  Let $G$ be a finite abelian group, let $0 < \alpha < 1$, and let $A \subseteq
G$ be a random set of size $|G|^\alpha$. We let $$ \mu(A) =
\max_{B,C:|B|=|C|=|A|}|\{(a,b,c) \in A \times B \times C : a = b + c \}|. $$
The issue is to determine upper bounds on $\mu(A)$ that hold with high
probability over the random choice of $A$. Mennink and Preneel \cite{BM}
conjecture that $\mu(A)$ should be close to $|A|$ (up to possible logarithmic
factors in $|G|$) for $\alpha \leq 1/2$ and that $\mu(A)$ should not much
exceed $|A|^{3/2}$ for $\alpha \leq 2/3$. We prove the second half of this
conjecture by showing that $$ \mu(A) \leq |A|^3/|G| + 4|A|^{3/2}\ln(|G|)^{1/2}
$$ with high probability, for all $0 < \alpha < 1$. We note that $3\alpha - 1
\leq (3/2)\alpha$ for $\alpha \leq 2/3$.
  In previous work, Alon et al$.$ have shown that $\mu(A) \leq O(1)|A|^3/|G|$
with high probability for $\alpha \geq 2/3$ while Kiltz, Pietrzak and Szegedy
show that $\mu(A) \leq |A|^{1 + 2\alpha}$ with high probability for $\alpha
\leq 1/4$. Current bounds on $\mu(A)$ are essentially sharp for the range $2/3
\leq \alpha \leq 1$. Finding better bounds remains an open problem for the
range $0 < \alpha < 2/3$ and especially for the range $1/4 < \alpha < 2/3$ 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 $\alpha \leq 1/2$ remains open.
