Exact 2-divisibility of exponential sums associated to boolean functions

The Covering Method for Exponential Sums and Some Applications

“…Her result is tight for the Boolean case and can be improved only by imposing additional conditions. The additional conditions to the covering method in Theorem 3 allowed us to obtain an improvement of her results [7].…”

“…Applications to cryptography and coding theory. The divisibility of exponential sums has been used to characterize and prove properties in coding theory and cryptography [3,7,18]. The computation of bounds or the exact 2divisibility of exponential sums of Boolean functions provides information on the Hamming weight of the function and can be used to obtain information on the covering radius and the weight distribution of certain codes.…”

“…It is not difficult to find sufficient conditions that can be used to determine a priori cosets of Reed-Muller codes that do not contain any balanced function, saving computational time. For example, we can use the covering method to identify by inspection 15 coset representatives (out of 32) in (3, 8)/ (2, 8) for which at least half of the functions in each coset are not balanced and provide constructions for these nonbalanced functions [7,10]. This can be used to determine a priori types of polynomials to avoid in the search for balanced functions.…”