Prime Numbers

180 Chapter 4 PRIMALITY PROVING
Let M(p) be the number of modular multiplications (with integers not
exceeding p) needed to prove p prime using Theorem 4.1.8 to traverse the
Lucastreeforp, and using binary addition chains for the exponentiations
(see Algorithm 2.1.5).
For example, consider p = 1279:
3 1278/2 ≡−1 (mod 1279), 3 1278/6 ≡ 775 (mod 1279),
2 2/2 ≡−1(mod3),
3 1278/142 ≡ 498 (mod 1279),
7 70/2 ≡−1 (mod 71), 7 70/10 ≡ 14 (mod 71),
2 4/2 ≡−1(mod5),
7 70/14 ≡ 51 (mod 71),
3 6/2 ≡−1(mod7), 3 6/6 ≡ 3(mod7),
2 2/2 ≡−1(mod3).
If we use the binary addition chain for each exponentiation, we have the
following number of modular multiplications:
1278/2 : 16
1278/6 : 11
1278/142 : 4
2/2 : 0
70/2 : 7
70/10 : 4
70/14 : 3
4/2 : 1
6/2 : 2
6/6 : 0
2/2 : 0.
Thus, using binary addition chains we have 48 modular multiplications, so
M(1279) = 48.
The following result is essentially due to [Pratt 1975]:
Theorem 4.1.9. For every odd prime p, M(p) < 2lg 2 p.
Proof. Let N(p) be the number of (not necessarily distinct) odd primes in
the Lucas tree for p. We first show that N(p) < lg p. Thisistrueforp =3.
Suppose it is true for every odd prime less than p. Ifp − 1isapowerof2,
then N(p) =1< lg p. Ifp − 1 has the odd prime factors q1,...,qk, then, by
the induction hypothesis,
k
k
p − 1
N(p) =1+ N(qi) < 1+ lg qi =1+lg(q1 ···qk) ≤ 1+lg < lg p.
2
i=1
i=1