According to smb@research.att.com:

If I understand this correctly, if p is not a prime, then n may not be unique.

Well, n isn't unique even if p is prime. Consider a=10,p=11. 10^2=10^4=10^6=10^8=10^10=1 mod 11. You only get a maximum-length cycle if ``a'' is a primitive root, hence the restriction I stated in the part I deleted...

That is, if a is a generator of G, or as close to one as possible. My thinking was obviously clowded... Not that I have a beer in me, I remember that for any element, a of group G, a will have order n, such that n|ord(G). This implies that there are n different (positive) powers of a which yield a particular number, b in our case. Each of which would qualify as a log. I think I understand.

It doesn't matter that n isn't unique, though you do want a good distribution. Primitive roots have a maximal distribution, which is

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