Polynomial Criterion for Abelian Difference Sets

Abstract

Difference sets are subsets of a group satisfying certain combinatorial property with respect to the group operation. They can be characterized using an equality in the group ring of the corresponding group. In this paper, we exploit the special structure of the group ring of an abelian group to establish a one-to one correspondence of the class of difference sets with specific parameters in that group with the set of all complex solutions of a specified system of polynomial equations. The correspondence also develops some tests for a Boolean function to be a bent function.

Algebraic methods in difference sets and bent functions

“…Further let ) is a bent function, then D (π,L) is a difference set. Thus by Ryser's condition (see Section 3 of [5]), it follows that parameters (v, k, λ) of D (π,L) are same as those of D, hence k − λ = 2 t−2 = 2 2(m−1) . Consequently, by (4) of Section 3, for any (ξ, η)…”

“…In Theorem 3.2 of [5], we have given a polynomial criterion for (v, k, λ) difference set in G as follows :…”

“…In [5], a criterion was developed for subsets of a finite abelian group to be difference sets with specified parameters. The criterion was in terms of polynomial conditions on their characteristic vectors.…”

“…In Section 3, the criterion of [5] is reformulated to characterize those exchanges of elements of a difference set which again lead to a difference set. The formulation is in terms of certain values of a polynomial function.…”

“…Thus we need another machinary to prove non-bentness of f (π,L) . As supports of bent functions are difference sets, the results of [5] become relevant.…”