Fully Leakage-Resilient Codes

Leakage resilient codes (LRCs) are probabilistic encoding schemes that guarantee message hiding even under some bounded leakage on the codeword. We introduce the notion of fully leakage resilient codes (FLRCs), where the adversary can leak λ0 bits from the encoding process, namely, the message and the randomness involved during the encoding process. In addition the adversary can as usual leak from the codeword. We give a simulation-based definition requiring that the adversary’s leakage from the encoding process and the codeword can be simulated given just λ0 bits of leakage from the message. For λ0 = 0 our new simulationbased notion is equivalent to the usual game-based definition. A FLRC would be interesting in its own right and would be useful in building other leakage-resilient primitives in a composable manner. Unfortunately, we give a fairly general impossibility result for FLRCs in the popular split-state model, where the codeword is broken into independent parts and where the leakage occurs independently on the parts. We show that if the leakage is allowed to be any poly-time function of the secret and if collision-resistant hash functions exist, then there is no FLRC for the split-state model. The result holds when the message length is at least linear in the security parameter. We can extend the impossibility result to FLRCs with constant-length messages under assumptions related to differing-input obfuscation. These results show that it is highly unlikely that we can build FLRCs for the split-state model when the leakage can be any poly-time function of the secret state. We then give two feasibility results for weaker models. First, we show that for NC-bounded leakage from the randomness and arbitrary poly-time leakage from the parts of the codeword the inner-product construction proposed by Daví et al. (SCN’10) and successively improved by Dziembowski and Faust (ASIACRYPT’11) is a FLRC for the split-state model. Second, we provide a compiler from any LRC to a FLRC in the common reference string model where the leakage on the encoding comes from a fixed leakage family of small cardinality. In particular, this compiler applies to the split-state model but also to other models.