A Generalization Of The Goresky-Klapper Conjecture, Part Ii

Suppose that f (x) = Ax(k) mod p is a permutation of the least residues mod p. With the exception of the maps f(x) = Ax and Ax((p+1)/2) mod p we show that for fixed n >= 2 the image of each residue class mod n contains elements from every residue class mod n, once p is sufficiently large. If f(x) = Ax mod p, then for each p and n there will be exactly (1 + o(1))6/pi(2) n(2) readily describable values of A for which the image of some residue class mod n misses at least one residue class mod n, even when p is large relative to n. A similar situation holds for f(x) = Ax((p+1)/2) mod p.