We show that, if p ? 3 is an odd prime satisfying p ? 5 (mod 8), 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 N + (frac(2, p)), where (frac(2, p)) 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. © 2006 Elsevier Inc. All rights reserved.
- Multiplicative subgroups of finite fields
- Quadratic residues