/**
* @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.ValueOf(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 = op2.Remainder(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.ShiftLeft(pow2Count));
}
/**
* 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);
}
/**
* It uses the sieve of Eratosthenes to discard several composite numbers in
* some appropriate range (at the moment {@code [this, this + 1024]}). After
* this process it applies the Miller-Rabin test to the numbers that were
* not discarded in the sieve.
*
* @see BigInteger#nextProbablePrime()
* @see #millerRabin(BigInteger, int)
*/
public static BigInteger NextProbablePrime(BigInteger n)
{
// PRE: n >= 0
int i, j;
int certainty;
int gapSize = 1024; // for searching of the next probable prime number
int[] modules = new int[primes.Length];
bool[] isDivisible = new bool[gapSize];
BigInteger startPoint;
BigInteger probPrime;
// If n < "last prime of table" searches next prime in the table
if ((n.numberLength == 1) && (n.Digits[0] >= 0) &&
(n.Digits[0] < primes[primes.Length - 1]))
{
for (i = 0; n.Digits[0] >= primes[i]; i++)
{
;
}
return(BIprimes[i]);
}
/*
* Creates a "N" enough big to hold the next probable prime Note that: N <
* "next prime" < 2*N
*/
startPoint = new BigInteger(1, n.numberLength,
new int[n.numberLength + 1]);
Array.Copy(n.Digits, 0, startPoint.Digits, 0, n.numberLength);
// To fix N to the "next odd number"
if (n.TestBit(0))
{
Elementary.inplaceAdd(startPoint, 2);
}
else
{
startPoint.Digits[0] |= 1;
}
// To set the improved certainly of Miller-Rabin
j = startPoint.BitLength;
for (certainty = 2; j < BITS[certainty]; certainty++)
{
;
}
// To calculate modules: N mod p1, N mod p2, ... for first primes.
for (i = 0; i < primes.Length; i++)
{
modules[i] = Division.Remainder(startPoint, primes[i]) - gapSize;
}
while (true)
{
// At this point, all numbers in the gap are initialized as
// probably primes
// Arrays.fill(isDivisible, false);
for (int k = 0; k < isDivisible.Length; k++)
{
isDivisible[k] = false;
}
// To discard multiples of first primes
for (i = 0; i < primes.Length; i++)
{
modules[i] = (modules[i] + gapSize) % primes[i];
j = (modules[i] == 0) ? 0 : (primes[i] - modules[i]);
for (; j < gapSize; j += primes[i])
{
isDivisible[j] = true;
}
}
// To execute Miller-Rabin for non-divisible numbers by all first
// primes
for (j = 0; j < gapSize; j++)
{
if (!isDivisible[j])
{
probPrime = startPoint.Copy();
Elementary.inplaceAdd(probPrime, j);
if (MillerRabin(probPrime, certainty))
{
return(probPrime);
}
}
}
Elementary.inplaceAdd(startPoint, gapSize);
}
}
