Hard Instance Generation for Sat Title: Hard Instance Generation for Sat

We consider the problem of generating hard instances for the Satisfying Assignment Search Problem (in short, SAT). It is not known whether SAT is di cult on average, while it has been believed that the Factorization Problem (in short, FACT) is hard on average. Thus, one can expect to generate hard-on-average instances by using a reduction from FACT to SAT. Although the asymptotically best reduction is obtained by using the Fast Fourier Transform [SS71] (in short, FFT), its constant factor is too big in practice. Here we propose to use the Chinese Remainder Theorem for constructing e cient yet simple reductions from FACT to SAT. First by using the Chinese Remainder Theorem recursively, we de ne a reduction that produces, from n bit FACT instances, SAT instances in the conjunctive normal form with O(n 1+ ) variables, where > 0 is any xed constant. (Cf. The reduction using FFT yields instances with O(n log n log log n) variables.) Next we demonstrate the e ciency of our approach with some concrete examples; we de ne a reduction that produces relatively small SAT instances. For example, it is possible to construct SAT instances with about 5,600 variables that is as hard as factorizing 100 bit integers. (Cf. The straightforward reduction yields SAT instances with 7,600 variables.)