Reaching the speed limit of classical block ciphers via quantum-like operator spreading

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. In particular, we show that in Feistel ciphers the entropy saturates its bound to exponential precision for ciphers with 4 or more rounds, consistent with the classic Luby-Rackoff result. 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 saturation of the entropy and the vanishing of OTOCs occurs to super-polynomial precision, implying that the cipher cannot be distinguished from a pseudorandom permutation with a polynomial number of queries. We argue that this criterion can be satisfied by $n$-bit block ciphers implemented via random reversible circuits with ${cal O}(n log n)$ gates. These circuits are composed of layers of $n/3$ non-overlapping non-local random 3-bit gates. In order to reach this speed limit we employ a two-stage circuit: this first stage deploys a set of linear inflationary gates that accelerate the growth of small individual strings; followed by a second stage implemented via universal gates that exponentially proliferate the number of macroscopic strings. We suggest that this two-stage construction would result in the scrambling of quantum states to similar precision and with circuits of similar size.