Time-Space Tradeoffs for Counting NP Solutions Modulo Integers

We prove the first time-space tradeoffs for counting the number of solutions to an NP problem modulo small integers, and also improve upon the known time-space tradeoffs for SAT. Let m be a positive integer, and define MOD_m- SAT to be the problem of determining if a given Boolean formula has exactly km satisfying assignments, for some integer k. We prove that for all primes p, except for possibly one of them, MOD_p-SAT is not solvable in nc time and n^o(1) space on RAMs, for c \geqslant 1 satisfying c^3 - c^2 - 2c + 1 \le 0 (c \le 1.801 suffices). That is, there is at most one prime p that does not satisfy the lower bound. Note that such a lower bound does not follow from the SAT time-space tradeoffs, as we do not know of an efficient deterministic reduction from SAT to MOD_p-SAT. The result is non-constructive, in that it does not provide an explicit prime for which the lower bound holds. However, we can prove that the same limitation holds for SAT and MOD_6-SAT, as well as MOD_m-SAT for any composite m that is not a prime power. Our main tool is a general method for rapidly simulating deterministic RAM computations with restricted space, by counting the number of solutions to NP predicates modulo primes. The simulation converts an ordinary RAM into a "canonical one" that runs in roughly the same amount of time and space, yet its configuration sequences have nice properties suitable for counting.