Construction of systems of polynomial equations with exact p-Divisibility via the covering method

Abstract

We present an elementary method to compute the exact p-divisibility of exponential sums of systems of polynomial equations over the prime field. Our results extend results by Carlitz and provide concrete and simple conditions to construct families of polynomial equations that are solvable over the prime field.

The Covering Method for Exponential Sums and Some Applications

“…The covering method to study the -divisibility of exponential sums is an elementary method introduced in [6] that lets us determine sufficient conditions to guarantee solvability and allows us to construct general families of solvable systems of polynomial equations [4,5]. Applications to cryptography and coding theory.…”

“…for 1 , … , ≥ 1, and such that 1 + ⋯ + is as small as possible [4]. If = 2, then = { 1 1 , 2 2 , … , } is a covering for , as some of the could be zero.…”

A divisibility approach to the open boundary cases of Cusick-Li-Stǎnicǎ’s conjecture

“…In this case, n = 2 a (20 + 32j 2 ) and k = 2 a (8 + 16j 2 ). It appears that as long as a > 1, then as j 2 runs through the non-negative integers, the sequence ν 2 (S(σ n,k )) is given by 5,6,6,7,6,7,7,8,6,7,7,8,7,8,8,9,6,7,7,8, · · · This appears to be the sequence w 2 (4j 2 + 3) + 3. The class 4j 2 + 1 does not have this behavior.…”

“…As before, consider the class 16j 4 +9. If a > 2, then the sequence ν 2 (S(σ n,k )) is given by 6,7,7,8,7,8,8,9,7,8,8,9,8,9,9,10,7,8,8,9, · · · This sequence seems to be w 2 (16j 4 + 9) + 4. Observe that now we need a > 2.…”