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Cryptographic Hardness of Learning

1994; Kearns, Valiant

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Encyclopedia of Algorithms

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Keywords and Synonyms

Representation-independent hardness for learning        

Problem Definition

This paper deals with proving negative results for distribution‐free PAC learning. The crux of the problem is proving that a polynomial-time algorithm for learning various concept classes in the PAC model implies that several well-known cryptosystems are insecure. Thus, if we assume a particular cryptosystem is secure we can conclude that it is impossible to efficiently learn a corresponding set of concept classes.

PAC Learning

We recall here the PAC learning model. Let C be a concept class (a set of functions over n variables), and let D be a distribution over the input space \( { \{0,1\}^{n} } \). With C we associate a size function size that measures the complexity of each \( { c \in C } \). For example if C is a class of Boolean circuits then size(c) is equal to the number of gates in c. Let A be a randomized algorithm that has access to an oracle which returns labeled examples \( { (x,c(x)) }...

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Recommended Reading

  1. Alekhnovich, M., Braverman, M., Feldman, V., Klivans, A. R., Pitassi, T.: Learnability and automatizability. In: Proceedings of the 45th Symposium on Foundations of Computer Science, 2004

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  2. Angluin, D., Kharitonov, M.: When Won't Membership Queries Help? J. Comput. Syst. Sci. 50, (1995)

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  3. Goldreich, O.: Foundations of Cryptography: Basic Tools. Cambridge University Press (2001)

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  4. Kearns, M., Valiant, L.: Cryptographic limitations on learning Boolean formulae and finite automata. J. ACM 41(1), 67–95 (1994)

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  5. Kearns, M., Vazirani, U.: An introduction to computational learning theory. MIT Press, Cambridge (1994)

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  6. Kharitonov, M.: Cryptographic hardness of distribution‐specific learning. In: Proceedings of the Twenty-Fifth Annual Symposium on Theory of Computing, 1993, pp. 372–381

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  7. Klivans, A. , Sherstov, A. A.: Cryptographic Hardness for Learning Intersections of Halfspaces. In: Proceedings of the 47th Symposium on Foundations of Computer Science, 2006

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  8. Klivans, A., Servedio, R.: Learning DNF in time \( { 2^{\tilde{O}(n^{1/3})} } \). In: Proceedings of the 33rd Annual Symposium on Theory of Computing, 2001

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  9. Regev, O.: New Lattice-Based Cryptographic Constructions. J. ACM 51, 899–942 (2004)

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  10. Valiant, L.: A theory of the learnable. Commun. ACM 27(11), 1134–1142 (1984)

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Author information

Authors and Affiliations

  1. Department of Computer Science, University of Texas at Austin, Austin, TX, USA

    Adam Klivans

Authors
  1. Adam Klivans
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Editor information

Editors and Affiliations

  1. Department of Electrical Engineering and Computer ScienceMcCormick School of Engineering and Applied Science, Northwestern University, Evanston, IL, 60208, USA

    Ming-Yang Kao Professor of Computer Science

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© 2008 Springer-Verlag

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Klivans, A. (2008). Cryptographic Hardness of Learning. In: Kao, MY. (eds) Encyclopedia of Algorithms. Springer, Boston, MA. https://doi.org/10.1007/978-0-387-30162-4_96

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