General Hardness Amplification of Predicates and Puzzles

TL;DR

This paper introduces new, simpler proofs for theorems on hardness amplification of predicates and puzzles, extending their applicability to cryptographic protocols and solving open problems in bit commitment.

Contribution

It provides generalized, simplified proofs for hardness amplification theorems, applicable in cryptography and interactive protocols, including solving a key open problem in bit commitment.

Findings

New proof of Yao's XOR-Lemma applicable to cryptography

Generalized hardness amplification for weakly verifiable puzzles

Application to strengthening cryptographic protocols and solving open problems

Abstract

We give new proofs for the hardness amplification of efficiently samplable predicates and of weakly verifiable puzzles which generalize to new settings. More concretely, in the first part of the paper, we give a new proof of Yao's XOR-Lemma that additionally applies to related theorems in the cryptographic setting. Our proof seems simpler than previous ones, yet immediately generalizes to statements similar in spirit such as the extraction lemma used to obtain pseudo-random generators from one-way functions [Hastad, Impagliazzo, Levin, Luby, SIAM J. on Comp. 1999]. In the second part of the paper, we give a new proof of hardness amplification for weakly verifiable puzzles, which is more general than previous ones in that it gives the right bound even for an arbitrary monotone function applied to the checking circuit of the underlying puzzle. Both our proofs are applicable in many settings of interactive cryptographic protocols because they satisfy a property that we call "non-rewinding". In particular, we show that any weak cryptographic protocol whose security is given by the unpredictability of single bits can be strengthened with a natural information theoretic protocol. As an example, we show how these theorems solve the main open question from [Halevi and Rabin, TCC2008] concerning bit commitment.