/// <summary>
/// It uses the sieve of Eratosthenes to discard several composite numbers in
/// some appropriate range (at the moment [this, this + 1024]).
/// <para>After this process it applies the Miller-Rabin test to the numbers that were not discarded in the sieve.</para>
/// </summary>
internal static BigInteger NextProbablePrime(BigInteger X)
{
// PRE: n >= 0
int i, j;
int certainty;
int gapSize = 1024; // for searching of the next probable prime number
int[] modules = new int[m_primes.Length];
bool[] isDivisible = new bool[gapSize];
BigInteger startPoint;
BigInteger probPrime;
// If n < "last prime of table" searches next prime in the table
if ((X.m_numberLength == 1) && (X.m_digits[0] >= 0) && (X.m_digits[0] < m_primes[m_primes.Length - 1]))
{
for (i = 0; X.m_digits[0] >= m_primes[i]; i++)
{
;
}
return(m_biPrimes[i]);
}
// Creates a "N" enough big to hold the next probable prime Note that: N < "next prime" < 2*N
startPoint = new BigInteger(1, X.m_numberLength, new int[X.m_numberLength + 1]);
Array.Copy(X.m_digits, 0, startPoint.m_digits, 0, X.m_numberLength);
// To fix N to the "next odd number"
if (X.TestBit(0))
{
Elementary.InplaceAdd(startPoint, 2);
}
else
{
startPoint.m_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 < m_primes.Length; i++)
{
modules[i] = Division.Remainder(startPoint, m_primes[i]) - gapSize;
}
while (true)
{
// At this point, all numbers in the gap are initialized as probably primes
for (int k = 0; k < isDivisible.Length; k++)
{
isDivisible[k] = false;
}
// To discard multiples of first primes
for (i = 0; i < m_primes.Length; i++)
{
modules[i] = (modules[i] + gapSize) % m_primes[i];
j = (modules[i] == 0) ? 0 : (m_primes[i] - modules[i]);
for (; j < gapSize; j += m_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 BigInteger ModInverseLorencz(BigInteger X, BigInteger Modulo)
{
// Based on "New Algorithm for Classical Modular Inverse" Róbert Lórencz. LNCS 2523 (2002)
// PRE: a is coprime with modulo, a < modulo
int max = System.Math.Max(X.m_numberLength, Modulo.m_numberLength);
int[] uDigits = new int[max + 1]; // enough place to make all the inplace operation
int[] vDigits = new int[max + 1];
Array.Copy(Modulo.m_digits, 0, uDigits, 0, Modulo.m_numberLength);
Array.Copy(X.m_digits, 0, vDigits, 0, X.m_numberLength);
BigInteger u = new BigInteger(Modulo.m_sign, Modulo.m_numberLength, uDigits);
BigInteger v = new BigInteger(X.m_sign, X.m_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.m_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.Signum() == v.Signum())
{
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.Signum() == 0 || u.Signum() == 0)
{
throw new ArithmeticException("BigInteger not invertible");
}
}
if (IsPowerOfTwo(v, coefV))
{
r = s;
if (v.Signum() != u.Signum())
{
u = u.Negate();
}
}
if (u.TestBit(n))
{
if (r.Signum() < 0)
{
r = r.Negate();
}
else
{
r = Modulo.Subtract(r);
}
}
if (r.Signum() < 0)
{
r = r.Add(Modulo);
}
return(r);
}
/// <summary>
/// See BigInteger#subtract(BigInteger)
/// </summary>
internal static BigInteger Subtract(BigInteger A, BigInteger B)
{
int resSign;
int[] resDigits;
int op1Sign = A._sign;
int op2Sign = B._sign;
if (op2Sign == 0)
{
return(A);
}
if (op1Sign == 0)
{
return(B.Negate());
}
int op1Len = A._numberLength;
int op2Len = B._numberLength;
if (op1Len + op2Len == 2)
{
long a = (A._digits[0] & 0xFFFFFFFFL);
long b = (B._digits[0] & 0xFFFFFFFFL);
if (op1Sign < 0)
{
a = -a;
}
if (op2Sign < 0)
{
b = -b;
}
return(BigInteger.ValueOf(a - b));
}
int cmp = ((op1Len != op2Len) ?
((op1Len > op2Len) ? 1 : -1) :
Elementary.CompareArrays(A._digits, B._digits, op1Len));
if (cmp == BigInteger.LESS)
{
resSign = -op2Sign;
resDigits = (op1Sign == op2Sign) ?
Subtract(B._digits, op2Len, A._digits, op1Len) :
Add(B._digits, op2Len, A._digits, op1Len);
}
else
{
resSign = op1Sign;
if (op1Sign == op2Sign)
{
if (cmp == BigInteger.EQUALS)
{
return(BigInteger.Zero);
}
resDigits = Subtract(A._digits, op1Len, B._digits, op2Len);
}
else
{
resDigits = Add(A._digits, op1Len, B._digits, op2Len);
}
}
BigInteger res = new BigInteger(resSign, resDigits.Length, resDigits);
res.CutOffLeadingZeroes();
return(res);
}
/// <summary>
/// Calculates x.modInverse(p) Based on: Savas, E; Koc, C "The Montgomery Modular Inverse - Revised"
/// </summary>
///
/// <param name="X">BigInteger X</param>
/// <param name="P">BigInteger P</param>
///
/// <returns>Returns <c>1/X Mod M</c></returns>
internal static BigInteger ModInverseMontgomery(BigInteger X, BigInteger P)
{
// ZERO hasn't inverse
if (X.m_sign == 0)
{
throw new ArithmeticException("BigInteger not invertible!");
}
// montgomery inverse require even modulo
if (!P.TestBit(0))
{
return(ModInverseLorencz(X, P));
}
int m = P.m_numberLength * 32;
// PRE: a \in [1, p - 1]
BigInteger u, v, r, s;
u = P.Copy(); // make copy to use inplace method
v = X.Copy();
int max = System.Math.Max(v.m_numberLength, u.m_numberLength);
r = new BigInteger(1, 1, new int[max + 1]);
s = new BigInteger(1, 1, new int[max + 1]);
s.m_digits[0] = 1;
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.m_sign = 1;
while (v.Signum() > 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.Signum() == 0)
{
break;
}
toShift = v.LowestSetBit;
BitLevel.InplaceShiftRight(v, toShift);
Elementary.InplaceAdd(s, r);
BitLevel.InplaceShiftLeft(r, toShift);
k += toShift;
}
}
// in u is stored the gcd
if (!u.IsOne())
{
throw new ArithmeticException("BigInteger not invertible.");
}
if (r.CompareTo(P) >= BigInteger.EQUALS)
{
Elementary.InplaceSubtract(r, P);
}
r = P.Subtract(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);
}