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On the Minimal Hardware Complexity of Pseudorandom Function Generators

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STACS 2001 (STACS 2001)

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Abstract

A set F of Boolean functions is called a pseudorandom function gen- erator(PRFG) if communicating with a randomly chosen secret function from F cannot be efficiently distinguished from communicating with a truly random function. We ask for the minimal hardware complexity of a PRFG. This ques- tion is motivated by design aspects of secure secret key cryptosystems. These should be efficient in hardware, but often are required to behave like PRFGs. By constructing efficient distinguishing schemes we show for a wide range of basic nonuniform complexity classes (including TC0 2 ), that they do not contain PRFGs. On the other hand we show that the PRFG proposed by Naor and Reingold in [24] consists of TC0 4 -functions. The question if TC0 3 -functions can form PRFGs re- mains as an interesting open problem. We further discuss relations of our results to previous work on cryptographic limitations of learning and Natural Proofs.

Supported by DFG grant Kr 1521/3-1.

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References

  1. N. Alon, J. Spencer, P. Erdös. The probabilistic method. Wiley & Sons 1992.

    Google Scholar 

  2. M. Bellare, S. Goldwasser. New paradigms for digital signatures and message authentication based on non-interactive zero knowledge proofs. Crypto’ 89, Springer LNCS, pp. 194–211.

    Google Scholar 

  3. M. Blaze, J. Feigenbaum, M. Naor. A Formal Treatment of Remotely Keyed Encryption. Eurocrypt’ 98, Springer LNCS, 1998.

    Google Scholar 

  4. A. Blum, M. Furst, M. Kearns, R.J. Lipton. Cryptographic primitives based on hard learning problems. Proc. CRYPTO 93, LNCS 773, 278–291.

    Google Scholar 

  5. A. Borodin, A. Razborov, R. Smolensky. On lower bounds for read k times branching programs. J. Computational Complexity 3, 1993, 1–13.

    Article  MATH  MathSciNet  Google Scholar 

  6. J. Bruck. Harmonic Analysis of polynomial threshold functions. SIAM Journal of Discrete Mathematics. 3:22, 1990, pp. 168–177.

    Article  MATH  MathSciNet  Google Scholar 

  7. O. Goldreich, S. Goldwasser, S. Micali. How to construct random functions. J. of the ACM, vol 33, pp. 792–807, 1986.

    Article  MathSciNet  Google Scholar 

  8. M. Goldmann, J. Hastad, A.A. Razborov. Majority gates versus general weighted Threshold gates. J. Computational Complexity 2, 1992, 277–300.

    Article  MATH  MathSciNet  Google Scholar 

  9. M. Goldmann, M. Karpinski. Simulating threshold circuits by majority circuits. Proc. 25th ACM Symp. on Theory of Computing (STOC),1993, 551–560.

    Google Scholar 

  10. A. Hajnal, W. Maass, P. Pudlak, M. Szegedy, G. Turan. Threshold circuits of bounded depth. FOCS’87, pp. 99–110.

    Google Scholar 

  11. J. Hastad. Almost optimal lower bounds for small depth circuits. STOC’86, pp. 6–20.

    Google Scholar 

  12. S. Jukna. A note on read-k time branching programs. Theoretical Informatics and Applications 29(1), 1995, 75–83.

    MATH  MathSciNet  Google Scholar 

  13. M. Kearns, L. Valiant. Cryptographic limitations on learning Boolean formulae and finite automata. J. of the ACM, vol. 41(1), 1994, pp. 67–95.

    Article  MATH  MathSciNet  Google Scholar 

  14. M. Krause, S. Lucks. On the minimal Hardware Complexity of Pseudorandom Function Generators. http://th.informatik.uni-mannheim.de/research/research.html.

  15. M. Krause, P. Pudlak. On the computational power of depth-2 circuits with threshold and modulo gates. J. Theoretical Computer Science 174, 1997, pp. 137–156. Prel. version in STOC’94, pp. 49-59.

    Article  MATH  MathSciNet  Google Scholar 

  16. M. Krause, P. Pudlak. Computing Boolean functions by polynomials and threshold circuits. J. Comput. complex. 7 (1998), pp. 346–370. Prel. version in FOCS’95, pp. 682-691.

    Article  MATH  MathSciNet  Google Scholar 

  17. M. Krause, P. Savicky, I. Wegener. Approximation by OBDDs, and the variable ordering problem. Lect. Notes Comp. Science 1644, Proc. of ICALP’99, pp. 493–502.

    Google Scholar 

  18. M. Krause, S. Waack. Variation ranks of communication matrices and lower bounds for depth two circuits having symmetric gates with unbounded fan-in. J. Mathematical System Theory 28, 1995, 553–564.

    Article  MathSciNet  Google Scholar 

  19. N. Linial, Y. Mansour, N. Nisan. Constant depth circuits, Fourier transform, and learnability. J. of the ACM, vol. 40(3), 1993, pp. 607–620. Prel. version in FOCS’89, pp. 574-579.

    Article  MATH  MathSciNet  Google Scholar 

  20. M. Luby, C. Rackoff. How to construct pseudorandom permutations from pseudorandom functions. SIAM J. Computing, Vol. 17, No.2, pp. 373–386, 1988.

    Article  MATH  MathSciNet  Google Scholar 

  21. S. Lucks. Faster Luby-Rackoff Ciphers. Fast Software Encryption 1996, Springer LNCS 1039, 189–203, 1996.

    Google Scholar 

  22. S. Lucks. On the Security of Remotely Keyed Encryption. Fast Software Encryption 1997, Springer LNCS 1267, 219–229, 1997.

    Chapter  Google Scholar 

  23. M. Naor, O. Reingold. Synthesizers and their application to the parallel construction of pseudo-random functions. Proc. 36th IEEE Symp. on Foundations of Computer Science, pp. 170–181, 1995.

    Google Scholar 

  24. M. Naor, O. Reingold. Number-theoretic constructions of efficient pseudo-random functions. Preliminary Version. Proc. 38th IEEE Symp. on Foundations of Computer Science, 1997.

    Google Scholar 

  25. M. Naor, O. Reingold. On the construction of pseudo-random permutations: Luby-Rackoff revisited. J. of Cryptology, Vol. 12, No1, 29–66, 1999.

    Article  MATH  MathSciNet  Google Scholar 

  26. E. Okolshnikova. On lower bounds for branching programs. Siberian Advances in Mathematics 3(1), 1993, 152–166.

    MathSciNet  Google Scholar 

  27. A. Razborov, S. Rudich. Natural Proofs. J. of Computer and System Science, vol. 55(1), 1997, pp. 24–35. Prel. version STOC’ 94, pp. 204-213.

    Article  MATH  MathSciNet  Google Scholar 

  28. K. Siu, J. Bruck, T. Kailath, T. Hofmeister. Depth efficient neural networks for division and related problems. IEEE Trans. of Inform. Theory, vol. 39, 1993, pp. 946–956

    Article  MATH  MathSciNet  Google Scholar 

  29. R. Smolensky. Algebraic methods in the theory of lower bounds for Boolean circuit complexity. STOC’87, pp. 77–82.

    Google Scholar 

  30. I. Wegener. The complexity of Boolean functions. John Wiley & Sons, 1987.

    Google Scholar 

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Authors and Affiliations

  1. Theoretische Informatik, Univ. Mannheim, 68131, Mannheim, Germany

    Matthias Krause & Stefan Lucks

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  1. Matthias Krause
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  2. Stefan Lucks
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Editors and Affiliations

  1. I3S & INRIA Sophia Antipolis INRIA, CNRS, 2004 Route des Lucioles, 06902, Sophia-Antipolis, France

    Afonso Ferreira

  2. Institut für Theoretische Informatik, Fakultät Informatik, TU Dresden, 01062, Dresden, Germany

    Horst Reichel

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Krause, M., Lucks, S. (2001). On the Minimal Hardware Complexity of Pseudorandom Function Generators. In: Ferreira, A., Reichel, H. (eds) STACS 2001. STACS 2001. Lecture Notes in Computer Science, vol 2010. Springer, Berlin, Heidelberg. https://doi.org/10.1007/3-540-44693-1_37

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  • DOI: https://doi.org/10.1007/3-540-44693-1_37

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