Particularly following the work of Whitfield Diffie and Martin Hellman in 1976 it became popular to consider cryptography systems based on mathematical problems that are easy to state but have been found difficult to solve. It was at first hoped that the problems could be NP-complete ones, which are universal in the sense that their solution can be used to provide a solution to any problem in the class NP (see page 1086). To date, however, no system has been devised whose cryptanalysis is known to be NP-complete. Indeed, essentially the only problem on which cryptography systems have so far successfully been based is factoring of integers (see below). And while this problem has resisted a fair number of attempts at solution, it is not known to be NP-complete (and indeed its ability to be solved in polynomial time on a formal quantum computer may suggest that it is not).
My cellular automaton cryptography system follows the principle of being based on a problem that is easy to state. And indeed the general problem of finding initial conditions for a cellular automaton is NP-complete (see page 767). But the problem is not known to be NP-complete for the specific case of, say, rule 30. Significantly less work has been done on the problem of finding initial conditions for rule 30 than on the problem of factoring integers. But the greater simplicity of rule 30 might make one already have almost as much confidence in the difficulty of solving this problem as of factoring integers.