Advertisement

Range Extension for Weak PRFs; The Good, the Bad, and the Ugly

  • Krzysztof Pietrzak
  • Johan Sjödin
Part of the Lecture Notes in Computer Science book series (LNCS, volume 4515)

Abstract

We investigate a general class of (black-box) constructions for range extension of weak pseudorandom functions: a construction based on m independent functions F 1,...,F m is given by a set of strings over {1,...,m}*, where for example {〈2〉, 〈1,2〉} corresponds to the function X ↦[F 2(X),F 2(F 1(X))]. All efficient constructions for range expansion of weak pseudorandom functions that we are aware of are of this form.

We completely classify such constructions as good, bad or ugly, where the good constructions are those whose security can be proven via a black-box reduction, the bad constructions are those whose insecurity can be proven via a black-box reduction, and the ugly constructions are those which are neither good nor bad.

Our classification shows that the range expansion from [10] is optimal, in the sense that it achieves the best possible expansion (2 m  − 1 when using m keys).

Along the way we show that for weak quasirandom functions (i.e. in the information theoretic setting), all constructions which are not bad – in particular all the ugly ones – are secure.

References

  1. 1.
    Bellare, M., Rogaway, P.: The security of triple encryption and a framework for code-based game-playing proofs. In: Vaudenay, S. (ed.) EUROCRYPT 2006. LNCS, vol. 4004, pp. 409–426. Springer, Heidelberg (2006)CrossRefGoogle Scholar
  2. 2.
    Damgård, I.B., Nielsen, J.B.: Expanding pseudorandom functions; or: From known-plaintext security to chosen-plaintext security. In: Yung, M. (ed.) CRYPTO 2002. LNCS, vol. 2442, pp. 449–464. Springer, Heidelberg (2002)CrossRefGoogle Scholar
  3. 3.
    Goldreich, O., Goldwasser, S., Micali, S.: How to construct random functions. J. ACM 33(4), 792–807 (1986)CrossRefMathSciNetGoogle Scholar
  4. 4.
    Håstad, J., Impagliazzo, R., Levin, L.A., Luby, M.: A pseudorandom generator from any one-way function. SIAM J. Comput. 28(4), 1364–1396 (1999)MATHCrossRefMathSciNetGoogle Scholar
  5. 5.
    Impagliazzo, R., Rudich, S.: Limits on the provable consequences of one-way permutations. In: Proc. 21th ACM Symposium on the Theory of Computing (STOC), pp. 44–61 (1989)Google Scholar
  6. 6.
    Luby, M., Rackoff, C.: Pseudo-random permutation generators and cryptographic composition. In: Proc. 18th ACM Symposium on the Theory of Computing (STOC), pp. 356–363 (1986)Google Scholar
  7. 7.
    Maurer, U.: Indistinguishability of random systems. In: Knudsen, L.R. (ed.) EUROCRYPT 2002. LNCS, vol. 2332, pp. 110–132. Springer, Heidelberg (2002)CrossRefGoogle Scholar
  8. 8.
    Maurer, U., Pietrzak, K.: Composition of random systems: When two weak make one strong. In: Naor, M. (ed.) TCC 2004. LNCS, vol. 2951, pp. 410–427. Springer, Heidelberg (2004)CrossRefGoogle Scholar
  9. 9.
    Maurer, U., Pietrzak, K., Renner, R.: Indistinguishability amplification. Cryptology ePrint Archive: Report 2006/456 (2006)Google Scholar
  10. 10.
    Maurer, U., Sjödin, J.: A fast and key-efficient reduction of chosen-ciphertext to known-plaintext security. In: Advances in Cryptology — EUROCRYPT ’07, this proceedings (2007)Google Scholar
  11. 11.
    Minematsu, K., Tsunoo, Y.: Expanding weak PRF with small key size. In: Won, D.H., Kim, S. (eds.) ICISC 2005. LNCS, vol. 3935, pp. 284–298. Springer, Heidelberg (2006)CrossRefGoogle Scholar
  12. 12.
    Naor, M., Reingold, O.: From unpredictability to indistinguishability: A simple construction of pseudo-random functions from MACs. In: Krawczyk, H. (ed.) CRYPTO 1998. LNCS, vol. 1462, pp. 267–282. Springer, Heidelberg (1998)Google Scholar
  13. 13.
    Pietrzak, K., Sjödin, J.: Weak pseudorandom functions in minicrypt. Manuscript (November 2006)Google Scholar

Copyright information

© Springer-Verlag Berlin Heidelberg 2007

Authors and Affiliations

  • Krzysztof Pietrzak
    • 1
  • Johan Sjödin
    • 2
  1. 1.CWIAmsterdam
  2. 2.Department of Computer ScienceETH ZurichZurichSwitzerland

Personalised recommendations