Privately Outsourcing Exponentiation to a Single Server: Cryptanalysis and Optimal Constructions

We address the problem of speeding up group computations in cryptography using a single untrusted computational resource. We analyze the security of an efficient protocol for securely outsourcing multiexponentiations proposed at ESORICS 2014. We show that this scheme does not achieve the claimed security guarantees and we present several practical polynomial-time attacks on the delegation protocol which allows the untrusted helper to recover part (or the whole) of the device secret inputs. We then provide simple constructions for outsourcing group exponentiations in different settings (e.g. public/secret, fixed/variable bases and public/secret exponents). Finally, we prove that our attacks on the ESORICS 2014 protocol are unavoidable if one wants to use a single untrusted computational resource and to limit the computational cost of the limited device to a constant number of (generic) group operations. In particular, we show that our constructions are actually optimal.