The k-subset sum problem over finite fields of characteristic 2

Moment subset sums over finite fields

“…The same argument as in the proof of Theorem 2.3 in [3] shows that if 3.05sm p = 3.05m p log 2 q < k ≤ |D|/2,…”

“…We will prove an analogue of Theorem 2.3 in [3]. Let D ⊆ F q , k ≤ |D|/2, By induction , one derives the following recursive formula for S D (k, ψ, f ) for all k > 1, which is the analogue of Lemma 2.1 [3].…”

“…In a sense, our set D is given as a black-box. As a supporting evidence, this conjecture has been proved to be true in the special case when the evaluation set D is the image of the monomial x n or Dickson polynomials of degree n: see [24] for the case p > 2 and [3] for the case p = 2. The aim of the present paper is to extend these results from m = 1 (k-SSP) to the higher m-moment k-SSP for each fixed m. Namely, our main result is Theorem 1.…”

“…To analyze the case p = 2 for large k, we rely on recent work by Choe and Choe [3] which examines the subset sum problem over finite fields of characteristic 2 . We adjust the definitions of this work to fit the higher moment subset sum problem over D which are images of monomials or Dickson polynomials.…”

“…To prove the above theorem, we will need to combine all the techniques available so far: dynamic programming for large n > q ǫ , Kayal's algorithm [13] for constant k, Brun sieve for medium k, the Li-Wan sieve for large k and p > 2, and the recent Choe-Choe argument [3] for large k and p = 2. In addition, we need to employ the Weil bound to prove a crucial new partial character sum estimate.…”