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/// Cast a type ulong value to type BigInteger value
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/// </summary>
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/// <param name="value">An unsigned long value</param>
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public static implicit operator BigInteger(ulong value)
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{
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return (new BigInteger(value));
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}
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/// <summary>
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/// Cast a type int value to type BigInteger value
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/// </summary>
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/// <param name="value">An int value</param>
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public static implicit operator BigInteger(int value)
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{
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return (new BigInteger((long)value));
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}
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/// <summary>
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/// Cast a type uint value to type BigInteger value
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/// </summary>
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/// <param name="value">An unsigned int value</param>
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public static implicit operator BigInteger(uint value)
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{
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return (new BigInteger((ulong)value));
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}
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/// <summary>
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/// Overloading of addition operator
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/// </summary>
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/// <param name="bi1">First BigInteger</param>
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/// <param name="bi2">Second BigInteger</param>
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/// <returns>Result of the addition of 2 BigIntegers</returns>
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public static BigInteger operator +(BigInteger bi1, BigInteger bi2)
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{
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BigInteger result = new BigInteger();
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result.dataLength = (bi1.dataLength > bi2.dataLength) ? bi1.dataLength : bi2.dataLength;
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long carry = 0;
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for (int i = 0; i < result.dataLength; i++)
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{
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long sum = (long)bi1.data[i] + (long)bi2.data[i] + carry;
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carry = sum >> 32;
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result.data[i] = (uint)(sum & 0xFFFFFFFF);
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}
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if (carry != 0 && result.dataLength < maxLength)
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{
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result.data[result.dataLength] = (uint)(carry);
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result.dataLength++;
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}
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while (result.dataLength > 1 && result.data[result.dataLength - 1] == 0)
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result.dataLength--;
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// overflow check
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int lastPos = maxLength - 1;
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if ((bi1.data[lastPos] & 0x80000000) == (bi2.data[lastPos] & 0x80000000) &&
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(result.data[lastPos] & 0x80000000) != (bi1.data[lastPos] & 0x80000000))
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{
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throw (new ArithmeticException());
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}
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return result;
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}
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/// <summary>
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/// Overloading of the unary ++ operator, which increments BigInteger by 1
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/// </summary>
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/// <param name="bi1">A BigInteger</param>
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/// <returns>Incremented BigInteger</returns>
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public static BigInteger operator ++(BigInteger bi1)
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{
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BigInteger result = new BigInteger(bi1);
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long val, carry = 1;
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int index = 0;
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while (carry != 0 && index < maxLength)
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{
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val = (long)(result.data[index]);
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val++;
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result.data[index] = (uint)(val & 0xFFFFFFFF);
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carry = val >> 32;
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index++;
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}
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if (index > result.dataLength)
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result.dataLength = index;
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else
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{
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while (result.dataLength > 1 && result.data[result.dataLength - 1] == 0)
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result.dataLength--;
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}
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// overflow check
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int lastPos = maxLength - 1;
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// overflow if initial value was +ve but ++ caused a sign
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// change to negative.
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if ((bi1.data[lastPos] & 0x80000000) == 0 &&
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(result.data[lastPos] & 0x80000000) != (bi1.data[lastPos] & 0x80000000))
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{
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throw (new ArithmeticException("Overflow in ++."));
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}
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return result;
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}
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/// <summary>
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/// Overloading of subtraction operator
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/// </summary>
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/// <param name="bi1">First BigInteger</param>
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/// <param name="bi2">Second BigInteger</param>
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/// <returns>Result of the subtraction of 2 BigIntegers</returns>
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public static BigInteger operator -(BigInteger bi1, BigInteger bi2)
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{
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BigInteger result = new BigInteger();
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result.dataLength = (bi1.dataLength > bi2.dataLength) ? bi1.dataLength : bi2.dataLength;
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long carryIn = 0;
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for (int i = 0; i < result.dataLength; i++)
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{
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long diff;
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diff = (long)bi1.data[i] - (long)bi2.data[i] - carryIn;
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result.data[i] = (uint)(diff & 0xFFFFFFFF);
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if (diff < 0)
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carryIn = 1;
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else
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carryIn = 0;
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}
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// roll over to negative
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if (carryIn != 0)
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{
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for (int i = result.dataLength; i < maxLength; i++)
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result.data[i] = 0xFFFFFFFF;
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result.dataLength = maxLength;
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}
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// fixed in v1.03 to give correct datalength for a - (-b)
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while (result.dataLength > 1 && result.data[result.dataLength - 1] == 0)
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result.dataLength--;
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// overflow check
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int lastPos = maxLength - 1;
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if ((bi1.data[lastPos] & 0x80000000) != (bi2.data[lastPos] & 0x80000000) &&
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(result.data[lastPos] & 0x80000000) != (bi1.data[lastPos] & 0x80000000))
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{
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throw (new ArithmeticException());
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}
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return result;
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}
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/// <summary>
|
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/// Overloading of the unary -- operator, decrements BigInteger by 1
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/// </summary>
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/// <param name="bi1">A BigInteger</param>
|
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/// <returns>Decremented BigInteger</returns>
|
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public static BigInteger operator --(BigInteger bi1)
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{
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BigInteger result = new BigInteger(bi1);
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|
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long val;
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bool carryIn = true;
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int index = 0;
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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;
|
|
}
|
|
|
|
|
|
/// <summary>
|
|
/// Overloading of multiplication operator
|
|
/// </summary>
|
|
/// <param name="bi1">First BigInteger</param>
|
|
/// <param name="bi2">Second BigInteger</param>
|
|
/// <returns>Result of the multiplication of 2 BigIntegers</returns>
|
|
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;
|
|
}
|
|
|
|
|
|
/// <summary>
|
|
/// Overloading of the unary << operator (left shift)
|
|
/// </summary>
|
|
/// <remarks>
|
|
/// Shifting by a negative number is an undefined behaviour (UB).
|
|
/// </remarks>
|
|
/// <param name="bi1">A BigInteger</param>
|
|
/// <param name="shiftVal">Left shift by shiftVal bit</param>
|
|
/// <returns>Left-shifted BigInteger</returns>
|
|
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;
|
|
|
|
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;
|
|
}
|
|
|
|
|
|
/// <summary>
|
|
/// Overloading of the unary >> operator (right shift)
|
|
/// </summary>
|
|
/// <remarks>
|
|
/// Shifting by a negative number is an undefined behaviour (UB).
|
|
/// </remarks>
|
|
/// <param name="bi1">A BigInteger</param>
|
|
/// <param name="shiftVal">Right shift by shiftVal bit</param>
|
|
/// <returns>Right-shifted BigInteger</returns>
|
|
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--;
|
|
|
|
for (int count = shiftVal; count > 0;)
|
|
{
|
|
if (count < shiftAmount)
|
|
{
|
|
shiftAmount = count;
|
|
invShift = 32 - 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) & 0xFFFFFFFF;
|
|
buffer[i] = (uint)(val);
|
|
}
|
|
|
|
count -= shiftAmount;
|
|
}
|
|
|
|
while (bufLen > 1 && buffer[bufLen - 1] == 0)
|
|
bufLen--;
|
|
|
|
return bufLen;
|
|
}
|
|
|
|
|
|
/// <summary>
|
|
/// Overloading of the bit-wise NOT operator (1's complement)
|
|
/// </summary>
|
|
/// <param name="bi1">A BigInteger</param>
|
|
/// <returns>Complemented BigInteger</returns>
|
|
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;
|
|
}
|
|
|
|
|
|
/// <summary>
|
|
/// Overloading of the NEGATE operator (2's complement)
|
|
/// </summary>
|
|
/// <param name="bi1">A BigInteger</param>
|
|
/// <returns>Negated BigInteger or default BigInteger value if bi1 is 0</returns>
|
|
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;
|
|
}
|
|
|
|
|
|
/// <summary>
|
|
/// Overloading of equality operator, allows comparing 2 BigIntegers with == operator
|
|
/// </summary>
|
|
/// <param name="bi1">First BigInteger</param>
|
|
/// <param name="bi2">Second BigInteger</param>
|
|
/// <returns>Boolean result of the comparison</returns>
|
|
public static bool operator ==(BigInteger bi1, BigInteger bi2)
|
|
{
|
|
return bi1.Equals(bi2);
|
|
}
|
|
|
|
|
|
/// <summary>
|
|
/// Overloading of not equal operator, allows comparing 2 BigIntegers with != operator
|
|
/// </summary>
|
|
/// <param name="bi1">First BigInteger</param>
|
|
/// <param name="bi2">Second BigInteger</param>
|
|
/// <returns>Boolean result of the comparison</returns>
|
|
public static bool operator !=(BigInteger bi1, BigInteger bi2)
|
|
{
|
|
return !(bi1.Equals(bi2));
|
|
}
|
|
|
|
|
|
/// <summary>
|
|
/// Overriding of Equals method, allows comparing BigInteger with an arbitary object
|
|
/// </summary>
|
|
/// <param name="o">Input object, to be casted into BigInteger type for comparison</param>
|
|
/// <returns>Boolean result of the comparison</returns>
|
|
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();
|
|
}
|
|
|
|
|
|
/// <summary>
|
|
/// Overloading of greater than operator, allows comparing 2 BigIntegers with > operator
|
|
/// </summary>
|
|
/// <param name="bi1">First BigInteger</param>
|
|
/// <param name="bi2">Second BigInteger</param>
|
|
/// <returns>Boolean result of the comparison</returns>
|
|
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;
|
|
}
|
|
|
|
|
|
/// <summary>
|
|
/// Overloading of greater than operator, allows comparing 2 BigIntegers with < operator
|
|
/// </summary>
|
|
/// <param name="bi1">First BigInteger</param>
|
|
/// <param name="bi2">Second BigInteger</param>
|
|
/// <returns>Boolean result of the comparison</returns>
|
|
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;
|
|
}
|
|
|
|
|
|
/// <summary>
|
|
/// Overloading of greater than or equal to operator, allows comparing 2 BigIntegers with >= operator
|
|
/// </summary>
|
|
/// <param name="bi1">First BigInteger</param>
|
|
/// <param name="bi2">Second BigInteger</param>
|
|
/// <returns>Boolean result of the comparison</returns>
|
|
public static bool operator >=(BigInteger bi1, BigInteger bi2)
|
|
{
|
|
return (bi1 == bi2 || bi1 > bi2);
|
|
}
|
|
|
|
|
|
/// <summary>
|
|
/// Overloading of less than or equal to operator, allows comparing 2 BigIntegers with <= operator
|
|
/// </summary>
|
|
/// <param name="bi1">First BigInteger</param>
|
|
/// <param name="bi2">Second BigInteger</param>
|
|
/// <returns>Boolean result of the comparison</returns>
|
|
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;
|
|
}
|
|
|
|
for (int i = 0; i < bi1.dataLength; i++)
|
|
remainder[i] = bi1.data[i];
|
|
shiftLeft(remainder, shift);
|
|
bi2 = bi2 << shift;
|
|
|
|
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];
|
|
|
|
ulong q_hat = dividend / firstDivisorByte;
|
|
ulong r_hat = dividend % firstDivisorByte;
|
|
|
|
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;
|
|
|
|
while (ss > kk)
|
|
{
|
|
q_hat--;
|
|
ss -= bi2;
|
|
}
|
|
BigInteger yy = kk - ss;
|
|
|
|
for (int h = 0; h < divisorLen; h++)
|
|
remainder[pos - h] = yy.data[bi2.dataLength - h];
|
|
|
|
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];
|
|
|
|
if (dividend >= divisor)
|
|
{
|
|
ulong quotient = dividend / divisor;
|
|
result[resultPos++] = (uint)quotient;
|
|
|
|
outRemainder.data[pos] = (uint)(dividend % divisor);
|
|
}
|
|
pos--;
|
|
|
|
while (pos >= 0)
|
|
{
|
|
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);
|
|
}
|
|
|
|
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--;
|
|
}
|
|
|
|
|
|
/// <summary>
|
|
/// Overloading of division operator
|
|
/// </summary>
|
|
/// <remarks>The dataLength of the divisor's absolute value must be less than maxLength</remarks>
|
|
/// <param name="bi1">Dividend</param>
|
|
/// <param name="bi2">Divisor</param>
|
|
/// <returns>Quotient of the division</returns>
|
|
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;
|
|
}
|
|
}
|
|
|
|
|
|
/// <summary>
|
|
/// Overloading of modulus operator
|
|
/// </summary>
|
|
/// <remarks>The dataLength of the divisor's absolute value must be less than maxLength</remarks>
|
|
/// <param name="bi1">Dividend</param>
|
|
/// <param name="bi2">Divisor</param>
|
|
/// <returns>Remainder of the division</returns>
|
|
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;
|
|
}
|
|
}
|
|
|
|
|
|
/// <summary>
|
|
/// Overloading of bitwise AND operator
|
|
/// </summary>
|
|
/// <param name="bi1">First BigInteger</param>
|
|
/// <param name="bi2">Second BigInteger</param>
|
|
/// <returns>BigInteger result after performing & operation</returns>
|
|
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;
|
|
}
|
|
|
|
|
|
/// <summary>
|
|
/// Overloading of bitwise OR operator
|
|
/// </summary>
|
|
/// <param name="bi1">First BigInteger</param>
|
|
/// <param name="bi2">Second BigInteger</param>
|
|
/// <returns>BigInteger result after performing | operation</returns>
|
|
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;
|
|
}
|
|
|
|
|
|
/// <summary>
|
|
/// Overloading of bitwise XOR operator
|
|
/// </summary>
|
|
/// <param name="bi1">First BigInteger</param>
|
|
/// <param name="bi2">Second BigInteger</param>
|
|
/// <returns>BigInteger result after performing ^ operation</returns>
|
|
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;
|
|
}
|
|
|
|
|
|
/// <summary>
|
|
/// Compare this and a BigInteger and find the maximum one
|
|
/// </summary>
|
|
/// <param name="bi">BigInteger to be compared with this</param>
|
|
/// <returns>The bigger value of this and bi</returns>
|
|
public BigInteger max(BigInteger bi)
|
|
{
|
|
if (this > bi)
|
|
return (new BigInteger(this));
|
|
else
|
|
return (new BigInteger(bi));
|
|
}
|
|
|
|
|
|
/// <summary>
|
|
/// Compare this and a BigInteger and find the minimum one
|
|
/// </summary>
|
|
/// <param name="bi">BigInteger to be compared with this</param>
|
|
/// <returns>The smaller value of this and bi</returns>
|
|
public BigInteger min(BigInteger bi)
|
|
{
|
|
if (this < bi)
|
|
return (new BigInteger(this));
|
|
else
|
|
return (new BigInteger(bi));
|
|
|
|
}
|
|
|
|
|
|
/// <summary>
|
|
/// Returns the absolute value
|
|
/// </summary>
|
|
/// <returns>Absolute value of this</returns>
|
|
public BigInteger abs()
|
|
{
|
|
if ((this.data[maxLength - 1] & 0x80000000) != 0)
|
|
return (-this);
|
|
else
|
|
return (new BigInteger(this));
|
|
}
|
|
|
|
|
|
/// <summary>
|
|
/// Returns a string representing the BigInteger in base 10
|
|
/// </summary>
|
|
/// <returns>string representation of the BigInteger</returns>
|
|
public override string ToString()
|
|
{
|
|
return ToString(10);
|
|
}
|
|
|
|
|
|
/// <summary>
|
|
/// Returns a string representing the BigInteger in [sign][magnitude] format in the specified radix
|
|
/// </summary>
|
|
/// <example>If the value of BigInteger is -255 in base 10, then ToString(16) returns "-FF"</example>
|
|
/// <param name="radix">Base</param>
|
|
/// <returns>string representation of the BigInteger in [sign][magnitude] format</returns>
|
|
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;
|
|
}
|
|
|
|
|
|
/// <summary>
|
|
/// Returns a hex string showing the contains of the BigInteger
|
|
/// </summary>
|
|
/// <example>
|
|
/// 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, thenToHexString() returns ".....FFFFFFFFFF01", which is the 2's complement representation of -255.
|
|
/// </example>
|
|
/// <returns></returns>
|
|
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;
|
|
}
|
|
|
|
|
|
/// <summary>
|
|
/// Modulo Exponentiation
|
|
/// </summary>
|
|
/// <param name="exp">Exponential</param>
|
|
/// <param name="n">Modulo</param>
|
|
/// <returns>BigInteger result of raising this to the power of exp and then modulo n </returns>
|
|
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;
|
|
|
|
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;
|
|
}
|
|
|
|
|
|
/// <summary>
|
|
/// Fast calculation of modular reduction using Barrett's reduction
|
|
/// </summary>
|
|
/// <remarks>
|
|
/// Requires x < b^(2k), where b is the base. In this case, base is 2^32 (uint).
|
|
///
|
|
/// Reference [4]
|
|
/// </remarks>
|
|
/// <param name="x"></param>
|
|
/// <param name="n"></param>
|
|
/// <param name="constant"></param>
|
|
/// <returns></returns>
|
|
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;
|
|
}
|
|
|
|
|
|
/// <summary>
|
|
/// Returns gcd(this, bi)
|
|
/// </summary>
|
|
/// <param name="bi"></param>
|
|
/// <returns>Greatest Common Divisor of this and bi</returns>
|
|
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;
|
|
}
|
|
|
|
|
|
/// <summary>
|
|
/// Populates "this" with the specified amount of random bits
|
|
/// </summary>
|
|
/// <param name="bits"></param>
|
|
/// <param name="rand"></param>
|
|
public void genRandomBits(int bits, Random rand)
|
|
{
|
|
int dwords = bits >> 5;
|
|
int remBits = bits & 0x1F;
|
|
|
|
if (remBits != 0)
|
|
dwords++;
|
|
|
|
if (dwords > maxLength || bits <= 0)
|
|
throw (new ArithmeticException("Number of required bits is not valid."));
|
|
|
|
byte[] randBytes = new byte[dwords * 4];
|
|
rand.NextBytes(randBytes);
|
|
|
|
for (int i = 0; i < dwords; i++)
|
|
data[i] = BitConverter.ToUInt32(randBytes, i * 4);
|
|
|
|
for (int i = dwords; i < maxLength; i++)
|
|
data[i] = 0;
|
|
|
|
if (remBits != 0)
|
|
{
|
|
uint mask;
|
|
|
|
if (bits != 1)
|
|
{
|
|
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;
|
|
}
|
|
|
|
|
|
/// <summary>
|
|
/// Populates "this" with the specified amount of random bits (secured version)
|
|
/// </summary>
|
|
/// <param name="bits"></param>
|
|
/// <param name="rng"></param>
|
|
public void genRandomBits(int bits, RNGCryptoServiceProvider rng)
|
|
{
|
|
int dwords = bits >> 5;
|
|
int remBits = bits & 0x1F;
|
|
|
|
if (remBits != 0)
|
|
dwords++;
|
|
|
|
if (dwords > maxLength || bits <= 0)
|
|
throw (new ArithmeticException("Number of required bits is not valid."));
|
|
|
|
byte[] randomBytes = new byte[dwords * 4];
|
|
rng.GetBytes(randomBytes);
|
|
|
|
for (int i = 0; i < dwords; i++)
|
|
data[i] = BitConverter.ToUInt32(randomBytes, i * 4);
|
|
|
|
for (int i = dwords; i < maxLength; i++)
|
|
data[i] = 0;
|
|
|
|
if (remBits != 0)
|
|
{
|
|
uint mask;
|
|
|
|
if (bits != 1)
|
|
{
|
|
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;
|
|
}
|
|
|
|
|
|
/// <summary>
|
|
/// Returns the position of the most significant bit in the BigInteger
|
|
/// </summary>
|
|
/// <example>
|
|
/// 1) The result is 1, if the value of BigInteger is 0...0000 0000
|
|
/// 2) The result is 1, if the value of BigInteger is 0...0000 0001
|
|
/// 3) The result is 2, if the value of BigInteger is 0...0000 0010
|
|
/// 4) The result is 2, if the value of BigInteger is 0...0000 0011
|
|
/// 5) The result is 5, if the value of BigInteger is 0...0001 0011
|
|
/// </example>
|
|
/// <returns></returns>
|
|
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 == 0 ? 1 : bits;
|
|
}
|
|
|
|
|
|
/// <summary>
|
|
/// Probabilistic prime test based on Fermat's little theorem
|
|
/// </summary>
|
|
/// <remarks>
|
|
/// 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).
|
|
///
|
|
/// This method is fast but fails for Carmichael numbers when the randomly chosen base is a factor of the number.
|
|
/// </remarks>
|
|
/// <param name="confidence">Number of chosen bases</param>
|
|
/// <returns>True if this is a pseudoprime to randomly chosen bases</returns>
|
|
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))
|
|
return false;
|
|
}
|
|
|
|
return true;
|
|
}
|
|
|
|
|
|
/// <summary>
|
|
/// Probabilistic prime test based on Rabin-Miller's
|
|
/// </summary>
|
|
/// <remarks>
|
|
/// 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.
|
|
/// </remarks>
|
|
/// <param name="confidence">Number of chosen bases</param>
|
|
/// <returns>True if this is a strong pseudoprime to randomly chosen bases</returns>
|
|
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);
|
|
|
|
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;
|
|
}
|
|
|
|
|
|
/// <summary>
|
|
/// Probabilistic prime test based on Solovay-Strassen (Euler Criterion)
|
|
/// </summary>
|
|
/// <remarks>
|
|
/// 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.
|
|
/// </remarks>
|
|
/// <param name="confidence">Number of chosen bases</param>
|
|
/// <returns>True if this is a Euler pseudoprime to randomly chosen bases</returns>
|
|
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);
|
|
|
|
// if they are different then it is not prime
|
|
if (expResult != jacob)
|
|
return false;
|
|
}
|
|
|
|
return true;
|
|
}
|
|
|
|
|
|
/// <summary>
|
|
/// Implementation of the Lucas Strong Pseudo Prime test
|
|
/// </summary>
|
|
/// <remarks>
|
|
/// 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.
|
|
/// </remarks>
|
|
/// <returns>True if number is a strong Lucus pseudo prime</returns>
|
|
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;
|
|
}
|
|
|
|
D = (Math.Abs(D) + 2) * sign;
|
|
sign = -sign;
|
|
}
|
|
dCount++;
|
|
}
|
|
|
|
long Q = (1 - D) >> 2;
|
|
|
|
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;
|
|
|
|
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;
|
|
}
|
|
|
|
|
|
/// <summary>
|
|
/// Determines whether a number is probably prime using the Rabin-Miller's test
|
|
/// </summary>
|
|
/// <remarks>
|
|
/// Before applying the test, the number is tested for divisibility by primes < 2000
|
|
/// </remarks>
|
|
/// <param name="confidence">Number of chosen bases</param>
|
|
/// <returns>True if this is probably prime</returns>
|
|
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)
|
|
return false;
|
|
}
|
|
|
|
if (thisVal.RabinMillerTest(confidence))
|
|
return true;
|
|
else
|
|
return false;
|
|
}
|
|
|
|
|
|
/// <summary>
|
|
/// Determines whether this BigInteger is probably prime using a combination of base 2 strong pseudoprime test and Lucas strong pseudoprime test
|
|
/// </summary>
|
|
/// <remarks>
|
|
/// 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.
|
|
///
|
|
/// For a detailed discussion of this primality test, see [6].
|
|
/// </remarks>
|
|
/// <returns>True if this is probably prime</returns>
|
|
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)
|
|
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;
|
|
}
|
|
|
|
|
|
/// <summary>
|
|
/// Returns the lowest 4 bytes of the BigInteger as an int
|
|
/// </summary>
|
|
/// <returns>Lowest 4 bytes as integer</returns>
|
|
public int IntValue()
|
|
{
|
|
return (int)data[0];
|
|
}
|
|
|
|
|
|
/// <summary>
|
|
/// Returns the lowest 8 bytes of the BigInteger as a long
|
|
/// </summary>
|
|
/// <returns>Lowest 8 bytes as long</returns>
|
|
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;
|
|
}
|
|
|
|
|
|
/// <summary>
|
|
/// Computes the Jacobi Symbol for 2 BigInteger a and b
|
|
/// </summary>
|
|
/// <remarks>
|
|
/// Algorithm adapted from [3] and [4] with some optimizations
|
|
/// </remarks>
|
|
/// <param name="a">Any BigInteger</param>
|
|
/// <param name="b">Odd BigInteger</param>
|
|
/// <returns>Jacobi Symbol</returns>
|
|
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));
|
|
}
|
|
|
|
|
|
/// <summary>
|
|
/// Generates a positive BigInteger that is probably prime
|
|
/// </summary>
|
|
/// <param name="bits">Number of bit</param>
|
|
/// <param name="confidence">Number of chosen bases</param>
|
|
/// <param name="rand">Random object</param>
|
|
/// <returns>A probably prime number</returns>
|
|
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;
|
|
}
|
|
|
|
|
|
/// <summary>
|
|
/// Generates a positive BigInteger that is probably prime (secured version)
|
|
/// </summary>
|
|
/// <param name="bits">Number of bit</param>
|
|
/// <param name="confidence">Number of chosen bases</param>
|
|
/// <param name="rand">RNGCryptoServiceProvider object</param>
|
|
/// <returns>A probably prime number</returns>
|
|
public static BigInteger genPseudoPrime(int bits, int confidence, RNGCryptoServiceProvider 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;
|
|
}
|
|
|
|
|
|
/// <summary>
|
|
/// Generates a random number with the specified number of bits such that gcd(number, this) = 1
|
|
/// </summary>
|
|
/// <remarks>
|
|
/// The number of bits must be greater than 0 and less than 2209
|
|
/// </remarks>
|
|
/// <param name="bits">Number of bit</param>
|
|
/// <param name="rand">Random object</param>
|
|
/// <returns>Relatively prime number of this</returns>
|
|
public BigInteger genCoPrime(int bits, Random rand)
|
|
{
|
|
bool done = false;
|
|
BigInteger result = new BigInteger();
|
|
|
|
while (!done)
|
|
{
|
|
result.genRandomBits(bits, rand);
|
|
|
|
// gcd test
|
|
BigInteger g = result.gcd(this);
|
|
if (g.dataLength == 1 && g.data[0] == 1)
|
|
done = true;
|
|
}
|
|
|
|
return result;
|
|
}
|
|
|
|
|
|
/// <summary>
|
|
/// Generates a random number with the specified number of bits such that gcd(number, this) = 1 (secured)
|
|
/// </summary>
|
|
/// <param name="bits">Number of bit</param>
|
|
/// <param name="rand">Random object</param>
|
|
/// <returns>Relatively prime number of this</returns>
|
|
public BigInteger genCoPrime(int bits, RNGCryptoServiceProvider rand)
|
|
{
|
|
bool done = false;
|
|
BigInteger result = new BigInteger();
|
|
|
|
while (!done)
|
|
{
|
|
result.genRandomBits(bits, rand);
|
|
|
|
// gcd test
|
|
BigInteger g = result.gcd(this);
|
|
if (g.dataLength == 1 && g.data[0] == 1)
|
|
done = true;
|
|
}
|
|
|
|
return result;
|
|
}
|
|
|
|
|
|
/// <summary>
|
|
/// Returns the modulo inverse of this
|
|
/// </summary>
|
|
/// <remarks>
|
|
/// Throws ArithmeticException if the inverse does not exist. (i.e. gcd(this, modulus) != 1)
|
|
/// </remarks>
|
|
/// <param name="modulus"></param>
|
|
/// <returns>Modulo inverse of this</returns>
|
|
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);
|
|
|
|
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;
|
|
}
|
|
|
|
|
|
/// <summary>
|
|
/// Returns the value of the BigInteger as a byte array
|
|
/// </summary>
|
|
/// <remarks>
|
|
/// The lowest index contains the MSB
|
|
/// </remarks>
|
|
/// <returns>Byte array containing value of the BigInteger</returns>
|
|
public byte[] getBytes()
|
|
{
|
|
int numBits = bitCount();
|
|
|
|
int numBytes = numBits >> 3;
|
|
if ((numBits & 0x7) != 0)
|
|
numBytes++;
|
|
|
|
byte[] result = new byte[numBytes];
|
|
|
|
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;
|
|
else if (pos > 0)
|
|
pos++;
|
|
|
|
if ((tempVal = (val >> 8 & 0xFF)) != 0)
|
|
result[pos++] = (byte)tempVal;
|
|
else if (pos > 0)
|
|
pos++;
|
|
|
|
if ((tempVal = (val & 0xFF)) != 0)
|
|
result[pos++] = (byte)tempVal;
|
|
else if (pos > 0)
|
|
pos++;
|
|
|
|
|
|
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;
|
|
}
|
|
|
|
|
|
/// <summary>
|
|
/// Sets the value of the specified bit to 1
|
|
/// </summary>
|
|
/// <remarks>
|
|
/// The Least Significant Bit position is 0
|
|
/// </remarks>
|
|
/// <param name="bitNum">The position of bit to be changed</param>
|
|
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;
|
|
}
|
|
|
|
|
|
/// <summary>
|
|
/// Sets the value of the specified bit to 0
|
|
/// </summary>
|
|
/// <remarks>
|
|
/// The Least Significant Bit position is 0
|
|
/// </remarks>
|
|
/// <param name="bitNum">The position of bit to be changed</param>
|
|
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--;
|
|
}
|
|
}
|
|
|
|
|
|
/// <summary>
|
|
/// Returns a value that is equivalent to the integer square root of this
|
|
/// </summary>
|
|
/// <remarks>
|
|
/// The integer square root of "this" is defined as the largest integer n, such that (n * n) <= this.
|
|
/// Square root of negative integer is an undefined behaviour (UB).
|
|
/// </remarks>
|
|
/// <returns>Integer square root of this</returns>
|
|
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;
|
|
}
|
|
|
|
|
|
/// <summary>
|
|
/// Returns the k_th number in the Lucas Sequence reduced modulo n
|
|
/// </summary>
|
|
/// <remarks>
|
|
/// 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
|
|
/// </remarks>
|
|
/// <param name="P"></param>
|
|
/// <param name="Q"></param>
|
|
/// <param name="k"></param>
|
|
/// <param name="n"></param>
|
|
/// <returns></returns>
|
|
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;
|
|
|
|
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
|
|
{
|
|
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,
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// (byte)0x8D, (byte)0x8C, (byte)0x44, (byte)0xCC, (byte)0xEE, (byte)0xEE,
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// (byte)0x3D, (byte)0x2C, (byte)0x9D, (byte)0x2C, (byte)0x12, (byte)0x41,
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// (byte)0x1E, (byte)0xF1, (byte)0xC5, (byte)0x32, (byte)0xC3, (byte)0xAA,
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// (byte)0x31, (byte)0x4A, (byte)0x52, (byte)0xD8, (byte)0xE8, (byte)0xAF,
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// (byte)0x42, (byte)0xF4, (byte)0x72, (byte)0xA1, (byte)0x2A, (byte)0x0D,
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// (byte)0x97, (byte)0xB1, (byte)0x31, (byte)0xB3,
|
|
// };
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|
|
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// byte[] pseudoPrime2 = { (byte)0x00,
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// (byte)0x99, (byte)0x98, (byte)0xCA, (byte)0xB8, (byte)0x5E, (byte)0xD7,
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// (byte)0xE5, (byte)0xDC, (byte)0x28, (byte)0x5C, (byte)0x6F, (byte)0x0E,
|
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// (byte)0x15, (byte)0x09, (byte)0x59, (byte)0x6E, (byte)0x84, (byte)0xF3,
|
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// (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);
|
|
|
|
//}
|
|
}
|
|
|