Three fundamental results of Levin involve algorithms or reductions
whose running time is exponential in the length of certain programs. We study the
question of whether such dependency can be made polynomial.
(1) Levin's ``optimal search algorithm'' performs at most a constant factor more slowly
than any other fixed algorithm. The constant, however, is exponential in the length
of the competing algorithm.
We note that the running time of a universal search cannot be made ``fully
polynomial'' (that is, the relation between slowdown and program length cannot
be made polynomial), unless P=NP.
(2) Levin's ``universal one-way function'' result has the following structure:
there is a polynomial time computable function $f_{Levin}$ such that if there is
a polynomial time computable adversary $A$ that inverts $f_{Levin}$ on an inverse
polynomial fraction of inputs, then for every polynomial time computable function $g$
there also is a polynomial time adversary $A_g$ that inverts $g$ on an inverse
polynomial fraction of inputs. Unfortunately, again the running time of $A_g$ depends
exponentially on the bit length of the program that computes $g$ in polynomial time.
We show that
a fully polynomial uniform reduction from an arbitrary one-way
function to a specific one-way function is not possible relative to an oracle that
we construct, and so no ``universal one-way function'' can have a fully polynomial
security analysis via relativizing techniques.
(3) Levin's completeness result for distributional NP problems implies that if a specific
problem in NP is easy on average under the uniform distribution, then every language $L$
in NP is also easy on average under any polynomial time computable distribution. The
running time of the implied algorithm for $L$, however, depends exponentially on the
bit length of the non-deterministic polynomial time Turing machine that decides $L$.
We show that if a completeness result for distributional NP can be proved
via a ``fully uniform'' and ``fully polynomial'' time reduction, then there is a worst-case
to average-case reduction for NP-complete problems. In particular, this means
that a fully polynomial completeness result for distributional NP is impossible,
even via randomized truth-table reductions, unless the polynomial hierarchy collapses.
