Sum uniform subsets of the integers modulo p and an application to finite fields

We show that, if p≠3 is an odd prime satisfying , then each nonzero element of GF(p) can be written as a sum of distinct quadratic residues in the same number of ways, N say, and that the number of ways of writing 0 as a sum of distinct quadratic residues is , where is the Legendre symbol. We actually prove a more general result on sum uniform subgroups of GF∗(p), which holds for any odd prime p≠3. These results are applied to the problem of determining subgroups H of the multiplicative group of a finite field, with the property that 1+h is a non-square of the field, for all h∈H.