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A Uniform Min-Max Theorem with Applications in Cryptography

  • Conference paper

Part of the Lecture Notes in Computer Science book series (LNSC,volume 8042)

Abstract

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)).

Keywords

  • Pseudorandom Generator
  • Common Reference String
  • Cryptographic Assumption
  • Uniform Algorithm
  • Universal Predictor

These keywords were added by machine and not by the authors. This process is experimental and the keywords may be updated as the learning algorithm improves.

Supported by NSF grant CCF-1116616 and US-Israel BSF grant 2010196. A full version of this paper [VZ2] to appear on the Cryptology ePrint Archive.

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© 2013 International Association for Cryptologic Research

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Vadhan, S., Zheng, C.J. (2013). A Uniform Min-Max Theorem with Applications in Cryptography. In: Canetti, R., Garay, J.A. (eds) Advances in Cryptology – CRYPTO 2013. CRYPTO 2013. Lecture Notes in Computer Science, vol 8042. Springer, Berlin, Heidelberg. https://doi.org/10.1007/978-3-642-40041-4_6

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  • DOI: https://doi.org/10.1007/978-3-642-40041-4_6

  • Publisher Name: Springer, Berlin, Heidelberg

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