/**
* 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 (BigInteger.TestBit(n, 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);
}
}
private static bool TryParse(string s, int radix, out BigInteger value, out Exception exception)
{
if (String.IsNullOrEmpty(s))
{
exception = new FormatException(Messages.math11);
value = null;
return(false);
}
if ((radix < CharHelper.MIN_RADIX) || (radix > CharHelper.MAX_RADIX))
{
// math.11=Radix out of range
exception = new FormatException(Messages.math12);
value = null;
return(false);
}
int sign;
int[] digits;
int numberLength;
int stringLength = s.Length;
int startChar;
int endChar = stringLength;
if (s[0] == '-')
{
sign = -1;
startChar = 1;
stringLength--;
}
else
{
sign = 1;
startChar = 0;
}
/*
* We use the following algorithm: split a string into portions of n
* char and convert each portion to an integer according to the
* radix. Then convert an exp(radix, n) based number to binary using the
* multiplication method. See D. Knuth, The Art of Computer Programming,
* vol. 2.
*/
try
{
int charsPerInt = Conversion.digitFitInInt[radix];
int bigRadixDigitsLength = stringLength / charsPerInt;
int topChars = stringLength % charsPerInt;
if (topChars != 0)
{
bigRadixDigitsLength++;
}
digits = new int[bigRadixDigitsLength];
// Get the maximal power of radix that fits in int
int bigRadix = Conversion.bigRadices[radix - 2];
// Parse an input string and accumulate the BigInteger's magnitude
int digitIndex = 0; // index of digits array
int substrEnd = startChar + ((topChars == 0) ? charsPerInt : topChars);
int newDigit;
for (int substrStart = startChar;
substrStart < endChar;
substrStart = substrEnd, substrEnd = substrStart
+ charsPerInt)
{
int bigRadixDigit = Convert.ToInt32(s.Substring(substrStart, substrEnd - substrStart), radix);
newDigit = Multiplication.MultiplyByInt(digits, digitIndex, bigRadix);
newDigit += Elementary.inplaceAdd(digits, digitIndex, bigRadixDigit);
digits[digitIndex++] = newDigit;
}
numberLength = digitIndex;
}
catch (Exception ex)
{
exception = ex;
value = null;
return(false);
}
value = new BigInteger();
value.sign = sign;
value.numberLength = numberLength;
value.digits = digits;
value.CutOffLeadingZeroes();
exception = null;
return(true);
}
/**
* 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);
}