Paper 2023/1973

Combinatorially Homomorphic Encryption

Abstract

Homomorphic encryption enables public computation over encrypted data. In the past few decades, homomorphic encryption has become a staple of both the theory and practice of cryptography. Nevertheless, while there is a general loose understanding of what it means for a scheme to be homomorphic, to date there is no single unifying minimal definition that captures all schemes. In this work, we propose a new definition, which we refer to as combinatorially homomorphic encryption, which attempts to give a broad base that captures the intuitive meaning of homomorphic encryption and draws a clear line between trivial and nontrivial homomorphism. Our notion relates the ability to accomplish some task when given a ciphertext, to accomplishing the same task without the ciphertext, in the context of communication complexity. Thus, we say that a scheme is combinatorially homomorphic if there exists a communication complexity problem $$ (where $$ is Alice's input and $$ is Bob's input) which requires communication $$, but can be solved with communication less than $$ when Alice is given in addition also an encryption $$ of Bob's input (using Bob's key $$). We show that this definition indeed captures pre-existing notions of homomorphic encryption and (suitable variants are) sufficiently strong to derive prior known implications of homomorphic encryption in a conceptually appealing way. These include constructions of (lossy) public-key encryption from homomorphic private-key encryption, as well as collision-resistant hash functions and private information retrieval schemes.