Prime Numbers

3.4 Pseudoprimes 131
}
g =gcd(s, x);
return “the largest divisor of x composed of primes from P is g”;
The Bernstein Algorithm 3.3.1 is an important addition to the repertoire
of computational number theory. It can profitably be used to speed up various
other algorithms where smoothness is desired. One example arises in the
step [Factor orders] of the Atkin–Morain primality test (Algorithm 7.6.3).
Algorithm 3.3.1 can even be useful in situations in which sieving is completely
appropriate, such as in the quadratic sieve and number field sieve factoring
algorithms (see Chapter 6). Indeed, in these algorithms, the yield rate of
smooth numbers can be so small, it is advantageous to sieve only partially
(forget about small primes in the factor base, which involve the most memory
retrievals), tune the sieve to report candidates with a large smooth divisor,
and then run Algorithm 3.3.1 on the much smaller, but still large, reported
set. This idea of removing small primes from a sieve can be found already in
[Pomerance 1985], but with Algorithm 3.3.1 it can be used more aggressively.
3.4 Pseudoprimes
Suppose we have a theorem, “If n is prime, then S is true about n,” where “S ”
is some easily checkable arithmetic statement. If we are presented with a large
number n, and we wish to decide whether n is prime or composite, we may
very well try out the arithmetic statement S and see whether it actually holds
for n. If the statement fails, we have proved the theorem that n is composite.
If the statement holds, however, it may be that n is prime, and it also may
be that n is composite. So we have the notion of S-pseudoprime, which is a
composite integer for which S holds.
One example might be the theorem, If n is prime, then n is 2 or n is
odd. Certainly this arithmetic property is easily checked for any given input
n. However, as one can readily see, this test is not very strong evidence of
primality, since there are many more pseudoprimes around for this test than
there are genuine primes. Thus, for the concept of “pseudoprime” to be useful,
it will have to be the case that there are, in some appropriate sense, few of
them.
3.4.1 Fermat pseudoprimes
The fact that the residue a b (mod n) may be rapidly computed (see Algorithm
2.1.5) is fundamental to many algorithms in number theory. Not least of these
is the exploitation of Fermat’s little theorem as a means to distinguish between
primes and composites.
Theorem 3.4.1 (Fermat’s little theorem). If n is prime, then for any
integer a, we have
a n ≡ a (mod n). (3.2)