org.bouncycastle.pqc.math.linearalgebra.IntegerFunctions Maven / Gradle / Ivy
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package org.bouncycastle.pqc.math.linearalgebra;
import java.math.BigInteger;
import java.security.SecureRandom;
import org.bouncycastle.crypto.CryptoServicesRegistrar;
import org.bouncycastle.util.BigIntegers;
/**
* Class of number-theory related functions for use with integers represented as
* int's or BigInteger 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
*
* (A|B) = 0 IF gcd(A,B) > 1
* (-1|B) = 1 IF n = 1 (mod 1)
* (-1|B) = -1 IF n = 3 (mod 4)
* (A|B) (C|B) = (AC|B)
* (A|B) (A|C) = (A|CB)
* (A|B) = (C|B) IF A = C (mod B)
* (2|B) = 1 IF N = 1 OR 7 (mod 8)
* (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 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 (g,u,v), where g = gcd(abs(a),abs(b)) = ua + vb
*/
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 ceil(log_256 n), the number of bytes needed to encode
* the integer n.
*
* @param n the integer
* @return the number of bytes needed to encode n
*/
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 ceil(log_256 n), the number of bytes needed to encode
* the long integer n.
*
* @param n the long integer
* @return the number of bytes needed to encode n
*/
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 h with 2^h | a if a!=0.
*
* @param a an integer
* @return the largest h with 2^h | a if a!=0,
* 0 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
* a
*/
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 k = 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 n reduced into [begin,end]
*/
public static BigInteger reduceInto(BigInteger n, BigInteger begin,
BigInteger end)
{
return n.subtract(begin).mod(end.subtract(begin)).add(begin);
}
/**
* Compute a e.
*
* @param a the base
* @param e the exponent
* @return a e
*/
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 a e.
*
* @param a the base
* @param e the exponent
* @return a e
*/
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 a e mod n.
*
* @param a the base
* @param e the exponent
* @param n the modulus
* @return a e mod n
*/
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 (d,u,v), where d = gcd(a,b) = ua + vb
*/
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 (a mod m). This method
* differs from % in that it always returns a non-negative
* integer.
*
* @param a value on which the modulo operation has to be performed.
* @param m the modulus.
* @return a mod m
*/
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 a -1 mod n
*/
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 a -1 mod n
*/
public static long modInverse(long a, long mod)
{
return BigInteger.valueOf(a).modInverse(BigInteger.valueOf(mod))
.longValue();
}
/**
* Tests whether an integer a is power of another integer
* p.
*
* @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 a.
*
* @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 true if the given integer is
* prime with probability 1 - 2 -20.
*
* @param n the integer to test for primality
* @return true if the given integer is prime with probability
* 2 -100, false 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 true if the number has no factor of the tested primes,
* false 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 n, or 1 if
* n <= 2
*/
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 n 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 n
*/
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 n with the default
* certainty (20).
*
* @param n a integer number
* @return the next prime greater than n
*/
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:
*
* if n !=0 and t != 0 then (n|t) = Mult(i=1, t): (n-(i-1))/i
* if t = 0 then (n|t) = 1
* if n = 0 and t > 0 then (n|t) = 0
*
*
* @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 = CryptoServicesRegistrar.getSecureRandom();
}
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 : CryptoServicesRegistrar.getSecureRandom();
}
for (int i = 0; i < 20; i++)
{
randomNum = BigIntegers.createRandomBigInteger(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 a
*/
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;
}
/**
* int power of a base float, only use for small ints
*
* @param f base float
* @param i power to be raised to.
* @return int power i of f
*/
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,
0.1699250014423123629074778878956330175196288153849621209115053090821964552970,
0.0874628412503394082540660108104043540112672823448206881266090643866965081686,
0.0443941193584534376531019906736094674630459333742491317685543002674288465967,
0.0223678130284545082671320837460849094932677948156179815932199216587899627785,
0.0112272554232541203378805844158839407281095943600297940811823651462712311786,
0.0056245491938781069198591026740666017211096815383520359072957784732489771013,
0.0028150156070540381547362547502839489729507927389771959487826944878598909400,
0.0014081943928083889066101665016890524233311715793462235597709051792834906001,
0.0007042690112466432585379340422201964456668872087249334581924550139514213168,
0.0003521774803010272377989609925281744988670304302127133979341729842842377649,
0.0001760994864425060348637509459678580940163670081839283659942864068257522373,
0.0000880524301221769086378699983597183301490534085738474534831071719854721939,
0.0000440268868273167176441087067175806394819146645511899503059774914593663365,
0.0000220136113603404964890728830697555571275493801909791504158295359319433723,
0.0000110068476674814423006223021573490183469930819844945565597452748333526464,
0.0000055034343306486037230640321058826431606183125807276574241540303833251704,
0.0000027517197895612831123023958331509538486493412831626219340570294203116559,
0.0000013758605508411382010566802834037147561973553922354232704569052932922954,
0.0000006879304394358496786728937442939160483304056131990916985043387874690617,
0.0000003439652607217645360118314743718005315334062644619363447395987584138324,
0.0000001719826406118446361936972479533123619972434705828085978955697643547921,
0.0000000859913228686632156462565208266682841603921494181830811515318381744650,
0.0000000429956620750168703982940244684787907148132725669106053076409624949917,
0.0000000214978311976797556164155504126645192380395989504741781512309853438587,
0.0000000107489156388827085092095702361647949603617203979413516082280717515504,
0.0000000053744578294520620044408178949217773318785601260677517784797554422804,
0.0000000026872289172287079490026152352638891824761667284401180026908031182361,
0.0000000013436144592400232123622589569799954658536700992739887706412976115422,
0.0000000006718072297764289157920422846078078155859484240808550018085324187007,
0.0000000003359036149273187853169587152657145221968468364663464125722491530858,
0.0000000001679518074734354745159899223037458278711244127245990591908996412262,
0.0000000000839759037391617577226571237484864917411614198675604731728132152582,
0.0000000000419879518701918839775296677020135040214077417929807824842667285938,
0.0000000000209939759352486932678195559552767641474249812845414125580747434389,
0.0000000000104969879676625344536740142096218372850561859495065136990936290929,
0.0000000000052484939838408141817781356260462777942148580518406975851213868092,
0.0000000000026242469919227938296243586262369156865545638305682553644113887909,
0.0000000000013121234959619935994960031017850191710121890821178731821983105443,
0.0000000000006560617479811459709189576337295395590603644549624717910616347038,
0.0000000000003280308739906102782522178545328259781415615142931952662153623493,
0.0000000000001640154369953144623242936888032768768777422997704541618141646683,
0.0000000000000820077184976595619616930350508356401599552034612281802599177300,
0.0000000000000410038592488303636807330652208397742314215159774270270147020117,
0.0000000000000205019296244153275153381695384157073687186580546938331088730952,
0.0000000000000102509648122077001764119940017243502120046885379813510430378661,
0.0000000000000051254824061038591928917243090559919209628584150482483994782302,
0.0000000000000025627412030519318726172939815845367496027046030028595094737777,
0.0000000000000012813706015259665053515049475574143952543145124550608158430592,
0.0000000000000006406853007629833949364669629701200556369782295210193569318434,
0.0000000000000003203426503814917330334121037829290364330169106716787999052925,
0.0000000000000001601713251907458754080007074659337446341494733882570243497196,
0.0000000000000000800856625953729399268240176265844257044861248416330071223615,
0.0000000000000000400428312976864705191179247866966320469710511619971334577509,
0.0000000000000000200214156488432353984854413866994246781519154793320684126179,
0.0000000000000000100107078244216177339743404416874899847406043033792202127070,
0.0000000000000000050053539122108088756700751579281894640362199287591340285355,
0.0000000000000000025026769561054044400057638132352058574658089256646014899499,
0.0000000000000000012513384780527022205455634651853807110362316427807660551208,
0.0000000000000000006256692390263511104084521222346348012116229213309001913762,
0.0000000000000000003128346195131755552381436585278035120438976487697544916191,
0.0000000000000000001564173097565877776275512286165232838833090480508502328437,
0.0000000000000000000782086548782938888158954641464170239072244145219054734086,
0.0000000000000000000391043274391469444084776945327473574450334092075712154016,
0.0000000000000000000195521637195734722043713378812583900953755962557525252782,
0.0000000000000000000097760818597867361022187915943503728909029699365320287407,
0.0000000000000000000048880409298933680511176764606054809062553340323879609794,
0.0000000000000000000024440204649466840255609083961603140683286362962192177597,
0.0000000000000000000012220102324733420127809717395445504379645613448652614939,
0.0000000000000000000006110051162366710063906152551383735699323415812152114058,
0.0000000000000000000003055025581183355031953399739107113727036860315024588989,
0.0000000000000000000001527512790591677515976780735407368332862218276873443537,
0.0000000000000000000000763756395295838757988410584167137033767056170417508383,
0.0000000000000000000000381878197647919378994210346199431733717514843471513618,
0.0000000000000000000000190939098823959689497106436628681671067254111334889005,
0.0000000000000000000000095469549411979844748553534196582286585751228071408728,
0.0000000000000000000000047734774705989922374276846068851506055906657137209047,
0.0000000000000000000000023867387352994961187138442777065843718711089344045782,
0.0000000000000000000000011933693676497480593569226324192944532044984865894525,
0.0000000000000000000000005966846838248740296784614396011477934194852481410926,
0.0000000000000000000000002983423419124370148392307506484490384140516252814304,
0.0000000000000000000000001491711709562185074196153830361933046331030629430117,
0.0000000000000000000000000745855854781092537098076934460888486730708440475045,
0.0000000000000000000000000372927927390546268549038472050424734256652501673274,
0.0000000000000000000000000186463963695273134274519237230207489851150821191330,
0.0000000000000000000000000093231981847636567137259618916352525606281553180093,
0.0000000000000000000000000046615990923818283568629809533488457973317312233323,
0.0000000000000000000000000023307995461909141784314904785572277779202790023236,
0.0000000000000000000000000011653997730954570892157452397493151087737428485431,
0.0000000000000000000000000005826998865477285446078726199923328593402722606924,
0.0000000000000000000000000002913499432738642723039363100255852559084863397344,
0.0000000000000000000000000001456749716369321361519681550201473345138307215067,
0.0000000000000000000000000000728374858184660680759840775119123438968122488047,
0.0000000000000000000000000000364187429092330340379920387564158411083803465567,
0.0000000000000000000000000000182093714546165170189960193783228378441837282509,
0.0000000000000000000000000000091046857273082585094980096891901482445902524441,
0.0000000000000000000000000000045523428636541292547490048446022564529197237262,
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])
{
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);
}
}