Composition of Random Systems: When Two Weak Make One Strong

  • Ueli Maurer
  • Krzysztof Pietrzak
Part of the Lecture Notes in Computer Science book series (LNCS, volume 2951)


A new technique for proving the adaptive indistinguishability of two systems, each composed of some component systems, is presented, using only the fact that corresponding component systems are non-adaptively indistinguishable. The main tool is the definition of a special monotone condition for a random system F, relative to another random system G, whose probability of occurring for a given distinguisher D is closely related to the distinguishing advantage ε of D for F and G, namely it is lower and upper bounded by ε and \(\epsilon(1 + {\rm ln}\frac{1}{\epsilon})\), respectively.

A concrete instantiation of this result shows that the cascade of two random permutations (with the second one inverted) is indistinguishable from a uniform random permutation by adaptive distinguishers which may query the system from both sides, assuming the components’ security only against non-adaptive one-sided distinguishers.

As applications we provide some results in various fields as almost k-wise independent probability spaces, decorrelation theory and computational indistinguishability (i.e., pseudo-randomness).


  1. 1.
    Alon, N., Goldreich, O., Hastad, J., Peralta, R.: Simple construction of almost k-wise independent random variables. Random Structures and Algorithms 3(3), 289–304 (1992)MATHCrossRefMathSciNetGoogle Scholar
  2. 2.
    Luby, M., Rackoff, C.: How to construct pseudo-random permutations from pseudo-random functions. SIAM J. on Computing 17(2), 373–386 (1988)MATHCrossRefMathSciNetGoogle Scholar
  3. 3.
    Maurer, U.: Indistinguishability of random systems. In: Knudsen, L.R. (ed.) EUROCRYPT 2002. LNCS, vol. 2332, pp. 110–132. Springer, Heidelberg (2002)CrossRefGoogle Scholar
  4. 4.
    Maurer, U., Pietrzak, K.: The security of many-round Luby-Rackoff pseudorandom permutations. In: Biham, E. (ed.) EUROCRYPT 2003. LNCS, vol. 2656, pp. 544–561. Springer, Heidelberg (2003)CrossRefGoogle Scholar
  5. 5.
    Naor, J., Naor, M.: Small-bias probability spaces: Efficient constructions and applications. SIAM Journal on Computing 22(4), 838–356 (1993)Google Scholar
  6. 6.
    Renner, R.: The Statistical Distance of Independently Repeated Experiments (manuscript), available at:
  7. 7.
    Vaudenay, S.: Provable security for block ciphers by decorrelation. In: Meinel, C., Morvan, M. (eds.) STACS 1998. LNCS, vol. 1373, pp. 249–275. Springer, Heidelberg (1998)CrossRefGoogle Scholar
  8. 8.
    Vaudenay, S.: Adaptive-attack norm for decorrelation and super-pseudorandomness. In: Heys, H.M., Adams, C.M. (eds.) SAC 1999. LNCS, vol. 1758, pp. 49–61. Springer, Heidelberg (2000)CrossRefGoogle Scholar

Copyright information

© Springer-Verlag Berlin Heidelberg 2004

Authors and Affiliations

  • Ueli Maurer
    • 1
  • Krzysztof Pietrzak
    • 1
  1. 1.Department of Computer ScienceETH Zürich 

Personalised recommendations