A New Approximate Min-Max Theorem with Applications in Cryptography

We propose a novel proof technique that can be applied to attack a broad class of problems in computational complexity, when switching the order of universal and existential quantifiers is helpful. Our approach combines the standard min-max theorem and convex approximation techniques, offering quantitative improvements over the standard way of using min-max theorems as well as more concise and elegant proofs.

💡 Research Summary

The paper introduces a novel proof technique that refines the classic von Neumann min‑max theorem by incorporating a carefully designed convex‑approximation step. The central problem addressed is the difficulty of swapping existential and universal quantifiers in complexity‑theoretic statements when the underlying sets of algorithms (A) and distributions (C) are not convex. The authors observe that while the standard min‑max theorem guarantees the equivalence of sup inf and inf sup when both sets are convex, most cryptographic applications involve non‑convex sets (e.g., polynomial‑size circuits). To bridge this gap they propose embedding the non‑convex sets into “almost” convex hulls A′ and C′, with a controlled approximation error δ. This is formalized in condition (3): for every A∈A and X∈C there exist A′∈conv(A′) and X′∈conv(C′) such that |v(A,X)−v(A′,X′)|≤δ, where v measures the payoff (e.g., prediction advantage). Under this condition they prove an Approximate Min‑Max Theorem: the weak statement “∀ A ∃ X : v(A,X)≤c” implies the dream statement “∃ X ∀ A : v(A,X)≤c+δ” after replacing A and C by their convex approximations.

The paper then applies this framework to two cornerstone results in cryptography:

1. Impagliazzo’s Hardcore Lemma – Traditionally proved via iterative boosting and a non‑standard “Nissan‑Levy” trick to improve hardcore density. The authors first establish the weak statement that for each circuit A of size s there exists a distribution X (with certain properties) on which A’s advantage is bounded. Using the Approximate Min‑Max Theorem with a Hölder‑based approximation, they construct A′ (circuits of size s′=Ω(s·δ²/ log(1/ε))) and C′ (conditional distributions) such that the error introduced is only δ. This yields a hardcore set of probability ε·O(log(1/ε)·δ⁻²) that works simultaneously for all circuits of size s′, matching the optimal parameters of prior work but with a dramatically simpler, modular proof that avoids boosting altogether.

2. Metric‑to‑HILL Pseudo‑Entropy Transformation – Metric pseudo‑entropy guarantees that any circuit of size s cannot distinguish a distribution Y from some high‑entropy Y′ by more than ε. HILL pseudo‑entropy requires a similar guarantee but with a min‑entropy lower bound k. Existing transformations suffered a loss proportional to n (the full length) and required a separate hardcore extraction step. By defining A as real‑valued circuits of size s and C as conditional high‑min‑entropy distributions X′|E (with Pr