Unprovability of leakage-resilient cryptography beyond the information-theoretic limit

In recent years, leakage-resilient cryptography—the design of cryptographic protocols resilient to bounded leakage of honest players’ secrets—has received significant attention. A major limitation of known provably-secure constructions (based on polynomial hardness assumptions) is that they require the secrets to have sufficient actual (i.e., information-theoretic), as opposed to comptutational, min-entropy even after the leakage. In this work, we present barriers to provably-secure constructions beyond the “information-theoretic barrier”: Assume the existence of collision-resistant hash functions. Then, no $$\mathcal{NP}$$ search problem with $$(2^{n^{\epsilon }})$$-bounded number of witnesses can be proven (even worst-case) hard in the presence of $$O(n^{\epsilon })$$ bits of computationally-efficient leakage of the witness, using a black-box reduction to any O(1)-round assumption. In particular, this implies that $$O(n^{\epsilon })$$-leakage resilient injective one-way functions, and more generally, one-way functions with at most $$2^{n^{\epsilon }}$$ pre-images, cannot be based on any “standard” complexity assumption using a black-box reduction.