Factoring Integers in the Tile Assembly Model

Tile assembly model is a highly distributed parallel computation model. In this paper, three tile assembly model systems are proposed on the theoretical level, which can be combined together to factor integers in a nondeterministical way. The combined factoring tile system randomly produces one divisor, then performs division and at last, checks whether the remainder equals to zero. All these proposed tile systems complete computation in Θ(1) distinct tile types. Each assembly process is executed in parallel, producing the solution in linear time with respect to the size of the known dividend. To find the successful solutions among the many parallel final configuration assembled by DNA tiles, a "yes" tile is used as a mark. It is proved that the probability of such a nondeterministic tile assembly system succeeding to find the prime factor is closed to 1.