Paper 2024/236

Public-Key Cryptography through the Lens of Monoid Actions

Abstract

We provide a novel view of public-key cryptography by showing full equivalences of certain primitives to "hard" monoid actions. More precisely, we show that key exchange and two-party computation are exactly equivalent to monoid actions with certain structural and hardness properties. To the best of our knowledge, this is the first "natural" characterization of the mathematical structure inherent to any key exchange or two-party computation protocol, and the first explicit proof of the necessity of mathematical structure for public-key cryptography. We then utilize these characterizations to show new black-box separation results. Concretely, we obtain the following results: - Two-Party Key Exchange: We show that that any two-party noninteractive key exchange protocol is equivalent to the existence of an abelian monoid action equipped with a natural hardness property, namely (distributional) unpredictability. More generally, we show that any $k$-round (two-party) key exchange protocol is essentially equivalent to the existence of a (distributional) unpredictable monoid action with certain commutator-like properties. Rudich (Crypto '91) shows a black-box separation of $k$-round and $(k+1)$-round key exchange for any $k$; we use our generic primitive here to formalize this result and extend it to efficient key exchange protocols (where communication is poly($k$)). - Two-Party Computation: We show that any maliciously secure two-party computation protocol is also equivalent to a monoid action with commutator-like properties and certain hardness guarantees. We then use a generic version of this primitive to show a black-box separation between $k$-round semi-honest secure two-party computation and $(k+1)$-round maliciously secure two-party computation. This yields the first black-box separation (to our knowledge) between $k$-round and $(k+1)$-round maliciously secure two-party computation protocols.

Note: The revised version contains updated and fine-tuned analyses of our separation results.