Matrioska: A Compiler for Multi-Key Homomorphic Signatures.

Multi-Key Homomorphic Signatures (MK-HS) enable clients in a system to sign and upload messages to an untrusted server. At any later point in time, the server can perform a computation C on data provided by t different clients, and return the output y and a short signature sigma(C,y) vouching for the correctness of y as the output of the function C on the signed data. Interestingly, MK-HS enable verifiers to check the validity of the signature using solely the public keys of the signers whose messages were used in the computation. Moreover, the signatures sC, y are succinct, namely their size depends at most linearly in the number of clients, and only logarithmically in the total number of inputs of C. Existing MK-HS are constructed based either on standard assumptions over lattices (Fiore et al. ASIACRYPT'16), or on non-falsifiable assumptions (SNARKs) (Lai et al., ePrint'16). In this paper, we investigate connections between single-key and multi-key homomorphic signatures. We propose a generic compiler, called Matrioska, which turns any (sufficiently expressive) single-key homomorphic signature scheme into a multi-key scheme. Matrioska establishes a formal connection between these two primitives and is the first alternative to the only known construction under standard falsifiable assumptions. Our result relies on a novel technique that exploits the homomorphic property of a single-key HS scheme to compress an arbitrary number of signatures from t different users into only t signatures.