Optimal and Parallel Online Memory Checking

Memory checking studies the problem of cryptographically verifying the correctness of untrusted indexed storage. After a series of results yielding checkers with O(log n) query complexity, Dwork, Naor, Ruthblum and Vaikuntanathan [9] derived an Ω(log n/ log logn) lower bound on the query complexity of any checker operating on memory words of polylogarithmic size, where n is the number of memory indices. In view of this lower bound, we make the following two contributions: 1. We construct an optimal online memory checker of Θ(log n/ log log n) query complexity, closing in this way the relevant complexity gap. Our construction employs pseudorandom functions and a simple data grouping technique inspired by I/O algorithms. 2. In our second and main result, we put forth the notion of parallel online memory checking and provide parallel checker constructions with O(1) query complexity and O(log n) processors. We initially show that checkers that use secret small memory, including our optimal checker, are easily parallelizable; However, checkers that use only reliable small memory cannot be naturally parallelized. We overcome this barrier by employing an algebraic hash function based on lattices assumptions and construct such parallel checkers with only reliable memory. To achieve our result, we establish and exploit a property that we call repeated linearity of lattice-based hash functions, that might be of independent interest. Applications of our checkers include update-optimal external memory authenticated data structures. We construct an authenticated B-tree data structure which can be updated with two I/Os, outperforming the logarithmic update complexity of hash-based external memory Merkle trees. ∗Email: [email protected]. †Email: [email protected].