Quantum statistical mechanics of encryption: reaching the speed limit of classical block ciphers

We cast encryption via classical block ciphers in terms of operator spreading in a dual space of Pauli strings, a formulation which allows us to characterize classical ciphers by using tools well known in the analysis of quantum many-body systems. We connect plaintext and ciphertext attacks to out-of-time order correlators (OTOCs) and quantify the quality of ciphers using measures of delocalization in string space such as participation ratios and corresponding entropies obtained from the wave function amplitudes in string space. The saturation of the string-space information entropy is accompanied by the vanishing of OTOCs. Together these signal irreversibility and chaos, which we take to be the defining properties of good classical ciphers. More precisely, we define a good cipher by requiring that the OTOCs vanish to exponential precision and that the string entropies saturate to the values associated with a random permutation, which are computed explicitly in the paper. We argue that these conditions can be satisfied by $n$-bit block ciphers implemented via random reversible circuits with ${\cal O}(n \log n)$ gates arranged on a tree structure, with layers of $n/3$ 3-bit gates, for which a "key" specifies uniquely the sequence of gates that comprise the circuit. We show that in order to reach this "speed limit" one must employ a three-stage circuit consisting of a stage implemented by layers of nonlinear gates that proliferate the number of strings, flanked by two other stages, each deploying layers of a special set of linear "inflationary" gates that accelerate the growth of small individual strings. A shallow, ${\cal O}(\log n)$-depth cipher of the type described here can be used in constructing a polynomial-overhead scheme for computation on encrypted data proposed in another publication as an alternative to Homomorphic Encryption.