We suggest a candidate one-way function using combinatorial constructs such as expander graphs. These graphs are used to determine a sequence of small overlapping subsets of input bits, to which a hard-wired random predicate is applied. Thus, the function is extremely easy to evaluate: All that is needed is to take multiple projections of the input bits, and to use these as entries to a look-up table. It is feasible for the adversary to scan the look-up table, but we believe it would be infeasible to find an input that fits a given sequence of values obtained for these overlapping projections.

The conjectured difficulty of inverting the suggested function does not seem to follow from any well-known assumption. Instead, we propose the study of the complexity of inverting this function as an interesting open problem, with the hope that further research will provide evidence to our belief that the inversion task is intractable.


One-Way Functions Expander Graphs 


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  1. 1.
    Applebaum, B., Barak, B., Wigderson, A.: Public-key cryptography from different assumptions. In: 42nd STOC, pp. 171–180 (2010)Google Scholar
  2. 2.
    Applebaum, B., Ishai, Y., Kushilevitz, E.: Cryptography in NC0. SICOMP 36(4), 845–888 (2006)CrossRefzbMATHGoogle Scholar
  3. 3.
    Alekhnovich, M., Ben-Sasson, E., Razborov, A., Wigderson, A.: Pseudorandom Generators in Propositional Proof Complexity. In: 41st FOCS, pp. 43–53 (2000)Google Scholar
  4. 4.
    Alon, N.: Eigenvalues, Geometric Expanders, Sorting in Rounds, and Ramsey Theory. Combinatorica 6, 207–219 (1986)MathSciNetCrossRefzbMATHGoogle Scholar
  5. 5.
    Alon, N., Milman, V.D.: λ 1, Isoperimetric Inequalities for Graphs and Superconcentrators. J. Combinatorial Theory, Ser. B 38, 73–88 (1985)MathSciNetCrossRefzbMATHGoogle Scholar
  6. 6.
    Bogdanov, A., Qiao, Y.: On the Security of Goldreich’s One-Way Function. In: Dinur, I., Jansen, K., Naor, J., Rolim, J. (eds.) APPROX 2009. LNCS, vol. 5687, pp. 392–405. Springer, Heidelberg (2009)CrossRefGoogle Scholar
  7. 7.
    Cook, J., Etesami, O., Miller, R., Trevisan, L.: Goldreich’s One-Way Function Candidate and Myopic Backtracking Algorithms. In: Reingold, O. (ed.) TCC 2009. LNCS, vol. 5444, pp. 521–538. Springer, Heidelberg (2009)CrossRefGoogle Scholar
  8. 8.
    Gaber, O., Galil, Z.: Explicit Constructions of Linear Size Superconcentrators. JCSS 22, 407–420 (1981)zbMATHGoogle Scholar
  9. 9.
    Goldreich, O., Goldwasser, S., Micali, S.: How to Construct Random Functions. JACM 33(4), 792–807 (1986)MathSciNetCrossRefzbMATHGoogle Scholar
  10. 10.
    Lubotzky, A., Phillips, R., Sarnak, P.: Ramanujan Graphs. Combinatorica 8, 261–277 (1988)MathSciNetCrossRefzbMATHGoogle Scholar
  11. 11.
    Nisan, N., Wigderson, A.: Hardness vs Randomness. JCSS 49(2), 149–167 (1994)MathSciNetzbMATHGoogle Scholar

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© Springer-Verlag Berlin Heidelberg 2011

Authors and Affiliations

  • Oded Goldreich

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