## Quantum Statistical Mechanics of Encryption: from Black Holes and Shallow Classical Ciphers to Computation on Encrypted Data

We examine the scrambling speed of information by random reversible classical circuits with long-range gates. In particular, we address the question of what is the most shallow cipher based on such circuits that cannot be distinguished from a random permutation by an adversary with polynomial resources. We make progress on this question by formulating encryption by block ciphers in terms of operator spreading in a dual space of Pauli strings, a formulation which allows us to characterize the security of classical ciphers by using tools well known from studies of chaos and irreversibility in quantum many-body systems. In particular, 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 string-space wave function amplitudes. The mapping to string space also allows us to draw conclusions about random quantum circuits which are used in simple models of scrambling of information by Black Holes, which are claimed to be the fastest scramblers in nature. Our shallow classical cipher scrambles as fast as a Black Hole but to a much stricter (cryptographic) standard! More interesting from a practical point of view is that such short ciphers enable an efficient (polynomial) scheme for computing directly on encrypted data, a physics-inspired alternative to homomorphic encryption.