Higher Moments Subset Sum Problem over Finite Fields

Let F_q be a finite field and let D ⊆ F_q. Let m be a positive integer and let k be an integer such that 1 ≤ k ≤ |D|. For b = (b_1,...,b_m) ∈ (F_q)^m , let N_m(k,b) denote the number of subsets S ⊆ D with cardinality k such that for i = 1,...,m, the sum, over a ∈ S, of a^i = b_i. The Moments Subset Sum Problem is to determine if N_m(k,b) > 0. There are many results for when m = 1, but not much is known about the higher moments. In this dissertation, we obtain a formula for N_m(k, b) when m = 2 and conditions on the solvability of the Moments Subset Sum Problem by using the Li-Wan sieve and properties of character sums and Gauss sums