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Exploring Naccache-Stern Knapsack Encryption

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Innovative Security Solutions for Information Technology and Communications (SecITC 2017)

Part of the book series: Lecture Notes in Computer Science ((LNSC,volume 10543))

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Abstract

The Naccache-Stern public-key cryptosystem (NS) relies on the conjectured hardness of the modular multiplicative knapsack problem: Given \(p,\{v_i\},\prod v_i^{m_i} \bmod p\), find the \(\{m_i\}\).

Given this scheme’s algebraic structure it is interesting to systematically explore its variants and generalizations. In particular it might be useful to enhance NS with features such as semantic security, re-randomizability or an extension to higher-residues.

This paper addresses these questions and proposes several such variants.

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Notes

  1. 1.

    p is usually prime but nothing prevents extending the problem to composite RSA moduli.

  2. 2.

    This can also be described as a modular variant of the “subset product” problem.

  3. 3.

    Alternatively, we can regard \(\mathsf {Setup}\) as a pro forma empty algorithm.

  4. 4.

    Note that this is obviously not be an issue with the original NS scheme.

References

  1. Adleman, L.M.: On breaking the iterated Merkle-Hellman public-key cryptosystem. In: Chaum, D., Rivest, R.L., Sherman, A.T. (eds.) Advances in Cryptology - CRYPTO 1982, pp. 303–308. Plenum Press, New York (1982)

    Google Scholar 

  2. Boneh, D., Goh, E.-J., Nissim, K.: Evaluating 2-DNF formulas on ciphertexts. In: Kilian, J. (ed.) TCC 2005. LNCS, vol. 3378, pp. 325–341. Springer, Heidelberg (2005). doi:10.1007/978-3-540-30576-7_18

    Chapter  Google Scholar 

  3. Brickell, E.F.: Breaking iterated Knapsacks. In: Blakley, G.R., Chaum, D. (eds.) CRYPTO 1984. LNCS, vol. 196, pp. 342–358. Springer, Heidelberg (1985). doi:10.1007/3-540-39568-7_27

    Chapter  Google Scholar 

  4. Canetti, R., Krawczyk, H., Nielsen, J.B.: Relaxing chosen-ciphertext security. In: Boneh, D. (ed.) CRYPTO 2003. LNCS, vol. 2729, pp. 565–582. Springer, Heidelberg (2003). doi:10.1007/978-3-540-45146-4_33

    Chapter  Google Scholar 

  5. Chee, Y.M., Joux, A., Stern, J.: The cryptanalysis of a new public-key cryptosystem based on modular Knapsacks. In: Feigenbaum, J. (ed.) CRYPTO 1991. LNCS, vol. 576, pp. 204–212. Springer, Heidelberg (1992). doi:10.1007/3-540-46766-1_15

    Google Scholar 

  6. Chevallier-Mames, B., Naccache, D., Stern, J.: Linear bandwidth Naccache-Stern encryption. In: Ostrovsky, R., De Prisco, R., Visconti, I. (eds.) SCN 2008. LNCS, vol. 5229, pp. 327–339. Springer, Heidelberg (2008). doi:10.1007/978-3-540-85855-3_22

    Chapter  Google Scholar 

  7. Cramer, R., Shoup, V.: A practical public key cryptosystem provably secure against adaptive chosen ciphertext attack. In: Krawczyk, H. (ed.) CRYPTO 1998. LNCS, vol. 1462, pp. 13–25. Springer, Heidelberg (1998). doi:10.1007/BFb0055717

    Google Scholar 

  8. Fujisaki, E., Okamoto, T., Pointcheval, D., Stern, J.: RSA-OAEP is secure under the RSA assumption. J. Cryptology 17(2), 81–104 (2004)

    Article  MATH  MathSciNet  Google Scholar 

  9. Goldwasser, S., Micali, S.: Probabilistic encryption and how to play mental poker keeping secret all partial information. In: Lewis, H.R., Simons, B.B., Burkhard, W.A., Landweber, L.H. (eds.) Proceedings of the 14th Annual ACM Symposium on Theory of Computing, 5–7 May 1982, San Francisco, California, USA, pp. 365–377. ACM (1982)

    Google Scholar 

  10. Groth, J.: Rerandomizable and replayable adaptive chosen ciphertext attack secure cryptosystems. In: Naor, M. (ed.) TCC 2004. LNCS, vol. 2951, pp. 152–170. Springer, Heidelberg (2004). doi:10.1007/978-3-540-24638-1_9

    Chapter  Google Scholar 

  11. Herold, G., Meurer, A.: New attacks for Knapsack based cryptosystems. In: Visconti, I., De Prisco, R. (eds.) SCN 2012. LNCS, vol. 7485, pp. 326–342. Springer, Heidelberg (2012). doi:10.1007/978-3-642-32928-9_18

    Chapter  Google Scholar 

  12. Joux, A., Stern, J.: Cryptanalysis of another Knapsack cryptosystem. In: Imai, H., Rivest, R.L., Matsumoto, T. (eds.) ASIACRYPT 1991. LNCS, vol. 739, pp. 470–476. Springer, Heidelberg (1993). doi:10.1007/3-540-57332-1_40

    Google Scholar 

  13. Lenstra, H.W.: On the Chor-Rivest Knapsack cryptosystem. J. Cryptology 3(3), 149–155 (1991)

    Article  MATH  MathSciNet  Google Scholar 

  14. Monier, L.: Evaluation and comparison of two efficient probabilistic primality testing algorithms. Theoret. Comput. Sci. 12(1), 97–108 (1980)

    Article  MATH  MathSciNet  Google Scholar 

  15. Naccache, D., Stern, J.: A new public-key cryptosystem. In: Fumy, W. (ed.) EUROCRYPT 1997. LNCS, vol. 1233, pp. 27–36. Springer, Heidelberg (1997). doi:10.1007/3-540-69053-0_3

    Google Scholar 

  16. Prabhakaran, M., Rosulek, M.: Rerandomizable RCCA encryption. In: Menezes, A. (ed.) CRYPTO 2007. LNCS, vol. 4622, pp. 517–534. Springer, Heidelberg (2007). doi:10.1007/978-3-540-74143-5_29

    Chapter  Google Scholar 

  17. Rabin, M.O.: Probabilistic algorithm for testing primality. J. Number Theory 12(1), 128–138 (1980)

    Article  MATH  MathSciNet  Google Scholar 

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Authors and Affiliations

  1. Ingenico Terminals, 9 Avenue de la Gare, 26300, Alixan, France

    Éric Brier

  2. École Normale Supérieure, 45 Rue d’Ulm, 75230, Paris Cedex 05, France

    Rémi Géraud & David Naccache

Authors
  1. Éric Brier
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  2. Rémi Géraud
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  3. David Naccache
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Correspondence to David Naccache .

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Editors and Affiliations

  1. École Normale Supérieure, Paris, France

    Pooya Farshim

  2. Polytechnic University of Bucharest, Bucharest, Romania

    Emil Simion

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Brier, É., Géraud, R., Naccache, D. (2017). Exploring Naccache-Stern Knapsack Encryption. In: Farshim, P., Simion, E. (eds) Innovative Security Solutions for Information Technology and Communications. SecITC 2017. Lecture Notes in Computer Science(), vol 10543. Springer, Cham. https://doi.org/10.1007/978-3-319-69284-5_6

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  • DOI: https://doi.org/10.1007/978-3-319-69284-5_6

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  • Online ISBN: 978-3-319-69284-5

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