/** * It requires that all parameters be positive. * * @return {@code base<sup>exponent</sup> mod (2<sup>j</sup>)}. * @see BigInteger#modPow(BigInteger, BigInteger) */ private static BigInteger Pow2ModPow(BigInteger b, BigInteger exponent, int j) { // PRE: (base > 0), (exponent > 0) and (j > 0) BigInteger res = BigInteger.One; BigInteger e = exponent.Copy(); BigInteger baseMod2toN = b.Copy(); BigInteger res2; /* * If 'base' is odd then it's coprime with 2^j and phi(2^j) = 2^(j-1); * so we can reduce reduce the exponent (mod 2^(j-1)). */ if (BigInteger.TestBit(b, 0)) { InplaceModPow2(e, j - 1); } InplaceModPow2(baseMod2toN, j); for (int i = e.BitLength - 1; i >= 0; i--) { res2 = res.Copy(); InplaceModPow2(res2, j); res = res * res2; if (BitLevel.TestBit(e, i)) { res = res * baseMod2toN; InplaceModPow2(res, j); } } InplaceModPow2(res, j); return(res); }
/** * Performs a<sup>2</sup> * @param a The number to square. * @param aLen The length of the number to square. */ static int[] square(int[] a, int aLen, int[] res) { long carry; for (int i = 0; i < aLen; i++) { carry = 0; for (int j = i + 1; j < aLen; j++) { carry = unsignedMultAddAdd(a[i], a[j], res[i + j], (int)carry); res[i + j] = (int)carry; carry = Utils.URShift(carry, 32); } res[i + aLen] = (int)carry; } BitLevel.shiftLeftOneBit(res, res, aLen << 1); carry = 0; for (int i = 0, index = 0; i < aLen; i++, index++) { carry = unsignedMultAddAdd(a[i], a[i], res[index], (int)carry); res[index] = (int)carry; carry = Utils.URShift(carry, 32); index++; carry += res[index] & 0xFFFFFFFFL; res[index] = (int)carry; carry = Utils.URShift(carry, 32); } return(res); }
/** * Returns a new {@code BigInteger} which has the same binary representation * as {@code this} but with the bit at position n cleared. The result is * equivalent to {@code this & ~(2^n)}. * <p> * <b>Implementation Note:</b> Usage of this method is not recommended as * the current implementation is not efficient. * * @param n * position where the bit in {@code this} has to be cleared. * @return {@code this & ~(2^n)}. * @throws ArithmeticException * if {@code n < 0}. */ public static BigInteger ClearBit(BigInteger value, int n) { if (TestBit(value, n)) { return(BitLevel.FlipBit(value, n)); } return(value); }
public static BigInteger ShiftLeft(BigInteger value, int n) { if ((n == 0) || (value.Sign == 0)) { return(value); } return((n > 0) ? BitLevel.ShiftLeft(value, n) : BitLevel.ShiftRight(value, -n)); }
/** * Returns a new {@code BigInteger} which has the same binary representation * as {@code this} but with the bit at position n flipped. The result is * equivalent to {@code this ^ 2^n}. * <p> * <b>Implementation Note:</b> Usage of this method is not recommended as * the current implementation is not efficient. * * @param n * position where the bit in {@code this} has to be flipped. * @return {@code this ^ 2^n}. * @throws ArithmeticException * if {@code n < 0}. */ public static BigInteger FlipBit(BigInteger value, int n) { if (n < 0) { // math.15=Negative bit address throw new ArithmeticException(Messages.math15); //$NON-NLS-1$ } return(BitLevel.FlipBit(value, n)); }
/** @see BigInteger#ToDouble() */ public static double BigInteger2Double(BigInteger val) { // val.bitLength() < 64 if ((val.numberLength < 2) || ((val.numberLength == 2) && (val.Digits[1] > 0))) { return(val.ToInt64()); } // val.bitLength() >= 33 * 32 > 1024 if (val.numberLength > 32) { return((val.Sign > 0) ? Double.PositiveInfinity : Double.NegativeInfinity); } int bitLen = BigMath.Abs(val).BitLength; long exponent = bitLen - 1; int delta = bitLen - 54; // We need 54 top bits from this, the 53th bit is always 1 in lVal. long lVal = (BigMath.Abs(val) >> delta).ToInt64(); /* * Take 53 bits from lVal to mantissa. The least significant bit is * needed for rounding. */ long mantissa = lVal & 0x1FFFFFFFFFFFFFL; if (exponent == 1023) { if (mantissa == 0X1FFFFFFFFFFFFFL) { return((val.Sign > 0) ? Double.PositiveInfinity : Double.NegativeInfinity); } if (mantissa == 0x1FFFFFFFFFFFFEL) { return((val.Sign > 0) ? Double.MaxValue : -Double.MaxValue); } } // Round the mantissa if (((mantissa & 1) == 1) && (((mantissa & 2) == 2) || BitLevel.NonZeroDroppedBits(delta, val.Digits))) { mantissa += 2; } mantissa >>= 1; // drop the rounding bit // long resSign = (val.sign < 0) ? 0x8000000000000000L : 0; long resSign = (val.Sign < 0) ? Int64.MinValue : 0; exponent = ((1023 + exponent) << 52) & 0x7FF0000000000000L; long result = resSign | exponent | mantissa; return(BitConverter.Int64BitsToDouble(result)); }
/*Implements the Montgomery modular exponentiation based in <i>The sliding windows algorithm and the Mongomery * Reduction</i>. *@ar.org.fitc.ref "A. Menezes,P. van Oorschot, S. Vanstone - Handbook of Applied Cryptography"; *@see #oddModPow(BigInteger, BigInteger, * BigInteger) */ private static BigInteger SlidingWindow(BigInteger x2, BigInteger a2, BigInteger exponent, BigInteger modulus, int n2) { // fill odd low pows of a2 BigInteger[] pows = new BigInteger[8]; BigInteger res = x2; int lowexp; BigInteger x3; int acc3; pows[0] = a2; x3 = MonPro(a2, a2, modulus, n2); for (int i = 1; i <= 7; i++) { pows[i] = MonPro(pows[i - 1], x3, modulus, n2); } for (int i = exponent.BitLength - 1; i >= 0; i--) { if (BitLevel.TestBit(exponent, i)) { lowexp = 1; acc3 = i; for (int j = System.Math.Max(i - 3, 0); j <= i - 1; j++) { if (BitLevel.TestBit(exponent, j)) { if (j < acc3) { acc3 = j; lowexp = (lowexp << (i - j)) ^ 1; } else { lowexp = lowexp ^ (1 << (j - acc3)); } } } for (int j = acc3; j <= i; j++) { res = MonPro(res, res, modulus, n2); } res = MonPro(pows[(lowexp - 1) >> 1], res, modulus, n2); i = acc3; } else { res = MonPro(res, res, modulus, n2); } } return(res); }
private static BigInteger SquareAndMultiply(BigInteger x2, BigInteger a2, BigInteger exponent, BigInteger modulus, int n2) { BigInteger res = x2; for (int i = exponent.BitLength - 1; i >= 0; i--) { res = MonPro(res, res, modulus, n2); if (BitLevel.TestBit(exponent, i)) { res = MonPro(res, a2, modulus, n2); } } return(res); }
/** * @param x an odd positive number. * @param n the exponent by which 2 is raised. * @return {@code x<sup>-1</sup> (mod 2<sup>n</sup>)}. */ private static BigInteger ModPow2Inverse(BigInteger x, int n) { // PRE: (x > 0), (x is odd), and (n > 0) BigInteger y = new BigInteger(1, new int[1 << n]); y.numberLength = 1; y.Digits[0] = 1; y.Sign = 1; for (int i = 1; i < n; i++) { if (BitLevel.TestBit(x * y, i)) { // Adding 2^i to y (setting the i-th bit) y.Digits[i >> 5] |= (1 << (i & 31)); } } return(y); }
internal BigInteger ShiftLeftOneBit() { return((sign == 0) ? this : BitLevel.ShiftLeftOneBit(this)); }
/** * * Based on "New Algorithm for Classical Modular Inverse" Róbert Lórencz. * LNCS 2523 (2002) * * @return a^(-1) mod m */ private static BigInteger ModInverseLorencz(BigInteger a, BigInteger modulo) { // PRE: a is coprime with modulo, a < modulo int max = System.Math.Max(a.numberLength, modulo.numberLength); int[] uDigits = new int[max + 1]; // enough place to make all the inplace operation int[] vDigits = new int[max + 1]; Array.Copy(modulo.Digits, 0, uDigits, 0, modulo.numberLength); Array.Copy(a.Digits, 0, vDigits, 0, a.numberLength); BigInteger u = new BigInteger(modulo.Sign, modulo.numberLength, uDigits); BigInteger v = new BigInteger(a.Sign, a.numberLength, vDigits); BigInteger r = new BigInteger(0, 1, new int[max + 1]); // BigInteger.ZERO; BigInteger s = new BigInteger(1, 1, new int[max + 1]); s.Digits[0] = 1; // r == 0 && s == 1, but with enough place int coefU = 0, coefV = 0; int n = modulo.BitLength; int k; while (!IsPowerOfTwo(u, coefU) && !IsPowerOfTwo(v, coefV)) { // modification of original algorithm: I calculate how many times the algorithm will enter in the same branch of if k = HowManyIterations(u, n); if (k != 0) { BitLevel.InplaceShiftLeft(u, k); if (coefU >= coefV) { BitLevel.InplaceShiftLeft(r, k); } else { BitLevel.InplaceShiftRight(s, System.Math.Min(coefV - coefU, k)); if (k - (coefV - coefU) > 0) { BitLevel.InplaceShiftLeft(r, k - coefV + coefU); } } coefU += k; } k = HowManyIterations(v, n); if (k != 0) { BitLevel.InplaceShiftLeft(v, k); if (coefV >= coefU) { BitLevel.InplaceShiftLeft(s, k); } else { BitLevel.InplaceShiftRight(r, System.Math.Min(coefU - coefV, k)); if (k - (coefU - coefV) > 0) { BitLevel.InplaceShiftLeft(s, k - coefU + coefV); } } coefV += k; } if (u.Sign == v.Sign) { if (coefU <= coefV) { Elementary.completeInPlaceSubtract(u, v); Elementary.completeInPlaceSubtract(r, s); } else { Elementary.completeInPlaceSubtract(v, u); Elementary.completeInPlaceSubtract(s, r); } } else { if (coefU <= coefV) { Elementary.completeInPlaceAdd(u, v); Elementary.completeInPlaceAdd(r, s); } else { Elementary.completeInPlaceAdd(v, u); Elementary.completeInPlaceAdd(s, r); } } if (v.Sign == 0 || u.Sign == 0) { // math.19: BigInteger not invertible throw new ArithmeticException(Messages.math19); } } if (IsPowerOfTwo(v, coefV)) { r = s; if (v.Sign != u.Sign) { u = -u; } } if (BigInteger.TestBit(u, n)) { if (r.Sign < 0) { r = -r; } else { r = modulo - r; } } if (r.Sign < 0) { r += modulo; } return(r); }
/** * Calculates a.modInverse(p) Based on: Savas, E; Koc, C "The Montgomery Modular * Inverse - Revised" */ public static BigInteger ModInverseMontgomery(BigInteger a, BigInteger p) { if (a.Sign == 0) { // ZERO hasn't inverse // math.19: BigInteger not invertible throw new ArithmeticException(Messages.math19); } if (!BigInteger.TestBit(p, 0)) { // montgomery inverse require even modulo return(ModInverseLorencz(a, p)); } int m = p.numberLength * 32; // PRE: a \in [1, p - 1] BigInteger u, v, r, s; u = p.Copy(); // make copy to use inplace method v = a.Copy(); int max = System.Math.Max(v.numberLength, u.numberLength); r = new BigInteger(1, 1, new int[max + 1]); s = new BigInteger(1, 1, new int[max + 1]); s.Digits[0] = 1; // s == 1 && v == 0 int k = 0; int lsbu = u.LowestSetBit; int lsbv = v.LowestSetBit; int toShift; if (lsbu > lsbv) { BitLevel.InplaceShiftRight(u, lsbu); BitLevel.InplaceShiftRight(v, lsbv); BitLevel.InplaceShiftLeft(r, lsbv); k += lsbu - lsbv; } else { BitLevel.InplaceShiftRight(u, lsbu); BitLevel.InplaceShiftRight(v, lsbv); BitLevel.InplaceShiftLeft(s, lsbu); k += lsbv - lsbu; } r.Sign = 1; while (v.Sign > 0) { // INV v >= 0, u >= 0, v odd, u odd (except last iteration when v is even (0)) while (u.CompareTo(v) > BigInteger.EQUALS) { Elementary.inplaceSubtract(u, v); toShift = u.LowestSetBit; BitLevel.InplaceShiftRight(u, toShift); Elementary.inplaceAdd(r, s); BitLevel.InplaceShiftLeft(s, toShift); k += toShift; } while (u.CompareTo(v) <= BigInteger.EQUALS) { Elementary.inplaceSubtract(v, u); if (v.Sign == 0) { break; } toShift = v.LowestSetBit; BitLevel.InplaceShiftRight(v, toShift); Elementary.inplaceAdd(s, r); BitLevel.InplaceShiftLeft(r, toShift); k += toShift; } } if (!u.IsOne) { // in u is stored the gcd // math.19: BigInteger not invertible. throw new ArithmeticException(Messages.math19); } if (r.CompareTo(p) >= BigInteger.EQUALS) { Elementary.inplaceSubtract(r, p); } r = p - r; // Have pair: ((BigInteger)r, (Integer)k) where r == a^(-1) * 2^k mod (module) int n1 = CalcN(p); if (k > m) { r = MonPro(r, BigInteger.One, p, n1); k = k - m; } r = MonPro(r, BigInteger.GetPowerOfTwo(m - k), p, n1); return(r); }
/** * Divides the array 'a' by the array 'b' and gets the quotient and the * remainder. Implements the Knuth's division algorithm. See D. Knuth, The * Art of Computer Programming, vol. 2. Steps D1-D8 correspond the steps in * the algorithm description. * * @param quot the quotient * @param quotLength the quotient's length * @param a the dividend * @param aLength the dividend's length * @param b the divisor * @param bLength the divisor's length * @return the remainder */ public static int[] Divide(int[] quot, int quotLength, int[] a, int aLength, int[] b, int bLength) { int[] normA = new int[aLength + 1]; // the normalized dividend // an extra byte is needed for correct shift int[] normB = new int[bLength + 1]; // the normalized divisor; int normBLength = bLength; /* * Step D1: normalize a and b and put the results to a1 and b1 the * normalized divisor's first digit must be >= 2^31 */ int divisorShift = Utils.NumberOfLeadingZeros(b[bLength - 1]); if (divisorShift != 0) { BitLevel.ShiftLeft(normB, b, 0, divisorShift); BitLevel.ShiftLeft(normA, a, 0, divisorShift); } else { Array.Copy(a, 0, normA, 0, aLength); Array.Copy(b, 0, normB, 0, bLength); } int firstDivisorDigit = normB[normBLength - 1]; // Step D2: set the quotient index int i = quotLength - 1; int j = aLength; while (i >= 0) { // Step D3: calculate a guess digit guessDigit int guessDigit = 0; if (normA[j] == firstDivisorDigit) { // set guessDigit to the largest unsigned int value guessDigit = -1; } else { long product = (((normA[j] & 0xffffffffL) << 32) + (normA[j - 1] & 0xffffffffL)); long res = Division.DivideLongByInt(product, firstDivisorDigit); guessDigit = (int)res; // the quotient of divideLongByInt int rem = (int)(res >> 32); // the remainder of // divideLongByInt // decrease guessDigit by 1 while leftHand > rightHand if (guessDigit != 0) { long leftHand = 0; long rightHand = 0; bool rOverflowed = false; guessDigit++; // to have the proper value in the loop // below do { guessDigit--; if (rOverflowed) { break; } // leftHand always fits in an unsigned long leftHand = (guessDigit & 0xffffffffL) * (normB[normBLength - 2] & 0xffffffffL); /* * rightHand can overflow; in this case the loop * condition will be true in the next step of the loop */ rightHand = ((long)rem << 32) + (normA[j - 2] & 0xffffffffL); long longR = (rem & 0xffffffffL) + (firstDivisorDigit & 0xffffffffL); /* * checks that longR does not fit in an unsigned int; * this ensures that rightHand will overflow unsigned * long in the next step */ if (Utils.NumberOfLeadingZeros((int)Utils.URShift(longR, 32)) < 32) { rOverflowed = true; } else { rem = (int)longR; } } while ((leftHand ^ Int64.MinValue) > (rightHand ^ Int64.MinValue)); //} while ((leftHand ^ Int64.MaxValue) > (rightHand ^ Int64.MaxValue)); // while (((leftHand ^ 0x8000000000000000L) > (rightHand ^ 0x8000000000000000L))) ; } } // Step D4: multiply normB by guessDigit and subtract the production // from normA. if (guessDigit != 0) { int borrow = Division.MultiplyAndSubtract(normA, j - normBLength, normB, normBLength, guessDigit); // Step D5: check the borrow if (borrow != 0) { // Step D6: compensating addition guessDigit--; long carry = 0; for (int k = 0; k < normBLength; k++) { carry += (normA[j - normBLength + k] & 0xffffffffL) + (normB[k] & 0xffffffffL); normA[j - normBLength + k] = (int)carry; carry = Utils.URShift(carry, 32); } } } if (quot != null) { quot[i] = guessDigit; } // Step D7 j--; i--; } /* * Step D8: we got the remainder in normA. Denormalize it id needed */ if (divisorShift != 0) { // reuse normB BitLevel.ShiftRight(normB, normBLength, normA, 0, divisorShift); return(normB); } Array.Copy(normA, 0, normB, 0, bLength); return(normA); }
/** * @param m a positive modulus * Return the greatest common divisor of op1 and op2, * * @param op1 * must be greater than zero * @param op2 * must be greater than zero * @see BigInteger#gcd(BigInteger) * @return {@code GCD(op1, op2)} */ public static BigInteger GcdBinary(BigInteger op1, BigInteger op2) { // PRE: (op1 > 0) and (op2 > 0) /* * Divide both number the maximal possible times by 2 without rounding * gcd(2*a, 2*b) = 2 * gcd(a,b) */ int lsb1 = op1.LowestSetBit; int lsb2 = op2.LowestSetBit; int pow2Count = System.Math.Min(lsb1, lsb2); BitLevel.InplaceShiftRight(op1, lsb1); BitLevel.InplaceShiftRight(op2, lsb2); BigInteger swap; // I want op2 > op1 if (op1.CompareTo(op2) == BigInteger.GREATER) { swap = op1; op1 = op2; op2 = swap; } do { // INV: op2 >= op1 && both are odd unless op1 = 0 // Optimization for small operands // (op2.bitLength() < 64) implies by INV (op1.bitLength() < 64) if ((op2.numberLength == 1) || ((op2.numberLength == 2) && (op2.Digits[1] > 0))) { op2 = BigInteger.FromInt64(Division.GcdBinary(op1.ToInt64(), op2.ToInt64())); break; } // Implements one step of the Euclidean algorithm // To reduce one operand if it's much smaller than the other one if (op2.numberLength > op1.numberLength * 1.2) { op2 = BigMath.Remainder(op2, op1); if (op2.Sign != 0) { BitLevel.InplaceShiftRight(op2, op2.LowestSetBit); } } else { // Use Knuth's algorithm of successive subtract and shifting do { Elementary.inplaceSubtract(op2, op1); // both are odd BitLevel.InplaceShiftRight(op2, op2.LowestSetBit); // op2 is even } while (op2.CompareTo(op1) >= BigInteger.EQUALS); } // now op1 >= op2 swap = op2; op2 = op1; op1 = swap; } while (op1.Sign != 0); return(op2 << pow2Count); }