Memory Checking Requires Logarithmic Overhead

We study the complexity of memory checkers with computational security and prove the first general tight lower bound. Memory checkers, first introduced over 30 years ago by Blum, Evans, Gemmel, Kannan, and Naor (FOCS 91, Algorithmica 94), allow a user to store and maintain a large memory on a remote and unreliable server by using small trusted local storage. The user can issue instructions to the server and after every instruction, obtain either the correct value or a failure (but not an incorrect answer) with high probability. The main complexity measure of interest is the size of the local storage and the number of queries the memory checker makes upon every logical instruction. The most efficient known construction has query complexity O(log n/log log n) and local space proportional to a computational security parameter, assuming one-way functions, where n is the logical memory size. Dwork, Naor, Rothblum, and Vaikuntanathan (TCC 09) showed that for a restricted class of deterministic and non-adaptive memory checkers, this construction is optimal, up to constant factors. However, going beyond the small class of deterministic and non-adaptive constructions has remained a major open problem. In this work, we fully resolve the complexity of memory checkers by showing that any construction with local space p and query complexity q must satisfy p >= n/(log n)(O(q)). This implies, as a special case, that q >= Omega(log n/log log n) in any scheme, assuming that p <= n(1-epsilon) for epsilon > 0. The bound applies to any scheme with computational security, completeness 2/3, and inverse polynomial in n soundness (all of which make our lower bound only stronger). We further extend the lower bound to schemes where the read complexity q(r) and write complexity q(w) differ. For instance, we show the tight bound that if q(r) = O(1) and p <= n(1-epsilon) for epsilon > 0, then q(w) >= n(Omega(1)). This is the first lower bound, for any non-trivial class of constructions, showing a read-write query complexity trade-off. Our proof is via a delicate compression argument showing that a too good to be true memory checker can be used to compress random bits of information. We draw inspiration from tools recently developed for lower bounds for relaxed locally decodable codes. However, our proof itself significantly departs from these works, necessitated by the differences between settings.