The Power of Negations in Cryptography

The study of monotonicity and negation complexity for Bool-ean functions has been prevalent in complexity theory as well as in computational learning theory, but little attention has been given to it in the cryptographic context. Recently, Goldreich and Izsak (2012) have initiated a study of whether cryptographic primitives can be monotone, and showed that one-way functions can be monotone (assuming they exist), but a pseudorandom generator cannot.

[1]  N. Alon,et al.  The Probabilistic Method: Alon/Probabilistic , 2008 .

[2]  Yuval Ishai,et al.  Cryptography in NC0 , 2004, SIAM J. Comput..

[3]  Leonid A. Levin,et al.  A hard-core predicate for all one-way functions , 1989, STOC '89.

[4]  Alon Rosen,et al.  Candidate weak pseudorandom functions in AC0 ○ MOD2 , 2014, ITCS.

[5]  Oded Goldreich,et al.  Foundations of Cryptography: Volume 1, Basic Tools , 2001 .

[6]  Nader H. Bshouty,et al.  On the Fourier spectrum of monotone functions , 1996, JACM.

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

[8]  Jayalal Sarma,et al.  Depth Lower Bounds against Circuits with Sparse Orientation , 2014, Fundam. Informaticae.

[9]  Thomas Hofmeister.  The power of negative thinking in constructing threshold circuits for addition , 1992, [1992] Proceedings of the Seventh Annual Structure in Complexity Theory Conference.

[10]  B. Applebaum.  Cryptography in NC0 , 2014 .

[11]  Michel Talagrand,et al.  How much are increasing sets positively correlated? , 1996, Comb..

[12]  Moni Naor,et al.  Small-Bias Probability Spaces: Efficient Constructions and Applications , 1993, SIAM J. Comput..

[13]  Rahul Santhanam,et al.  Uniform Hardness Amplification in NP via Monotone Codes , 2006, Electron. Colloquium Comput. Complex..

[14]  E. Kushilevitz,et al.  Communication Complexity: Basics , 1996 .

[15]  Rocco A. Servedio,et al.  Learning circuits with few negations , 2014, Electron. Colloquium Comput. Complex..

[16]  T. Sanders,et al.  Analysis of Boolean Functions , 2012, ArXiv.

[17]  Michael J. Fischer.  Hauptvortrag: The complexity of negation-limited networks - A brief survey , 1975, Automata Theory and Formal Languages.

[18]  A. A. Markov.  On the Inversion Complexity of a System of Functions , 1958, JACM.

[19]  Silvio Micali,et al.  How to construct random functions , 1986, JACM.

[20]  Ryan O'Donnell,et al.  KKL, Kruskal-Katona, and Monotone Nets , 2009, 2009 50th Annual IEEE Symposium on Foundations of Computer Science.

[21]  Rocco A. Servedio,et al.  Optimal Cryptographic Hardness of Learning Monotone Functions , 2009, Theory Comput..

[22]  Tetsuro Nishino,et al.  More on the complexity of negation-limited circuits , 1995, STOC '95.

[23]  Nathan Linial,et al.  The influence of variables on Boolean functions , 1988, [Proceedings 1988] 29th Annual Symposium on Foundations of Computer Science.

[24]  Siyao Guo,et al.  Sparse extractor families for all the entropy , 2012, ITCS '13.

[25]  Shao Chin Sung,et al.  An exponential gap with the removal of one negation gate , 2002, Inf. Process. Lett..

[26]  Shao Chin Sung,et al.  Limiting Negations in Bounded-Depth Circuits: An Extension of Markov's Theorem , 2003, ISAAC.

[27]  Miklos Santha,et al.  Limiting Negations in Constant Depth Circuits , 1993, SIAM J. Comput..

[28]  C. Fortuin,et al.  Correlation inequalities on some partially ordered sets , 1971 .

[29]  Hiroki Morizumi.  Limiting Negations in Formulas , 2009, ICALP.

[30]  Akira Maruoka,et al.  A Superpolynomial Lower Bound for a Circuit Computing the Clique Function with At Most (1/6) log log n Negation Gates , 1998, MFCS.

[31]  Tetsuro Nishino,et al.  On the Complexity of Negation-Limited Boolean Networks , 1998, SIAM J. Comput..

[32]  ApplebaumBenny,et al.  Cryptography in $NC^0$ , 2006 .

[33]  Oded Goldreich,et al.  Monotone Circuits: One-Way Functions versus Pseudorandom Generators , 2012, Theory Comput..

[34]  Hiroki Morizumi.  Limiting negations in non-deterministic circuits , 2009, Theor. Comput. Sci..

[35]  Alexander Russell,et al.  Spectral Bounds on General Hard Core Predicates , 2000, STACS.

[36]  Éva Tardos,et al.  The gap between monotone and non-monotone circuit complexity is exponential , 1988, Comb..

[37]  Avi Wigderson,et al.  Monotone circuits for connectivity require super-logarithmic depth , 1990, STOC '88.

[38]  John Langford,et al.  On learning monotone Boolean functions , 1998, Proceedings 39th Annual Symposium on Foundations of Computer Science (Cat. No.98CB36280).

[39]  Stasys Jukna,et al.  Boolean Function Complexity Advances and Frontiers , 2012, Bull. EATCS.

[40]  Noga Alon,et al.  The Probabilistic Method , 2015, Fundamentals of Ramsey Theory.