Information Theoretic Reasons for Computational Difficulty

We give an intuitive account of the concepts in the title, by considering the following simple number-theoretic example. Imagine two distant players who communicate by exchanging binary messages (bits). One player is given a prime number x, and the second a composite number y, where x,y < 2". The players' task is to find a prime number p, with p < 2n, such that x ^ y (mod p). The existence of such a small prime p is guaranteed by the prime number theorem and the Chinese remainder theorem. The players agree beforehand on a "protocol" for exchanging messages. The protocol dictates to each player what message to send at each point, based on his input and the messages he received so far. It also dictates when to stop, and how do determine the answer from the information received. There is no limit on the computational complexity of these decisions, which are free of charge. The cost of the protocol is the number of bits they have to exchange on the worst case choice of inputs. We shall be interested in the cost of the best protocol under this measure, which we denote by t(n). There is a trivial protocol in which one player sends his input to the second (n bits), who computes the answer and sends it (log n bits) back to the first. This shows that t(n) <,n + log n. How small can t(n) be? Is it possible that t(n) = 0(log n), which is (essentially) the trivial lower bound? At present, these trivial upper and lower bounds are the best known ! Why should anyone take the time to think about this problem, besides its innocently simple statement and the challenge of the exponential gap in our knowledge? The reason is that this information theoretic problem encodes the computational difficulty of primality testing! Answering it is extremely important for computational number theory and theoretical computer science as follows: