  The promise of quantum computation and its consequences for
complexity-theoretic cryptography motivates an immediate search for
cryptosystems which can be implemented with current technology, but which
remain secure even in the presence of quantum computers. Inspired by recent
negative results pertaining to the nonabelian hidden subgroup problem, we
present here a classical algebraic function $f_V(M)$ of a matrix $M$ which we
believe is a one-way function secure against quantum attacks. Specifically,
inverting $f_V$ reduces naturally to solving a hidden subgroup problem over the
general linear group (which is at least as hard as the hidden subgroup problem
over the symmetric group). We also demonstrate a reduction from Graph
Isomorphism to the problem of inverting $f_V$; unlike Graph Isomorphism,
however, the function $f_V$ is random self-reducible and therefore uniformly
hard.
  These results suggest that, unlike Shor's algorithm for the discrete
logarithm--which is, so far, the only successful quantum attack on a classical
one-way function--quantum attacks based on the hidden subgroup problem are
unlikely to work. We also show that reconstructing any entry of $M$, or the
trace of $M$, with nonnegligible advantage is essentially as hard as inverting
$f_V$. Finally, $f_V$ can be efficiently computed and the number of output bits
is less than $1+\epsilon$ times the number of input bits for any $\epsilon >
0$.
