bool IsPrime(const Integer &p)
{
if (p <= s_lastSmallPrime)
return IsSmallPrime(p);
else if (p <= Singleton<Integer, NewLastSmallPrimeSquared>().Ref())
return SmallDivisorsTest(p);
else
return SmallDivisorsTest(p) && IsStrongProbablePrime(p, 3) && IsStrongLucasProbablePrime(p);
}
bool IsPrime(const Integer &p)
{
static const Integer lastSmallPrimeSquared = Integer(lastSmallPrime).Squared();
if (p <= lastSmallPrime)
return IsSmallPrime(p);
else if (p <= lastSmallPrimeSquared)
return SmallDivisorsTest(p);
else
return SmallDivisorsTest(p) && IsStrongProbablePrime(p, 3) && IsStrongLucasProbablePrime(p);
}
void PrimeAndGenerator::Generate(signed int delta, RandomNumberGenerator &rng, unsigned int pbits, unsigned int qbits)
{
// no prime exists for delta = -1, qbits = 4, and pbits = 5
assert(qbits > 4);
assert(pbits > qbits);
if (qbits+1 == pbits)
{
Integer minP = Integer::Power2(pbits-1);
Integer maxP = Integer::Power2(pbits) - 1;
bool success = false;
while (!success)
{
p.Randomize(rng, minP, maxP, Integer::ANY, 6+5*delta, 12);
PrimeSieve sieve(p, STDMIN(p+PrimeSearchInterval(maxP)*12, maxP), 12, delta);
while (sieve.NextCandidate(p))
{
assert(IsSmallPrime(p) || SmallDivisorsTest(p));
q = (p-delta) >> 1;
assert(IsSmallPrime(q) || SmallDivisorsTest(q));
if (FastProbablePrimeTest(q) && FastProbablePrimeTest(p) && IsPrime(q) && IsPrime(p))
{
success = true;
break;
}
}
}
if (delta == 1)
{
// find g such that g is a quadratic residue mod p, then g has order q
// g=4 always works, but this way we get the smallest quadratic residue (other than 1)
for (g=2; Jacobi(g, p) != 1; ++g) {}
// contributed by Walt Tuvell: g should be the following according to the Law of Quadratic Reciprocity
assert((p%8==1 || p%8==7) ? g==2 : (p%12==1 || p%12==11) ? g==3 : g==4);
}
else
{
assert(delta == -1);
// find g such that g*g-4 is a quadratic non-residue,
// and such that g has order q
for (g=3; ; ++g)
if (Jacobi(g*g-4, p)==-1 && Lucas(q, g, p)==2)
break;
}
}
