Prime Numbers

194 Chapter 4 PRIMALITY PROVING
The reason for assuming x > 16 is to ensure that the triple-logarithm is
positive. It is not necessary in the results so far that I be squarefree, but
because of an algorithm in the next section, this condition is included in the
above result.
Corollary 4.3.6. There is a positive number c ′ such that the expected
running time for Algorithm 4.3.4 to declare a prime input n to be prime is
less than (ln n) c′ ln ln ln n .
Since the triple log function grows so slowly, this running-time bound is
“almost” ln O(1) n, and so is “almost” polynomial time.
4.4 Gauss and Jacobi sums
In 1983, Adleman, Pomerance, and Rumely [Adleman et al. 1983] published a
primality test with the running-time bound of (ln n) c ln ln ln n for prime inputs
n and some positive constant c. The proof rested on Theorem 4.3.5 and on
arithmetic properties of Jacobi sums. Two versions of the test were presented,
a somewhat simpler and more practical version that was probabilistic, and a
deterministic test. Both versions had the same complexity estimate. As with
some of the other algorithms in this chapter, a declaration of primality in the
probabilistic APR test definitely implies that the number is prime. The only
thing in doubt is a prediction of the running time.
Shortly afterwards, there were two types of developments. In one direction,
more practical versions of the test were found, and in the other, less practical,
but simpler versions of the test were found. In the next section we shall discuss
one of the second variety, the deterministic Gauss sums test of H. Lenstra
[Lenstra 1981].
4.4.1 Gauss sums test
In Section 2.3.1 we introduced Gauss sums for quadratic characters. Here we
consider Gauss sums for arbitrary Dirichlet characters. If q is a prime with
primitive root g and if ζ is a complex number with ζ q−1 =1,thenwecan
“construct” a character χ to the modulus q via χ(g k )=ζ k for every integer
k (and of course, χ(m) =0ifm is a multiple of q). (See Section 1.4.3 for a
discussion of characters.) We may also “construct” the Gauss sum τ(χ). With
the notation ζn = e 2πi/n (which is a primitive n-th root of 1), we define
τ(χ) =
q−1
χ(m)ζ
m=1
m q−1
q = χ(g
k=1
k )ζ gk
q−1
q = ζ
k=1
k ζ gk
q .
As a character mod q, the order of χ is a divisor of q − 1. Suppose p is
a prime factor of q − 1 and we wish the order of χ to be exactly p. Wemay
concretely construct such a character χp,q as follows. Suppose g = gq is the
least positive primitive root for q, andletχp,q(g k q )=ζ k p for every integer k.
As in the above paragraph, we have thus defined a character mod q since