Prime Numbers

4.1 The n − 1 test 179
4.1.3 Succinct certificates
The goal in primality testing is to quickly find a short proof of primality for
prime inputs p. But how do we know that a short proof exists? Any search
will necessarily be in vain if p does not have a short primality proof. We now
show that every prime p has a short proof of primality, or what V. Pratt has
called a “succinct certificate.”
In fact, there is always a short proof that is based on the Lucas Theorem
4.1.1. This might appear obvious, for once you have somehow found the
complete prime factorization of p − 1 and the primitive root a, the conditions
(4.1) may be quickly verified.
However, for the proof to be complete, one needs a demonstration that we
indeed have the complete factorization of p − 1; that is, that the numbers q
appearing in (4.1) really are prime. This suggests an iteration of the method,
but then arises the possibility that there may be a proliferation of cases. The
heart of the proof is to show in the worst case, not too much proliferation can
occur.
It is convenient to make a small, and quite practical, modification in the
Lucas Theorem 4.1.1. The idea is to treat the prime q = 2 differently from
the other primes q dividing p − 1. In fact, we know what a (p−1)/2 should be
congruent to (mod p) ifitisnot1,namely−1. And if a (p−1)/2 ≡−1(modp),
we do not need to check that a p−1 ≡ 1(modp). Further, if q is an odd prime
factor of p − 1, let m = a (p−1)/2q .Ifm q ≡−1(modp) andm 2 ≡ 1(modp),
then m ≡−1(modp) (regardless of whether p is prime or composite). Thus,
to show that a (p−1)/q ≡ 1(modp) it suffices to show a (p−1)/2q ≡−1(modp).
Thus we have the following result.
Theorem 4.1.8. Suppose p>1 is an odd integer and
a (p−1)/2 ≡−1(modp),
a (p−1)/2q ≡−1(modp) for every odd prime q|p − 1.
(4.11)
Then p is prime. Conversely, if p is an odd prime, then every primitive root
a of p satisfies conditions (4.11).
We now describe what might be called a “Lucas tree.” It is a rooted tree
with odd primes at the vertices, p at the root (level 0), and for each positive
level k, aprimer at level k is connected to a prime q at level k − 1ifandonly
if r|q − 1.Forexample,hereistheLucastreeforp = 1279:
1279
✓❙
✓✓ ❙❙71
3
✓❙
✓✓ ❙❙7
5
✓
level 0
level 1
level 2
✓✓ 3 level 3