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Generalized Compact Knapsacks Are Collision Resistant

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Automata, Languages and Programming (ICALP 2006)

Part of the book series: Lecture Notes in Computer Science ((LNTCS,volume 4052))

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Abstract

In (Micciancio, FOCS 2002), it was proved that solving the generalized compact knapsack problem on the average is as hard as solving certain worst-case problems for cyclic lattices. This result immediately yielded very efficient one-way functions whose security was based on worst-case hardness assumptions. In this work, we show that, while the function proposed by Micciancio is not collision resistant, it can be easily modified to achieve collision resistance under essentially the same complexity assumptions on cyclic lattices. Our modified function is obtained as a special case of a more general result, which yields efficient collision-resistant hash functions based on the worst-case hardness of various new problems. These include new problems from algebraic number theory as well as classic lattice problems (e.g., the shortest vector problem) over ideal lattices, a class of lattices that includes cyclic lattices as a special case.

The full version of this extended abstract appears in ECCC TR05-142. Research supported by NSF CAREER 0093029 and NSF ITR 0313241.

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References

  1. Aharonov, D., Regev, O.: Lattice problems in NP ∩ coNP. Journal of the ACM 52(5), 749–765 (2005)

    Article  MathSciNet  Google Scholar 

  2. M. Ajtai. Generating hard instances of lattice problems. In STOC, pages 99–108, 1996.

    Google Scholar 

  3. Ajtai, M., Kumar, R., Sivakumar, D.: A sieve algorithm for the shortest lattice vector problem. In: STOC, pp. 601–610 (2001)

    Google Scholar 

  4. Biham, E., Chen, R., Joux, A., Carribault, P., Jalby, W., Lemuet, C.: Collisions of SHA-0 and Reduced SHA-1. In: EUROCRYPT (2005)

    Google Scholar 

  5. Cai, J., Nerurkar, A.: An improved worst-case to average-case connection for lattice problems. In: FOCS, pp. 468–477 (1997)

    Google Scholar 

  6. Chor, B., Rivest, R.L.: A knapsack type public-key cryptosystem based on arithmetic in finite fields. IEEE Trans. Inform. Theory 34(5), 901–909 (1988)

    Article  MathSciNet  Google Scholar 

  7. Damgard, I.: A design principle for hash functions. In: CRYPTO 1989, pp. 416–427 (1989)

    Google Scholar 

  8. Dinur, I.: Approximating SVP  ∞  to within almost-polynomial factors is NP-hard. Theor. Comput. Sci. 285(1), 55–71 (2002)

    Article  MATH  MathSciNet  Google Scholar 

  9. Goldreich, O., Goldwasser, S.: On the limits of nonapproximability of lattice problems. J. Comput. Syst. Sci. 60(3) (2000)

    Google Scholar 

  10. Hoffstein, J., Pipher, J., Silverman, J.H.: Ntru: A ring-based public key cryptosystem. In: ANTS, pp. 267–288 (1998)

    Google Scholar 

  11. Joux, A., Granboulan, L.: A practical attack against knapsack based hash functions. In: EUROCRYPT 1994, pp. 58–66 (1994)

    Google Scholar 

  12. Lenstra, A.K., Lenstra, J.H.W., Lovasz, L.: Factoring polynomials with rational coefficients. Mathematische Annalen 261, 513–534 (1982)

    Article  MathSciNet  Google Scholar 

  13. Merkle, R.C., Hellman, M.E.: Hiding information and signatures in trapdoor knapsacks. IEEE Transactions on Information Theory IT-24, 525–530 (1978)

    Article  Google Scholar 

  14. Micciancio, D.: Generalized compact knapsacks, cyclic lattices, and efficient one-way functions from worst-case complexity assumptions. In: Computational Complexity (to appear preliminary version in FOCS 2002).

    Google Scholar 

  15. Micciancio, D.: Almost perfect lattices, the covering radius problem, and applications to Ajtai’s connection factor. SIAM J. on Computing 34(1), 118–169 (2004)

    Article  MATH  MathSciNet  Google Scholar 

  16. Micciancio, D., Goldwasser, S.: Complexity Of Lattice Problems: A Cryptographic Perspective. Kluwer Academic Publishers, Dordrecht (2002)

    MATH  Google Scholar 

  17. Micciancio, D., Regev, O.: Worst-case to average-case reductions based on Gaussian measures. In: SIAM J. on Computing, (to appear preliminary version in FOCS 2004)

    Google Scholar 

  18. Peikert, C., Rosen, A.: Efficient Collision-Resistant Hashing from Worst-Case Assumptions on Cyclic Lattices. In: TCC (2006)

    Google Scholar 

  19. Prasolov, V.V.: Polynomials. Algorithms and Computation in Mathematics, vol. 11. Springer, Heidelberg (2004)

    MATH  Google Scholar 

  20. Schnorr, C.P.: A hierarchy of polynomial time basis reduction algorithms. Theoretical Computer Science 53, 201–224 (1987)

    Article  MATH  MathSciNet  Google Scholar 

  21. Shamir, A.: A polynomial time algorithm for breaking the basic Merkle-Hellman cryptosystem. IEEE Transactions on Information Theory IT-30(5), 699–704 (1984)

    Article  MathSciNet  Google Scholar 

  22. Vaudenay, S.: Cryptanalysis of the Chor–Rivest cryptosystem. Journal of Cryptology 14(2), 87–100 (2001)

    Article  MATH  MathSciNet  Google Scholar 

  23. Wang, X., Lai, X., Feng, D., Chen, H., Yu, X.: Cryptanalysis for Hash Functions MD4 and RIPEMD. In: EUROCRYPT (2005)

    Google Scholar 

  24. Wang, X., Yu, H.: How to Break MD5 and Other Hash Functions. In: EUROCRYPT (2005)

    Google Scholar 

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Authors and Affiliations

  1. University of California, San Diego, 9500 Gilman Drive, La Jolla, CA, 92093-0404, USA

    Vadim Lyubashevsky & Daniele Micciancio

Authors
  1. Vadim Lyubashevsky
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  2. Daniele Micciancio
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Editor information

Editors and Affiliations

  1. Dipartimento di Informatica, Università Ca’ Foscari, Venice,  

    Michele Bugliesi

  2. Interdisciplinary Institute for BroadBand Technology (IBBT), Belgium

    Bart Preneel

  3. ECS, University of Southampton, UK

    Vladimiro Sassone

  4. FB Informatik, LS2, Univ. Dortmund, 44221, Dortmund, Germany

    Ingo Wegener

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© 2006 Springer-Verlag Berlin Heidelberg

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Lyubashevsky, V., Micciancio, D. (2006). Generalized Compact Knapsacks Are Collision Resistant. In: Bugliesi, M., Preneel, B., Sassone, V., Wegener, I. (eds) Automata, Languages and Programming. ICALP 2006. Lecture Notes in Computer Science, vol 4052. Springer, Berlin, Heidelberg. https://doi.org/10.1007/11787006_13

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  • DOI: https://doi.org/10.1007/11787006_13

  • Publisher Name: Springer, Berlin, Heidelberg

  • Print ISBN: 978-3-540-35907-4

  • Online ISBN: 978-3-540-35908-1

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