Generalized environmental security from number theoretic assumptions

We address the problem of realizing concurrently composable secure computation without setup assumptions. While provably impossible in the UC framework of [Can01], Prabhakaran and Sahai had recently suggested a relaxed framework called generalized Environmental Security (gES) [PS04], as well as a restriction of it to a “client-server” setting based on monitored functionalities [PS05]. In these settings, the impossibility results do not apply, and they provide secure protocols relying on new non-standard assumptions regarding the existence of hash functions with certain properties. In this paper, we first provide gES protocols for general secure computation, based on a new, concrete number theoretic assumption called the relativized discrete log assumption (rDLA). Second, we provide secure protocols for functionalities in the (limited) client-server framework of [PS05], replacing their hash function assumption with the standard discrete log assumption. Both our results (like previous work) also use (standard) super-polynomially strong trapdoor permutations. We believe this is an important step towards obtaining positive results for efficient secure computation in a concurrent environment based on well studied assumptions. Furthermore, the new assumption we put forward is of independent interest, and may prove useful for other cryptographic applications.