We show that approximating the *shortest vector problem* (in
any *L*_{p} norm) to within any constant factor less than
*2*^{1/p} is hard for NP under *reverse unfaithful
random* reductions with inverse polynomial error probability. In
particular, approximating the *shortest vector problem* is not
in RP (random polynomial time), unless NP equals RP. We also prove a
proper NP-hardness result (i.e., hardness under deterministic many-one
reductions) under a reasonable number theoretic conjecture on the
distribution of square-free smooth numbers. As part of our proof, we
give an alternative construction of Ajtai's constructive variant of
Sauer's lemma, that greatly simplifies Ajtai's original proof.