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Diffstat (limited to 'core/src/main/java/org/spongycastle/pqc/math/linearalgebra/IntegerFunctions.java')
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1 files changed, 1423 insertions, 0 deletions
diff --git a/core/src/main/java/org/spongycastle/pqc/math/linearalgebra/IntegerFunctions.java b/core/src/main/java/org/spongycastle/pqc/math/linearalgebra/IntegerFunctions.java new file mode 100644 index 00000000..a6d5a181 --- /dev/null +++ b/core/src/main/java/org/spongycastle/pqc/math/linearalgebra/IntegerFunctions.java @@ -0,0 +1,1423 @@ +package org.spongycastle.pqc.math.linearalgebra; + +import java.math.BigInteger; +import java.security.SecureRandom; + +/** + * Class of number-theory related functions for use with integers represented as + * <tt>int</tt>'s or <tt>BigInteger</tt> objects. + */ +public final class IntegerFunctions +{ + + private static final BigInteger ZERO = BigInteger.valueOf(0); + + private static final BigInteger ONE = BigInteger.valueOf(1); + + private static final BigInteger TWO = BigInteger.valueOf(2); + + private static final BigInteger FOUR = BigInteger.valueOf(4); + + private static final int[] SMALL_PRIMES = {3, 5, 7, 11, 13, 17, 19, 23, + 29, 31, 37, 41}; + + private static final long SMALL_PRIME_PRODUCT = 3L * 5 * 7 * 11 * 13 * 17 + * 19 * 23 * 29 * 31 * 37 * 41; + + private static SecureRandom sr = null; + + // the jacobi function uses this lookup table + private static final int[] jacobiTable = {0, 1, 0, -1, 0, -1, 0, 1}; + + private IntegerFunctions() + { + // empty + } + + /** + * Computes the value of the Jacobi symbol (A|B). The following properties + * hold for the Jacobi symbol which makes it a very efficient way to + * evaluate the Legendre symbol + * <p> + * (A|B) = 0 IF gcd(A,B) > 1<br> + * (-1|B) = 1 IF n = 1 (mod 1)<br> + * (-1|B) = -1 IF n = 3 (mod 4)<br> + * (A|B) (C|B) = (AC|B)<br> + * (A|B) (A|C) = (A|CB)<br> + * (A|B) = (C|B) IF A = C (mod B)<br> + * (2|B) = 1 IF N = 1 OR 7 (mod 8)<br> + * (2|B) = 1 IF N = 3 OR 5 (mod 8) + * + * @param A integer value + * @param B integer value + * @return value of the jacobi symbol (A|B) + */ + public static int jacobi(BigInteger A, BigInteger B) + { + BigInteger a, b, v; + long k = 1; + + k = 1; + + // test trivial cases + if (B.equals(ZERO)) + { + a = A.abs(); + return a.equals(ONE) ? 1 : 0; + } + + if (!A.testBit(0) && !B.testBit(0)) + { + return 0; + } + + a = A; + b = B; + + if (b.signum() == -1) + { // b < 0 + b = b.negate(); // b = -b + if (a.signum() == -1) + { + k = -1; + } + } + + v = ZERO; + while (!b.testBit(0)) + { + v = v.add(ONE); // v = v + 1 + b = b.divide(TWO); // b = b/2 + } + + if (v.testBit(0)) + { + k = k * jacobiTable[a.intValue() & 7]; + } + + if (a.signum() < 0) + { // a < 0 + if (b.testBit(1)) + { + k = -k; // k = -k + } + a = a.negate(); // a = -a + } + + // main loop + while (a.signum() != 0) + { + v = ZERO; + while (!a.testBit(0)) + { // a is even + v = v.add(ONE); + a = a.divide(TWO); + } + if (v.testBit(0)) + { + k = k * jacobiTable[b.intValue() & 7]; + } + + if (a.compareTo(b) < 0) + { // a < b + // swap and correct intermediate result + BigInteger x = a; + a = b; + b = x; + if (a.testBit(1) && b.testBit(1)) + { + k = -k; + } + } + a = a.subtract(b); + } + + return b.equals(ONE) ? (int)k : 0; + } + + /** + * Computes the square root of a BigInteger modulo a prime employing the + * Shanks-Tonelli algorithm. + * + * @param a value out of which we extract the square root + * @param p prime modulus that determines the underlying field + * @return a number <tt>b</tt> such that b<sup>2</sup> = a (mod p) if + * <tt>a</tt> is a quadratic residue modulo <tt>p</tt>. + * @throws NoQuadraticResidueException if <tt>a</tt> is a quadratic non-residue modulo <tt>p</tt> + */ + public static BigInteger ressol(BigInteger a, BigInteger p) + throws IllegalArgumentException + { + + BigInteger v = null; + + if (a.compareTo(ZERO) < 0) + { + a = a.add(p); + } + + if (a.equals(ZERO)) + { + return ZERO; + } + + if (p.equals(TWO)) + { + return a; + } + + // p = 3 mod 4 + if (p.testBit(0) && p.testBit(1)) + { + if (jacobi(a, p) == 1) + { // a quadr. residue mod p + v = p.add(ONE); // v = p+1 + v = v.shiftRight(2); // v = v/4 + return a.modPow(v, p); // return a^v mod p + // return --> a^((p+1)/4) mod p + } + throw new IllegalArgumentException("No quadratic residue: " + a + ", " + p); + } + + long t = 0; + + // initialization + // compute k and s, where p = 2^s (2k+1) +1 + + BigInteger k = p.subtract(ONE); // k = p-1 + long s = 0; + while (!k.testBit(0)) + { // while k is even + s++; // s = s+1 + k = k.shiftRight(1); // k = k/2 + } + + k = k.subtract(ONE); // k = k - 1 + k = k.shiftRight(1); // k = k/2 + + // initial values + BigInteger r = a.modPow(k, p); // r = a^k mod p + + BigInteger n = r.multiply(r).remainder(p); // n = r^2 % p + n = n.multiply(a).remainder(p); // n = n * a % p + r = r.multiply(a).remainder(p); // r = r * a %p + + if (n.equals(ONE)) + { + return r; + } + + // non-quadratic residue + BigInteger z = TWO; // z = 2 + while (jacobi(z, p) == 1) + { + // while z quadratic residue + z = z.add(ONE); // z = z + 1 + } + + v = k; + v = v.multiply(TWO); // v = 2k + v = v.add(ONE); // v = 2k + 1 + BigInteger c = z.modPow(v, p); // c = z^v mod p + + // iteration + while (n.compareTo(ONE) == 1) + { // n > 1 + k = n; // k = n + t = s; // t = s + s = 0; + + while (!k.equals(ONE)) + { // k != 1 + k = k.multiply(k).mod(p); // k = k^2 % p + s++; // s = s + 1 + } + + t -= s; // t = t - s + if (t == 0) + { + throw new IllegalArgumentException("No quadratic residue: " + a + ", " + p); + } + + v = ONE; + for (long i = 0; i < t - 1; i++) + { + v = v.shiftLeft(1); // v = 1 * 2^(t - 1) + } + c = c.modPow(v, p); // c = c^v mod p + r = r.multiply(c).remainder(p); // r = r * c % p + c = c.multiply(c).remainder(p); // c = c^2 % p + n = n.multiply(c).mod(p); // n = n * c % p + } + return r; + } + + /** + * Computes the greatest common divisor of the two specified integers + * + * @param u - first integer + * @param v - second integer + * @return gcd(a, b) + */ + public static int gcd(int u, int v) + { + return BigInteger.valueOf(u).gcd(BigInteger.valueOf(v)).intValue(); + } + + /** + * Extended euclidian algorithm (computes gcd and representation). + * + * @param a the first integer + * @param b the second integer + * @return <tt>(g,u,v)</tt>, where <tt>g = gcd(abs(a),abs(b)) = ua + vb</tt> + */ + public static int[] extGCD(int a, int b) + { + BigInteger ba = BigInteger.valueOf(a); + BigInteger bb = BigInteger.valueOf(b); + BigInteger[] bresult = extgcd(ba, bb); + int[] result = new int[3]; + result[0] = bresult[0].intValue(); + result[1] = bresult[1].intValue(); + result[2] = bresult[2].intValue(); + return result; + } + + public static BigInteger divideAndRound(BigInteger a, BigInteger b) + { + if (a.signum() < 0) + { + return divideAndRound(a.negate(), b).negate(); + } + if (b.signum() < 0) + { + return divideAndRound(a, b.negate()).negate(); + } + return a.shiftLeft(1).add(b).divide(b.shiftLeft(1)); + } + + public static BigInteger[] divideAndRound(BigInteger[] a, BigInteger b) + { + BigInteger[] out = new BigInteger[a.length]; + for (int i = 0; i < a.length; i++) + { + out[i] = divideAndRound(a[i], b); + } + return out; + } + + /** + * Compute the smallest integer that is greater than or equal to the + * logarithm to the base 2 of the given BigInteger. + * + * @param a the integer + * @return ceil[log(a)] + */ + public static int ceilLog(BigInteger a) + { + int result = 0; + BigInteger p = ONE; + while (p.compareTo(a) < 0) + { + result++; + p = p.shiftLeft(1); + } + return result; + } + + /** + * Compute the smallest integer that is greater than or equal to the + * logarithm to the base 2 of the given integer. + * + * @param a the integer + * @return ceil[log(a)] + */ + public static int ceilLog(int a) + { + int log = 0; + int i = 1; + while (i < a) + { + i <<= 1; + log++; + } + return log; + } + + /** + * Compute <tt>ceil(log_256 n)</tt>, the number of bytes needed to encode + * the integer <tt>n</tt>. + * + * @param n the integer + * @return the number of bytes needed to encode <tt>n</tt> + */ + public static int ceilLog256(int n) + { + if (n == 0) + { + return 1; + } + int m; + if (n < 0) + { + m = -n; + } + else + { + m = n; + } + + int d = 0; + while (m > 0) + { + d++; + m >>>= 8; + } + return d; + } + + /** + * Compute <tt>ceil(log_256 n)</tt>, the number of bytes needed to encode + * the long integer <tt>n</tt>. + * + * @param n the long integer + * @return the number of bytes needed to encode <tt>n</tt> + */ + public static int ceilLog256(long n) + { + if (n == 0) + { + return 1; + } + long m; + if (n < 0) + { + m = -n; + } + else + { + m = n; + } + + int d = 0; + while (m > 0) + { + d++; + m >>>= 8; + } + return d; + } + + /** + * Compute the integer part of the logarithm to the base 2 of the given + * integer. + * + * @param a the integer + * @return floor[log(a)] + */ + public static int floorLog(BigInteger a) + { + int result = -1; + BigInteger p = ONE; + while (p.compareTo(a) <= 0) + { + result++; + p = p.shiftLeft(1); + } + return result; + } + + /** + * Compute the integer part of the logarithm to the base 2 of the given + * integer. + * + * @param a the integer + * @return floor[log(a)] + */ + public static int floorLog(int a) + { + int h = 0; + if (a <= 0) + { + return -1; + } + int p = a >>> 1; + while (p > 0) + { + h++; + p >>>= 1; + } + + return h; + } + + /** + * Compute the largest <tt>h</tt> with <tt>2^h | a</tt> if <tt>a!=0</tt>. + * + * @param a an integer + * @return the largest <tt>h</tt> with <tt>2^h | a</tt> if <tt>a!=0</tt>, + * <tt>0</tt> otherwise + */ + public static int maxPower(int a) + { + int h = 0; + if (a != 0) + { + int p = 1; + while ((a & p) == 0) + { + h++; + p <<= 1; + } + } + + return h; + } + + /** + * @param a an integer + * @return the number of ones in the binary representation of an integer + * <tt>a</tt> + */ + public static int bitCount(int a) + { + int h = 0; + while (a != 0) + { + h += a & 1; + a >>>= 1; + } + + return h; + } + + /** + * determines the order of g modulo p, p prime and 1 < g < p. This algorithm + * is only efficient for small p (see X9.62-1998, p. 68). + * + * @param g an integer with 1 < g < p + * @param p a prime + * @return the order k of g (that is k is the smallest integer with + * g<sup>k</sup> = 1 mod p + */ + public static int order(int g, int p) + { + int b, j; + + b = g % p; // Reduce g mod p first. + j = 1; + + // Check whether g == 0 mod p (avoiding endless loop). + if (b == 0) + { + throw new IllegalArgumentException(g + " is not an element of Z/(" + + p + "Z)^*; it is not meaningful to compute its order."); + } + + // Compute the order of g mod p: + while (b != 1) + { + b *= g; + b %= p; + if (b < 0) + { + b += p; + } + j++; + } + + return j; + } + + /** + * Reduces an integer into a given interval + * + * @param n - the integer + * @param begin - left bound of the interval + * @param end - right bound of the interval + * @return <tt>n</tt> reduced into <tt>[begin,end]</tt> + */ + public static BigInteger reduceInto(BigInteger n, BigInteger begin, + BigInteger end) + { + return n.subtract(begin).mod(end.subtract(begin)).add(begin); + } + + /** + * Compute <tt>a<sup>e</sup></tt>. + * + * @param a the base + * @param e the exponent + * @return <tt>a<sup>e</sup></tt> + */ + public static int pow(int a, int e) + { + int result = 1; + while (e > 0) + { + if ((e & 1) == 1) + { + result *= a; + } + a *= a; + e >>>= 1; + } + return result; + } + + /** + * Compute <tt>a<sup>e</sup></tt>. + * + * @param a the base + * @param e the exponent + * @return <tt>a<sup>e</sup></tt> + */ + public static long pow(long a, int e) + { + long result = 1; + while (e > 0) + { + if ((e & 1) == 1) + { + result *= a; + } + a *= a; + e >>>= 1; + } + return result; + } + + /** + * Compute <tt>a<sup>e</sup> mod n</tt>. + * + * @param a the base + * @param e the exponent + * @param n the modulus + * @return <tt>a<sup>e</sup> mod n</tt> + */ + public static int modPow(int a, int e, int n) + { + if (n <= 0 || (n * n) > Integer.MAX_VALUE || e < 0) + { + return 0; + } + int result = 1; + a = (a % n + n) % n; + while (e > 0) + { + if ((e & 1) == 1) + { + result = (result * a) % n; + } + a = (a * a) % n; + e >>>= 1; + } + return result; + } + + /** + * Extended euclidian algorithm (computes gcd and representation). + * + * @param a - the first integer + * @param b - the second integer + * @return <tt>(d,u,v)</tt>, where <tt>d = gcd(a,b) = ua + vb</tt> + */ + public static BigInteger[] extgcd(BigInteger a, BigInteger b) + { + BigInteger u = ONE; + BigInteger v = ZERO; + BigInteger d = a; + if (b.signum() != 0) + { + BigInteger v1 = ZERO; + BigInteger v3 = b; + while (v3.signum() != 0) + { + BigInteger[] tmp = d.divideAndRemainder(v3); + BigInteger q = tmp[0]; + BigInteger t3 = tmp[1]; + BigInteger t1 = u.subtract(q.multiply(v1)); + u = v1; + d = v3; + v1 = t1; + v3 = t3; + } + v = d.subtract(a.multiply(u)).divide(b); + } + return new BigInteger[]{d, u, v}; + } + + /** + * Computation of the least common multiple of a set of BigIntegers. + * + * @param numbers - the set of numbers + * @return the lcm(numbers) + */ + public static BigInteger leastCommonMultiple(BigInteger[] numbers) + { + int n = numbers.length; + BigInteger result = numbers[0]; + for (int i = 1; i < n; i++) + { + BigInteger gcd = result.gcd(numbers[i]); + result = result.multiply(numbers[i]).divide(gcd); + } + return result; + } + + /** + * Returns a long integer whose value is <tt>(a mod m</tt>). This method + * differs from <tt>%</tt> in that it always returns a <i>non-negative</i> + * integer. + * + * @param a value on which the modulo operation has to be performed. + * @param m the modulus. + * @return <tt>a mod m</tt> + */ + public static long mod(long a, long m) + { + long result = a % m; + if (result < 0) + { + result += m; + } + return result; + } + + /** + * Computes the modular inverse of an integer a + * + * @param a - the integer to invert + * @param mod - the modulus + * @return <tt>a<sup>-1</sup> mod n</tt> + */ + public static int modInverse(int a, int mod) + { + return BigInteger.valueOf(a).modInverse(BigInteger.valueOf(mod)) + .intValue(); + } + + /** + * Computes the modular inverse of an integer a + * + * @param a - the integer to invert + * @param mod - the modulus + * @return <tt>a<sup>-1</sup> mod n</tt> + */ + public static long modInverse(long a, long mod) + { + return BigInteger.valueOf(a).modInverse(BigInteger.valueOf(mod)) + .longValue(); + } + + /** + * Tests whether an integer <tt>a</tt> is power of another integer + * <tt>p</tt>. + * + * @param a - the first integer + * @param p - the second integer + * @return n if a = p^n or -1 otherwise + */ + public static int isPower(int a, int p) + { + if (a <= 0) + { + return -1; + } + int n = 0; + int d = a; + while (d > 1) + { + if (d % p != 0) + { + return -1; + } + d /= p; + n++; + } + return n; + } + + /** + * Find and return the least non-trivial divisor of an integer <tt>a</tt>. + * + * @param a - the integer + * @return divisor p >1 or 1 if a = -1,0,1 + */ + public static int leastDiv(int a) + { + if (a < 0) + { + a = -a; + } + if (a == 0) + { + return 1; + } + if ((a & 1) == 0) + { + return 2; + } + int p = 3; + while (p <= (a / p)) + { + if ((a % p) == 0) + { + return p; + } + p += 2; + } + + return a; + } + + /** + * Miller-Rabin-Test, determines wether the given integer is probably prime + * or composite. This method returns <tt>true</tt> if the given integer is + * prime with probability <tt>1 - 2<sup>-20</sup></tt>. + * + * @param n the integer to test for primality + * @return <tt>true</tt> if the given integer is prime with probability + * 2<sup>-100</sup>, <tt>false</tt> otherwise + */ + public static boolean isPrime(int n) + { + if (n < 2) + { + return false; + } + if (n == 2) + { + return true; + } + if ((n & 1) == 0) + { + return false; + } + if (n < 42) + { + for (int i = 0; i < SMALL_PRIMES.length; i++) + { + if (n == SMALL_PRIMES[i]) + { + return true; + } + } + } + + if ((n % 3 == 0) || (n % 5 == 0) || (n % 7 == 0) || (n % 11 == 0) + || (n % 13 == 0) || (n % 17 == 0) || (n % 19 == 0) + || (n % 23 == 0) || (n % 29 == 0) || (n % 31 == 0) + || (n % 37 == 0) || (n % 41 == 0)) + { + return false; + } + + return BigInteger.valueOf(n).isProbablePrime(20); + } + + /** + * Short trial-division test to find out whether a number is not prime. This + * test is usually used before a Miller-Rabin primality test. + * + * @param candidate the number to test + * @return <tt>true</tt> if the number has no factor of the tested primes, + * <tt>false</tt> if the number is definitely composite + */ + public static boolean passesSmallPrimeTest(BigInteger candidate) + { + final int[] smallPrime = {2, 3, 5, 7, 11, 13, 17, 19, 23, 29, 31, 37, + 41, 43, 47, 53, 59, 61, 67, 71, 73, 79, 83, 89, 97, 101, 103, + 107, 109, 113, 127, 131, 137, 139, 149, 151, 157, 163, 167, + 173, 179, 181, 191, 193, 197, 199, 211, 223, 227, 229, 233, + 239, 241, 251, 257, 263, 269, 271, 277, 281, 283, 293, 307, + 311, 313, 317, 331, 337, 347, 349, 353, 359, 367, 373, 379, + 383, 389, 397, 401, 409, 419, 421, 431, 433, 439, 443, 449, + 457, 461, 463, 467, 479, 487, 491, 499, 503, 509, 521, 523, + 541, 547, 557, 563, 569, 571, 577, 587, 593, 599, 601, 607, + 613, 617, 619, 631, 641, 643, 647, 653, 659, 661, 673, 677, + 683, 691, 701, 709, 719, 727, 733, 739, 743, 751, 757, 761, + 769, 773, 787, 797, 809, 811, 821, 823, 827, 829, 839, 853, + 857, 859, 863, 877, 881, 883, 887, 907, 911, 919, 929, 937, + 941, 947, 953, 967, 971, 977, 983, 991, 997, 1009, 1013, 1019, + 1021, 1031, 1033, 1039, 1049, 1051, 1061, 1063, 1069, 1087, + 1091, 1093, 1097, 1103, 1109, 1117, 1123, 1129, 1151, 1153, + 1163, 1171, 1181, 1187, 1193, 1201, 1213, 1217, 1223, 1229, + 1231, 1237, 1249, 1259, 1277, 1279, 1283, 1289, 1291, 1297, + 1301, 1303, 1307, 1319, 1321, 1327, 1361, 1367, 1373, 1381, + 1399, 1409, 1423, 1427, 1429, 1433, 1439, 1447, 1451, 1453, + 1459, 1471, 1481, 1483, 1487, 1489, 1493, 1499}; + + for (int i = 0; i < smallPrime.length; i++) + { + if (candidate.mod(BigInteger.valueOf(smallPrime[i])).equals( + ZERO)) + { + return false; + } + } + return true; + } + + /** + * Returns the largest prime smaller than the given integer + * + * @param n - upper bound + * @return the largest prime smaller than <tt>n</tt>, or <tt>1</tt> if + * <tt>n <= 2</tt> + */ + public static int nextSmallerPrime(int n) + { + if (n <= 2) + { + return 1; + } + + if (n == 3) + { + return 2; + } + + if ((n & 1) == 0) + { + n--; + } + else + { + n -= 2; + } + + while (n > 3 & !isPrime(n)) + { + n -= 2; + } + return n; + } + + /** + * Compute the next probable prime greater than <tt>n</tt> with the + * specified certainty. + * + * @param n a integer number + * @param certainty the certainty that the generated number is prime + * @return the next prime greater than <tt>n</tt> + */ + public static BigInteger nextProbablePrime(BigInteger n, int certainty) + { + + if (n.signum() < 0 || n.signum() == 0 || n.equals(ONE)) + { + return TWO; + } + + BigInteger result = n.add(ONE); + + // Ensure an odd number + if (!result.testBit(0)) + { + result = result.add(ONE); + } + + while (true) + { + // Do cheap "pre-test" if applicable + if (result.bitLength() > 6) + { + long r = result.remainder( + BigInteger.valueOf(SMALL_PRIME_PRODUCT)).longValue(); + if ((r % 3 == 0) || (r % 5 == 0) || (r % 7 == 0) + || (r % 11 == 0) || (r % 13 == 0) || (r % 17 == 0) + || (r % 19 == 0) || (r % 23 == 0) || (r % 29 == 0) + || (r % 31 == 0) || (r % 37 == 0) || (r % 41 == 0)) + { + result = result.add(TWO); + continue; // Candidate is composite; try another + } + } + + // All candidates of bitLength 2 and 3 are prime by this point + if (result.bitLength() < 4) + { + return result; + } + + // The expensive test + if (result.isProbablePrime(certainty)) + { + return result; + } + + result = result.add(TWO); + } + } + + /** + * Compute the next probable prime greater than <tt>n</tt> with the default + * certainty (20). + * + * @param n a integer number + * @return the next prime greater than <tt>n</tt> + */ + public static BigInteger nextProbablePrime(BigInteger n) + { + return nextProbablePrime(n, 20); + } + + /** + * Computes the next prime greater than n. + * + * @param n a integer number + * @return the next prime greater than n + */ + public static BigInteger nextPrime(long n) + { + long i; + boolean found = false; + long result = 0; + + if (n <= 1) + { + return BigInteger.valueOf(2); + } + if (n == 2) + { + return BigInteger.valueOf(3); + } + + for (i = n + 1 + (n & 1); (i <= n << 1) && !found; i += 2) + { + for (long j = 3; (j <= i >> 1) && !found; j += 2) + { + if (i % j == 0) + { + found = true; + } + } + if (found) + { + found = false; + } + else + { + result = i; + found = true; + } + } + return BigInteger.valueOf(result); + } + + /** + * Computes the binomial coefficient (n|t) ("n over t"). Formula: + * <ul> + * <li>if n !=0 and t != 0 then (n|t) = Mult(i=1, t): (n-(i-1))/i</li> + * <li>if t = 0 then (n|t) = 1</li> + * <li>if n = 0 and t > 0 then (n|t) = 0</li> + * </ul> + * + * @param n - the "upper" integer + * @param t - the "lower" integer + * @return the binomialcoefficient "n over t" as BigInteger + */ + public static BigInteger binomial(int n, int t) + { + + BigInteger result = ONE; + + if (n == 0) + { + if (t == 0) + { + return result; + } + return ZERO; + } + + // the property (n|t) = (n|n-t) be used to reduce numbers of operations + if (t > (n >>> 1)) + { + t = n - t; + } + + for (int i = 1; i <= t; i++) + { + result = (result.multiply(BigInteger.valueOf(n - (i - 1)))) + .divide(BigInteger.valueOf(i)); + } + + return result; + } + + public static BigInteger randomize(BigInteger upperBound) + { + if (sr == null) + { + sr = new SecureRandom(); + } + return randomize(upperBound, sr); + } + + public static BigInteger randomize(BigInteger upperBound, + SecureRandom prng) + { + int blen = upperBound.bitLength(); + BigInteger randomNum = BigInteger.valueOf(0); + + if (prng == null) + { + prng = sr != null ? sr : new SecureRandom(); + } + + for (int i = 0; i < 20; i++) + { + randomNum = new BigInteger(blen, prng); + if (randomNum.compareTo(upperBound) < 0) + { + return randomNum; + } + } + return randomNum.mod(upperBound); + } + + /** + * Extract the truncated square root of a BigInteger. + * + * @param a - value out of which we extract the square root + * @return the truncated square root of <tt>a</tt> + */ + public static BigInteger squareRoot(BigInteger a) + { + int bl; + BigInteger result, remainder, b; + + if (a.compareTo(ZERO) < 0) + { + throw new ArithmeticException( + "cannot extract root of negative number" + a + "."); + } + + bl = a.bitLength(); + result = ZERO; + remainder = ZERO; + + // if the bit length is odd then extra step + if ((bl & 1) != 0) + { + result = result.add(ONE); + bl--; + } + + while (bl > 0) + { + remainder = remainder.multiply(FOUR); + remainder = remainder.add(BigInteger.valueOf((a.testBit(--bl) ? 2 + : 0) + + (a.testBit(--bl) ? 1 : 0))); + b = result.multiply(FOUR).add(ONE); + result = result.multiply(TWO); + if (remainder.compareTo(b) != -1) + { + result = result.add(ONE); + remainder = remainder.subtract(b); + } + } + + return result; + } + + /** + * Takes an approximation of the root from an integer base, using newton's + * algorithm + * + * @param base the base to take the root from + * @param root the root, for example 2 for a square root + */ + public static float intRoot(int base, int root) + { + float gNew = base / root; + float gOld = 0; + int counter = 0; + while (Math.abs(gOld - gNew) > 0.0001) + { + float gPow = floatPow(gNew, root); + while (Float.isInfinite(gPow)) + { + gNew = (gNew + gOld) / 2; + gPow = floatPow(gNew, root); + } + counter += 1; + gOld = gNew; + gNew = gOld - (gPow - base) / (root * floatPow(gOld, root - 1)); + } + return gNew; + } + + /** + * Calculation of a logarithmus of a float param + * + * @param param + * @return + */ + public static float floatLog(float param) + { + double arg = (param - 1) / (param + 1); + double arg2 = arg; + int counter = 1; + float result = (float)arg; + + while (arg2 > 0.001) + { + counter += 2; + arg2 *= arg * arg; + result += (1. / counter) * arg2; + } + return 2 * result; + } + + /** + * int power of a base float, only use for small ints + * + * @param f + * @param i + * @return + */ + public static float floatPow(float f, int i) + { + float g = 1; + for (; i > 0; i--) + { + g *= f; + } + return g; + } + + /** + * calculate the logarithm to the base 2. + * + * @param x any double value + * @return log_2(x) + * @deprecated use MathFunctions.log(double) instead + */ + public static double log(double x) + { + if (x > 0 && x < 1) + { + double d = 1 / x; + double result = -log(d); + return result; + } + + int tmp = 0; + double tmp2 = 1; + double d = x; + + while (d > 2) + { + d = d / 2; + tmp += 1; + tmp2 *= 2; + } + double rem = x / tmp2; + rem = logBKM(rem); + return tmp + rem; + } + + /** + * calculate the logarithm to the base 2. + * + * @param x any long value >=1 + * @return log_2(x) + * @deprecated use MathFunctions.log(long) instead + */ + public static double log(long x) + { + int tmp = floorLog(BigInteger.valueOf(x)); + long tmp2 = 1 << tmp; + double rem = (double)x / (double)tmp2; + rem = logBKM(rem); + return tmp + rem; + } + + /** + * BKM Algorithm to calculate logarithms to the base 2. + * + * @param arg a double value with 1<= arg<= 4.768462058 + * @return log_2(arg) + * @deprecated use MathFunctions.logBKM(double) instead + */ + private static double logBKM(double arg) + { + double ae[] = // A_e[k] = log_2 (1 + 0.5^k) + { + 1.0000000000000000000000000000000000000000000000000000000000000000000000000000, + 0.5849625007211561814537389439478165087598144076924810604557526545410982276485, + 0.3219280948873623478703194294893901758648313930245806120547563958159347765589, + 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0.0000000000000000000000000000022761714318270646273745024223029238091160103901}; + int n = 53; + double x = 1; + double y = 0; + double z; + double s = 1; + int k; + + for (k = 0; k < n; k++) + { + z = x + x * s; + if (z <= arg) + { + x = z; + y += ae[k]; + } + s *= 0.5; + } + return y; + } + + public static boolean isIncreasing(int[] a) + { + for (int i = 1; i < a.length; i++) + { + if (a[i - 1] >= a[i]) + { + System.out.println("a[" + (i - 1) + "] = " + a[i - 1] + " >= " + + a[i] + " = a[" + i + "]"); + return false; + } + } + return true; + } + + public static byte[] integerToOctets(BigInteger val) + { + byte[] valBytes = val.abs().toByteArray(); + + // check whether the array includes a sign bit + if ((val.bitLength() & 7) != 0) + { + return valBytes; + } + // get rid of the sign bit (first byte) + byte[] tmp = new byte[val.bitLength() >> 3]; + System.arraycopy(valBytes, 1, tmp, 0, tmp.length); + return tmp; + } + + public static BigInteger octetsToInteger(byte[] data, int offset, + int length) + { + byte[] val = new byte[length + 1]; + + val[0] = 0; + System.arraycopy(data, offset, val, 1, length); + return new BigInteger(val); + } + + public static BigInteger octetsToInteger(byte[] data) + { + return octetsToInteger(data, 0, data.length); + } + + public static void main(String[] args) + { + System.out.println("test"); + // System.out.println(intRoot(37, 5)); + // System.out.println(floatPow((float)2.5, 4)); + System.out.println(floatLog(10)); + System.out.println("test2"); + } +} |