  It has been conjectured by Sarkozy that with finitely many exceptions, the
set of quadratic residues modulo a prime $p$ cannot be represented as a sumset
$\{a+b\colon a\in A, b\in B\}$ with non-singleton sets $A,B\subset F_p$. The
case $A=B$ of this conjecture has been recently established by Shkredov. The
analogous problem for differences remains open: is it true that for all
sufficiently large primes $p$, the set of quadratic residues modulo $p$ is not
of the form $\{a'-a"\colon a',a"\in A,\,a'\ne a"\}$ with $A\subset F_p$?
  We attack here a presumably more tractable variant of this problem, which is
to show that there is no $A\subset F_p$ such that every quadratic residue has a
\emph{unique}representation as $a'-a"$ with $a',a"\in A$, and no non-residue is
represented in this form. We give a number of necessary conditions for the
existence of such $A$, involving for the most part the behavior of primes
dividing $p-1$. These conditions enable us to rule out all primes $p$ in the
range $13<p<10^{18}$ (the primes $p=5$ and $p=13$ being conjecturally the only
exceptions).
