General Hardness Amplification of Predicates and Puzzles

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.

💡 Research Summary

The paper presents two unified, non‑rewinding hardness‑amplification frameworks—one for efficiently samplable predicates and another for weakly verifiable puzzles—and shows how both can be leveraged to strengthen a broad class of cryptographic protocols. In the first part the authors revisit Yao’s classic XOR‑Lemma. Traditional proofs rely on intricate recursive arguments and rewinding techniques that are ill‑suited for interactive settings. By focusing on the reduction of “predictability” rather than on combinatorial constructions, they give a clean information‑theoretic proof: if a predicate f can be guessed with advantage ε over random guessing, then the XOR of k independent copies of f can be guessed with advantage at most ε·2⁻ᵏ. This proof not only simplifies the argument but also immediately yields the extraction lemma used by Håstad, Impagliazzo, Levin, and Luby to build pseudorandom generators from one‑way functions. The authors show that the same reasoning works for any efficiently samplable predicate, making the result modular and readily composable with other cryptographic primitives.

The second part addresses weakly verifiable puzzles, a model where a generator produces a challenge and a checking circuit verifies a solution, but the adversary only sees the checking circuit. Prior work proved hardness amplification only for simple Boolean combinations (AND, OR) of puzzle instances. The new proof works for an arbitrary monotone function g applied to the checking circuit, i.e., the amplified puzzle’s verifier is Check′(x)=g(Check(x)). The authors model each puzzle instance’s success as an independent Bernoulli variable and avoid the usual rewinding by constructing a non‑rewinding martingale that tracks the adversary’s cumulative advantage. Using this martingale together with a refined Markov‑type bound, they show that the success probability of solving the amplified puzzle is bounded by the original success probability transformed by the sensitivity of g. Consequently, the amplification bound is tight for any monotone composition, a substantial generalization over earlier results.

Finally, the paper demonstrates how these two amplification tools can be combined to reinforce cryptographic protocols whose security reduces to the unpredictability of single bits. The authors introduce an “information‑theoretic amplification protocol” that runs multiple independent instances of the underlying primitive in parallel and aggregates their outputs without ever rewinding the adversary. As a concrete application they resolve the open problem posed by Halevi and Rabin (TCC 2008) concerning bit‑commitment: starting from a weak commitment scheme whose binding property follows from the hardness of a single predicate, the authors apply their XOR‑amplification to obtain a new commitment that is both statistically hiding and computationally binding, even against adversaries that can adaptively query the commitment phase.

Overall, the work provides a conceptually simpler, more general, and non‑rewinding‑friendly approach to hardness amplification. By abstracting away from protocol‑specific rewinding tricks, the authors open the door to applying amplification techniques in a wide range of interactive cryptographic constructions, including zero‑knowledge proofs, secure multi‑party computation, and commitment schemes, thereby offering a powerful new tool for both theoreticians and practitioners.