/* In the _vast_ majority of cases this simply checks that your chosen random
* number is >= KLastSmallPrimeSquared and return EFalse and lets the normal
* prime generation routines handle the situation. In the case where it is
* smaller, it generates a provable prime and returns ETrue. The algorithm for
* finding a provable prime < KLastPrimeSquared is not the most efficient in the
* world, but two points come to mind
* 1) The two if statements hardly _ever_ evaluate to ETrue in real life.
* 2) Even when it is, the distribution of primes < KLastPrimeSquared is pretty
* dense, so you aren't going to have check many.
* This function is essentially here for two reasons:
* 1) Ensures that it is possible to generate primes < KLastPrimeSquared (the
* test code does this)
* 2) Ensures that if you request a prime of a large bit size that there is an
* even probability distribution across all integers < 2^aBits
*/
TBool TInteger::SmallPrimeRandomizeL(void)
{
TBool foundPrime = EFalse;
//If the random number we've chosen is less than KLastSmallPrime,
//testing for primality is easy.
if(*this <= KLastSmallPrime)
{
//If Zero or One, or two, next prime number is two
if(IsZero() || *this == One() || *this == Two())
{
CopyL(TInteger::Two());
foundPrime = ETrue;
}
else
{
//Make sure any number we bother testing is at least odd
SetBit(0);
//Binary search the small primes.
while(!IsSmallPrime(ConvertToUnsignedLong()))
{
//If not prime, add two and try the next odd number.
//will never carry as the minimum size of an RInteger is 2
//words. Much bigger than KLastSmallPrime on 32bit
//architectures.
IncrementNoCarry(Ptr(), Size(), 2);
}
assert(IsSmallPrime(ConvertToUnsignedLong()));
foundPrime = ETrue;
}
}
else if(*this <= KLastSmallPrimeSquared)
{
//Make sure any number we bother testing is at least odd
SetBit(0);
while(HasSmallDivisorL(*this) && *this <= KLastSmallPrimeSquared)
{
//If not prime, add two and try the next odd number.
//will never carry as the minimum size of an RInteger is 2
//words. Much bigger than KLastSmallPrime on 32bit
//architectures.
IncrementNoCarry(Ptr(), Size(), 2);
}
//If we exited while loop because it had no small divisor then it is
//prime. Otherwise, we've exceeded the limit of what we can provably
//generate. Therefore the normal prime gen routines will be run on it
//now.
if(*this < KLastSmallPrimeSquared)
{
foundPrime = ETrue;
}
}
//This doesn't mean there is no such prime, simply means that the number
//wasn't less than KSmallPrimeSquared and needs to be handled by the normal
//prime generation routines.
return foundPrime;
}
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;
}
}
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);
}
TBool TInteger::IsPrimeL(void) const
{
if( NotPositive() )
{
return EFalse;
}
else if( IsEven() )
{
return *this == Two();
}
else if( *this <= KLastSmallPrime )
{
assert(KLastSmallPrime < KMaxTUint);
return IsSmallPrime(this->ConvertToUnsignedLong());
}
else if( *this <= KLastSmallPrimeSquared )
{
return !HasSmallDivisorL(*this);
}
else
{
return !HasSmallDivisorL(*this) && IsStrongProbablePrimeL(*this);
}
}
