# A uniform min-max theorem with applications in cryptography

### Citation:

Vadhan, Salil, and Colin Jia Zheng. “A uniform min-max theorem with applications in cryptography.” In Ran Canetti and Juan Garay, editors, Advances in Cryptology—CRYPTO ‘13, Lecture Notes on Computer Science, 8042:93-110. Springer Verlag, Lecture Notes in Computer Science, 2013.
 CRYPTO2013.pdf 307 KB ECCC2013.pdf 567 KB

### Abstract:

Version History:
Full version published on ECCC2013 and IACR ePrint 2013.

We present a new, more constructive proof of von Neumann’s Min-Max Theorem for two-player zero-sum game — specifically, an algorithm that builds a near-optimal mixed strategy for the second player from several best-responses of the second player to mixed strategies of the first player. The algorithm extends previous work of Freund and Schapire (Games and Economic Behavior ’99) with the advantage that the algorithm runs in poly$$(n)$$ time even when a pure strategy for the first player is a distribution chosen from a set of distributions over $$\{0,1\}^n$$. This extension enables a number of additional applications in cryptography and complexity theory, often yielding uniform security versions of results that were previously only proved for nonuniform security (due to use of the non-constructive Min-Max Theorem).

We describe several applications, including a more modular and improved uniform version of Impagliazzo’s Hardcore Theorem (FOCS ’95), showing impossibility of constructing succinct non-interactive arguments (SNARGs) via black-box reductions under uniform hardness assumptions (using techniques from Gentry and Wichs (STOC ’11) for the nonuniform setting), and efficiently simulating high entropy distributions within any sufficiently nice convex set (extending a result of Trevisan, Tulsiani and Vadhan (CCC ’09)).

Publisher's Version

Last updated on 07/07/2021