Inverted Leftover Hash Lemma

Universal hashing found a lot of applications in computer science. In cryptography the most important fact about universal families is the so called Leftover Hash Lemma, proved by Impagliazzo, Levin and Luby. In the language of modern cryptography it states that almost universal families are good extractors. In this work we provide a somewhat surprising characterization in the opposite direction. Namely, every extractor with sufficiently good parameters yields a universal family on a noticeable fraction of its inputs. Our proof technique is based on tools from extremal graph theory applied to the “collision graph” induced by the extractor, and may be of independent interest. We discuss possible applications to the theory of randomness extractors and non-malleable codes.

[1]  Sergei Vassilvitskii,et al.  A model of computation for MapReduce , 2010, SODA '10.

[2]  Bart Preneel,et al.  Key-Recovery Attacks on Universal Hash Function Based MAC Algorithms , 2008, CRYPTO.

[3]  Hugo Krawczyk,et al.  MMH: Software Message Authentication in the Gbit/Second Rates , 1997, FSE.

[4]  Jaikumar Radhakrishnan,et al.  Bounds for Dispersers, Extractors, and Depth-Two Superconcentrators , 2000, SIAM J. Discret. Math..

[5]  Russell Impagliazzo,et al.  How to recycle random bits , 1989, 30th Annual Symposium on Foundations of Computer Science.

[6]  Ran Raz,et al.  Non-malleable Extractors with Short Seeds and Applications to Privacy Amplification , 2012, 2012 IEEE 27th Conference on Computational Complexity.

[7]  Xin Li,et al.  Improved non-malleable extractors, non-malleable codes and independent source extractors , 2016, Electron. Colloquium Comput. Complex..

[8]  Douglas R. Stinson,et al.  On the Connections Between Universal Hashing, Combinatorial Designs and Error-Correcting Codes , 1995, Electron. Colloquium Comput. Complex..

[9]  Rafail Ostrovsky,et al.  Fuzzy Extractors: How to Generate Strong Keys from Biometrics and Other Noisy Data , 2004, SIAM J. Comput..

[10]  Anna Pagh,et al.  Uniform hashing in constant time and linear space , 2003, STOC '03.

[11]  Renato Renner,et al.  Simple and Tight Bounds for Information Reconciliation and Privacy Amplification , 2005, ASIACRYPT.

[12]  Larry Carter,et al.  Universal Classes of Hash Functions , 1979, J. Comput. Syst. Sci..

[13]  Yevgeniy Dodis,et al.  Privacy Amplification and Non-malleable Extractors via Character Sums , 2011, 2011 IEEE 52nd Annual Symposium on Foundations of Computer Science.

[14]  Wen-Guey Tzeng,et al.  Extracting randomness from multiple independent sources , 2005, IEEE Transactions on Information Theory.

[15]  Larry Carter,et al.  New Hash Functions and Their Use in Authentication and Set Equality , 1981, J. Comput. Syst. Sci..

[16]  Stefan Dziembowski,et al.  Leakage-Resilient Circuits without Computational Assumptions , 2012, TCC.

[17]  Shachar Lovett,et al.  Non-malleable codes from additive combinatorics , 2014, STOC.

[18]  Hugo Krawczyk,et al.  UMAC: Fast and Secure Message Authentication , 1999, CRYPTO.

[19]  Leonid A. Levin,et al.  Pseudo-random Generation from one-way functions (Extended Abstracts) , 1989, STOC 1989.

[20]  Charles E. Leiserson,et al.  Deterministic parallel random-number generation for dynamic-multithreading platforms , 2012, PPoPP '12.

[21]  Stefan Dziembowski,et al.  Non-Malleable Codes from Two-Source Extractors , 2013, IACR Cryptol. ePrint Arch..

[22]  Thomas Johansson,et al.  On Families of Hash Functions via Geometric Codes and Concatenation , 1993, CRYPTO.

[23]  Yevgeniy Dodis,et al.  Non-malleable extractors and symmetric key cryptography from weak secrets , 2009, STOC '09.

[24]  Noam Nisan,et al.  Pseudorandom generators for space-bounded computation , 1992, Comb..

[25]  Stefan Dziembowski,et al.  Leakage-Resilient Storage , 2010, SCN.

[26]  Nico Döttling,et al.  Information Theoretic Continuously Non-Malleable Codes in the Constant Split-State Model , 2017, Electron. Colloquium Comput. Complex..

[27]  Stefan Dziembowski,et al.  Leakage-Resilient Non-malleable Codes , 2015, TCC.

[28]  Himanshu Tyagi,et al.  Universal Hashing for Information-Theoretic Security , 2014, Proceedings of the IEEE.

[29]  Michael Sipser,et al.  A complexity theoretic approach to randomness , 1983, STOC.

[30]  Masahito Hayashi,et al.  Exponential Decreasing Rate of Leaked Information in Universal Random Privacy Amplification , 2009, IEEE Transactions on Information Theory.

[31]  Ueli Maurer,et al.  Generalized privacy amplification , 1994, Proceedings of 1994 IEEE International Symposium on Information Theory.

[32]  Alan Siegel,et al.  On Universal Classes of Extremely Random Constant-Time Hash Functions , 1995, SIAM J. Comput..