Enumeration of power sums modulo a prime

We consider, for odd primes p, the function N(p, m, α) which equals the number of subsets S⊆{1,...,p - 1} with the property that Σ_∞∈Sx^m ≡ α (mod p). We obtain a closed form expression for N(p, m, α). We give simple explicit formulas for N(p, 2, α) (which in some cases involve class numbers and fundamental units), and show that for a fixed m, the difference between N(p, m, α) and its average value p^-12^p-1 is of the order of exp(p ^{1 2}) or less. Finally, we obtain the curious result that if p - 1 does not divide m, then N(p, m, 0) > N(p, m, α) for all α ≢ 0 (mod p).