The sum-capture problem for abelian groups

Let G be a finite abelian group, let 0 < α < 1, and let A ⊆ G be a random set of size |G|. We let μ(A) = max B,C:|B|=|C|=|A| |{(a, b, c) ∈ A×B × C : a = b+ c}|. The issue is to determine upper bounds on μ(A) that hold with high probability over the random choice of A. Mennink and Preneel [4] conjecture that μ(A) should be close to |A| (up to possible logarithmic factors in |G|) for α ≤ 1/2 and that μ(A) should not much exceed |A| for α ≤ 2/3. We prove the second half of this conjecture by showing that μ(A) ≤ |A|/|G|+ 4|A| ln(|G|) with high probability, for all 0 < α < 1. We note that 3α− 1 ≤ (3/2)α for α ≤ 2/3. In previous work, Alon et al. have shown that μ(A) ≤ O(1)|A|/|G| with high probability for α ≥ 2/3 while Kiltz, Pietrzak and Szegedy show that μ(A) ≤ |A| with high probability for α ≤ 1/4. Current bounds on μ(A) are essentially sharp for the range 2/3 ≤ α ≤ 1. Finding better bounds remains an open problem for the range 0 < α < 2/3 and especially for the range 1/4 < α < 2/3 in which the bound of Kiltz et al. doesn’t improve on the bound given in this paper (even if that bound applied). Moreover the conjecture of Mennink and Preneel for α ≤ 1/2 remains open.