On the Cryptographic Complexity of the Worst Functions

We study the complexity of realizing the “worst” functions in several standard models of information-theoretic cryptography. In particular, for the case of security against passive adversaries, we obtain the following main results.

[1]  Yuval Ishai,et al.  Extending Oblivious Transfers Efficiently , 2003, CRYPTO.

[2]  Martijn Stam.  Beyond Uniformity: Better Security/Efficiency Tradeoffs for Compression Functions , 2008, CRYPTO.

[3]  Anna Gál,et al.  Lower bounds for monotone span programs , 1994, Proceedings of IEEE 36th Annual Foundations of Computer Science.

[4]  Douglas R. Stinson,et al.  Decomposition constructions for secret-sharing schemes , 1994, IEEE Trans. Inf. Theory.

[5]  Yuval Ishai,et al.  Share Conversion and Private Information Retrieval , 2012, 2012 IEEE 27th Conference on Computational Complexity.

[6]  Silvio Micali,et al.  How to play ANY mental game , 1987, STOC.

[7]  Yuval Ishai,et al.  On the power of nonlinear secret-sharing , 2001, Proceedings 16th Annual IEEE Conference on Computational Complexity.

[8]  Yuval Ishai,et al.  How Many Oblivious Transfers Are Needed for Secure Multiparty Computation? , 2007, CRYPTO.

[9]  Avi Wigderson,et al.  Completeness theorems for non-cryptographic fault-tolerant distributed computation , 1988, STOC '88.

[10]  Oded Goldreich,et al.  A randomized protocol for signing contracts , 1985, CACM.

[11]  Vinod M. Prabhakaran,et al.  Assisted Common Information With an Application to Secure Two-Party Sampling , 2014, IEEE Transactions on Information Theory.

[12]  Zeev Dvir,et al.  Matching Vector Codes , 2010, 2010 IEEE 51st Annual Symposium on Foundations of Computer Science.

[13]  Silvio Micali,et al.  Parallel Reducibility for Information-Theoretically Secure Computation , 2000, CRYPTO.

[14]  Vladimir Kolesnikov.  Gate Evaluation Secret Sharing and Secure One-Round Two-Party Computation , 2005, ASIACRYPT.

[15]  Donald Beaver,et al.  Precomputing Oblivious Transfer , 1995, CRYPTO.

[16]  Ran Canetti,et al.  Advances in Cryptology – CRYPTO 2012 , 2012, Lecture Notes in Computer Science.

[17]  Claude E. Shannon,et al.  The synthesis of two-terminal switching circuits , 1949, Bell Syst. Tech. J..

[18]  Richard Cleve.  Towards optimal simulations of formulas by bounded-width programs , 1990, STOC '90.

[19]  Klim Efremenko,et al.  3-Query Locally Decodable Codes of Subexponential Length , 2008 .

[20]  Yuval Ishai,et al.  Founding Cryptography on Oblivious Transfer - Efficiently , 2008, CRYPTO.

[21]  Alfredo De Santis,et al.  Tight Bounds on the Information Rate of Secret Sharing Schemes , 1997, Des. Codes Cryptogr..

[22]  Rafail Ostrovsky,et al.  Extracting Correlations , 2009, 2009 50th Annual IEEE Symposium on Foundations of Computer Science.

[23]  László Csirmaz.  Secret sharing schemes on graphs , 2005, IACR Cryptol. ePrint Arch..

[24]  Moti Yung,et al.  Non-interactive cryptocomputing for NC/sup 1/ , 1999, 40th Annual Symposium on Foundations of Computer Science (Cat. No.99CB37039).

[25]  Yuval Ishai,et al.  OT-Combiners via Secure Computation , 2008, TCC.

[26]  Avi Wigderson,et al.  Partial Derivatives in Arithmetic Complexity and Beyond , 2011, Found. Trends Theor. Comput. Sci..

[27]  Ivan Damgård,et al.  Constant-Overhead Secure Computation of Boolean Circuits using Preprocessing , 2013, TCC.

[28]  Carl Pomerance,et al.  Advances in Cryptology — CRYPTO ’87 , 2000, Lecture Notes in Computer Science.

[29]  Hung-Min Sun,et al.  Secret sharing in graph-based prohibited structures , 1997, Proceedings of INFOCOM '97.

[30]  David P. Woodruff,et al.  A geometric approach to information-theoretic private information retrieval , 2005, 20th Annual IEEE Conference on Computational Complexity (CCC'05).

[31]  Mihir Bellare.  Advances in Cryptology — CRYPTO 2000 , 2000, Lecture Notes in Computer Science.

[32]  Yuval Ishai,et al.  On the Hardness of Information-Theoretic Multiparty Computation , 2004, EUROCRYPT.

[33]  Mitsuru Ito,et al.  Secret sharing scheme realizing general access structure , 1989 .

[34]  Tal Malkin,et al.  A Quantitative Approach to Reductions in Secure Computation , 2004, TCC.

[35]  Iordanis Kerenidis,et al.  Non-local box complexity and secure function evaluation , 2011, Quantum Inf. Comput..

[36]  Joe Kilian.  More general completeness theorems for secure two-party computation , 2000, STOC '00.

[37]  Kenneth G. Paterson.  Advances in Cryptology - EUROCRYPT 2011 - 30th Annual International Conference on the Theory and Applications of Cryptographic Techniques, Tallinn, Estonia, May 15-19, 2011. Proceedings , 2011, EUROCRYPT.

[38]  Aggelos Kiayias,et al.  Traceable Signatures , 2004, EUROCRYPT.

[39]  Gilles Brassard,et al.  Information theoretic reductions among disclosure problems , 1986, 27th Annual Symposium on Foundations of Computer Science (sfcs 1986).

[40]  Yuval Ishai,et al.  General constructions for information-theoretic private information retrieval , 2005, J. Comput. Syst. Sci..

[41]  Yuval Ishai,et al.  On Locally Decodable Codes, Self-correctable Codes, and t -Private PIR , 2007, APPROX-RANDOM.

[42]  Michael O. Rabin,et al.  How To Exchange Secrets with Oblivious Transfer , 2005, IACR Cryptol. ePrint Arch..

[43]  Joe Kilian,et al.  Achieving Oblivious Transfer Using Weakened Security Assumptions (Extended Abstract) , 1988, FOCS 1988.

[44]  Vince Grolmusz,et al.  Superpolynomial Size Set-systems with Restricted Intersections mod 6 and Explicit Ramsey Graphs , 2000, Comb..

[45]  Anat Paskin-Cherniavsky,et al.  On the Power of Correlated Randomness in Secure Computation , 2013, TCC.

[46]  Eyal Kushilevitz,et al.  Communication Complexity: Index of Notation , 1996 .

[47]  Amit Sahai,et al.  Secure Multi-Party Computation , 2013 .

[48]  Eyal Kushilevitz,et al.  Communication Complexity , 1997, Adv. Comput..

[49]  Hung-Min Sun,et al.  Decomposition Construction for Secret Sharing Schemes with Graph Access Structures in Polynomial Time , 2010, SIAM J. Discret. Math..

[50]  Rudolf Ahlswede,et al.  Founding Cryptography on Oblivious Transfer , 2016 .

[51]  Marten van Dijk.  On the information rate of perfect secret sharing schemes , 1995, Des. Codes Cryptogr..

[52]  Oded Goldreich,et al.  How to Solve any Protocol Problem - An Efficiency Improvement , 1987, CRYPTO.

[53]  Yuval Ishai,et al.  Perfect Constant-Round Secure Computation via Perfect Randomizing Polynomials , 2002, ICALP.

[54]  Ivan Damgård,et al.  Semi-Homomorphic Encryption and Multiparty Computation , 2011, IACR Cryptol. ePrint Arch..

[55]  David Chaum,et al.  Multiparty unconditionally secure protocols , 1988, STOC '88.

[56]  Joan Feigenbaum,et al.  Security with Low Communication Overhead , 1990, CRYPTO.

[57]  A. Maximov,et al.  Fast computation of large distributions and its cryptographic applications , 2005 .

[58]  Oded Goldreich,et al.  The Foundations of Cryptography - Volume 2: Basic Applications , 2001 .

[59]  Yuval Ishai,et al.  Randomization Techniques for Secure Computation , 2013, Secure Multi-Party Computation.

[60]  Donald Beaver,et al.  Correlated pseudorandomness and the complexity of private computations , 1996, STOC '96.

[61]  Severin Winkler,et al.  On the Efficiency of Classical and Quantum Oblivious Transfer Reductions , 2010, IACR Cryptol. ePrint Arch..

[62]  Amos Beimel,et al.  Secret-Sharing Schemes for Very Dense Graphs , 2014, Journal of Cryptology.

[63]  Michael Wiener,et al.  Advances in Cryptology — CRYPTO’ 99 , 1999 .

[64]  Joe Kilian,et al.  Founding crytpography on oblivious transfer , 1988, STOC '88.

[65]  Oded Goldreich,et al.  Foundations of Cryptography: Volume 2, Basic Applications , 2004 .

[66]  Joe Kilian,et al.  Achieving oblivious transfer using weakened security assumptions , 1988, [Proceedings 1988] 29th Annual Symposium on Foundations of Computer Science.

[67]  Tal Rabin.  Advances in Cryptology - CRYPTO 2010, 30th Annual Cryptology Conference, Santa Barbara, CA, USA, August 15-19, 2010. Proceedings , 2010, CRYPTO.

[68]  Yuval Ishai,et al.  Randomizing polynomials: A new representation with applications to round-efficient secure computation , 2000, Proceedings 41st Annual Symposium on Foundations of Computer Science.

[69]  Anna Gál,et al.  Lower bounds for monotone span programs , 2005, computational complexity.

[70]  Paul Erdös,et al.  Covering a graph by complete bipartite graphs , 1997, Discret. Math..

[71]  Moni Naor,et al.  A minimal model for secure computation (extended abstract) , 1994, STOC '94.

[72]  Alfredo De Santis,et al.  On the Information Rate of Secret Sharing Schemes , 1996, Theor. Comput. Sci..

[73]  Yuval Ishai,et al.  Private simultaneous messages protocols with applications , 1997, Proceedings of the Fifth Israeli Symposium on Theory of Computing and Systems.

[74]  Amos Beimel,et al.  Secret Sharing Schemes for Very Dense Graphs , 2012, CRYPTO.

[75]  A. J. Menezes,et al.  Advances in Cryptology - CRYPTO 2007, 27th Annual International Cryptology Conference, Santa Barbara, CA, USA, August 19-23, 2007, Proceedings , 2007, CRYPTO.

[76]  Yehuda Lindell,et al.  More Efficient Constant-Round Multi-Party Computation from BMR and SHE , 2016, IACR Cryptol. ePrint Arch..

[77]  Dan Boneh,et al.  Advances in Cryptology - CRYPTO 2003 , 2003, Lecture Notes in Computer Science.

[78]  Yuval Ishai,et al.  Computationally Private Randomizing Polynomials and Their Applications , 2005, Computational Complexity Conference.

[79]  Siegfried Bublitz,et al.  Decomposition of graphs and monotone formula size of homogeneous functions , 1986, Acta Informatica.

[80]  Eyal Kushilevitz,et al.  Computing Functions of a Shared Secret , 2000, SIAM J. Discret. Math..

[81]  Sergey Yekhanin.  Towards 3-query locally decodable codes of subexponential length , 2007, STOC '07.

[82]  Eyal Kushilevitz,et al.  Private information retrieval , 1995, Proceedings of IEEE 36th Annual Foundations of Computer Science.

[83]  Joan Feigenbaum,et al.  Locally random reductions: Improvements and applications , 1997, Journal of Cryptology.

[84]  Josh Benaloh,et al.  Generalized Secret Sharing and Monotone Functions , 1990, CRYPTO.