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Hash Functions Based on Ramanujan Graphs

  • Hyungrok JoEmail author
Chapter
Part of the Mathematics for Industry book series (MFI, volume 29)

Abstract

Cayley hash functions are a family of cryptographic hash functions constructed from Cayley graphs, with appealing properties such as a natural parallelism and a security reduction to a clean, well-defined mathematical problem. As this problem involves non-Abelian groups, it is a priori resistant to quantum period finding algorithms and Cayley hash functions may therefore be a good foundation for post-quantum cryptography. Four particular parameter sets for Cayley hash functions have been proposed in the past, and so far dedicated preimage algorithms have been found for all of them. These algorithms do however not seem to extend to generic parameters, and as a result it is still an open problem to determine the security of Cayley hash functions in general. In this chapter, we introduce how to design hash functions based on Ramanujan graphs, which can be considered as an optimal expander graphs in a sense of qualities of transmission network schemes. We introduce a polynomial time preimage attack against Cayley hash functions based on two explicit Ramanujan graphs. We suggest some possible ways to construct the Cayley hash functions that may not be affected by this type of attacks as open problems, which can contribute to a better understanding of the hard problems underlying the security of Cayley hash functions.

Keywords

Expander graphs Ramanujan graphs LPS Ramanujan graphs Cubic Ramanujan graphs Cayley graphs Cayley hash functions Lifting attacks 

References

  1. 1.
    N. Alon, V. Milman, \(\lambda _1\), isoperimetric inequalities for graphs, and superconcentrators. J. Comb. Theory B 38(1), 73–88 (1985)CrossRefzbMATHGoogle Scholar
  2. 2.
    J. Basilla, On the solution of \(x^2+dy^2=m\). Proc. Jpn. Acad. A Math. 80(5), 40–41 (2004)MathSciNetCrossRefzbMATHGoogle Scholar
  3. 3.
    J.F. Biasse, D. Jao, A. Sankar, A quantum algorithm for computing isogenies between supersingular elliptic curves, in INDOCRYPT, LNCS, vol. 8885 (2014), pp. 428–442Google Scholar
  4. 4.
    D. Charles, K. Lauter, E. Goren, Cryptographic hash functions from expander graphs. J. Cryptol. 22(1), 93–113 (2009)MathSciNetCrossRefzbMATHGoogle Scholar
  5. 5.
    P. Chiu, Cubic Ramanujan graphs. Combinatorica 12(3), 275–285 (1992)MathSciNetCrossRefzbMATHGoogle Scholar
  6. 6.
    G. Davidoff, P. Sarnak, A. Valette, Elementary Number Theory, Group Theory and Ramanujan Graphs (Cambridge University Press, Cambridge, 2003)CrossRefzbMATHGoogle Scholar
  7. 7.
    L. De Feo, D. Jao, J. Plût, Towards quantum-resistant cryptosystems from supersingular elliptic curve isogenies. J. Math. Cryptol. 8(3), 209–247 (2014)MathSciNetzbMATHGoogle Scholar
  8. 8.
    J. Dodziuk, Difference equations, isoperimetric inequality and transience of certain random walks. Trans. Am. Math. Soc. 284(2), 787–794 (1984)MathSciNetCrossRefzbMATHGoogle Scholar
  9. 9.
    M. Eichler, The basis problem for modular forms and the traces of the Hecke operators, in Modular Functions of One Variable, vol. 320, ed. by W. Kuyk (Springer, Heidelberg, 1973), pp. 75–152CrossRefGoogle Scholar
  10. 10.
    M. Eichler, S. Sundaravaradan, Lectures on modular correspondences. Tata Institute of Fundamental Research (1956), http://www.math.tifr.res.in/~publ/ln/tifr09.pdf
  11. 11.
    O. Goldreich, Foundations of Cryptography (Cambridge University Press, Cambridge, 2004)CrossRefzbMATHGoogle Scholar
  12. 12.
    M. Hirschhorn, A simple proof of Jacobi’s four-square theorem. Proc. Am. Math. Soc. 101(3), 436–438 (1987)MathSciNetzbMATHGoogle Scholar
  13. 13.
    H. Hoory, N. Linial, A. Wigderson, Expander graphs and their applications. Bull. Am. Math. Soc. 43(4), 439–561 (2006)MathSciNetCrossRefzbMATHGoogle Scholar
  14. 14.
    T. Ibukiyama, On maximal orders of division quaternion algebras over the rational number field with certain optimal embeddings. Nagoya. Math. J. 88, 181–195 (1982)MathSciNetCrossRefzbMATHGoogle Scholar
  15. 15.
    A. Lubotzky, R. Phillips, P. Sarnak, Ramanujan graphs. Combinatorica 8(3), 261–277 (1988)MathSciNetCrossRefzbMATHGoogle Scholar
  16. 16.
    G. Margulis, Explicit group-theoretical constructions of combinatorial schemes and their application to the design of expanders and concentrators. Probl. Peredachi Inf. 24(1), 51–60 (1988)zbMATHGoogle Scholar
  17. 17.
    C. Petit, K. Lauter, J. Quisquater, Full cryptanalysis of LPS and Morgenstern hash functions, in SCN, LNCS, vol. 5229 (2008), pp. 263–277Google Scholar
  18. 18.
    A.K. Pizer, Ramanujan graphs and Hecke operators. Bull. Am. Math. Soc. 23(1), 127–137 (1990)MathSciNetCrossRefzbMATHGoogle Scholar
  19. 19.
    P. Sarnak, Some Applications of Modular Forms (Cambridge University Press, Cambridge, 1999)zbMATHGoogle Scholar
  20. 20.
    J. Tillich, G. Zèmor, Collisions for the LPS expander graph hash function, in EUROCRYPT, LNCS, vol. 3027 (2008), pp. 254–269Google Scholar

Copyright information

© Springer Nature Singapore Pte Ltd. 2018

Authors and Affiliations

  1. 1.Graduate School of MathematicsKyushu UniversityFukuokaJapan

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