DDH-Like Assumptions Based on Extension Rings

  • Ronald Cramer
  • Ivan Damgård
  • Eike Kiltz
  • Sarah Zakarias
  • Angela Zottarel
Part of the Lecture Notes in Computer Science book series (LNCS, volume 7293)

Abstract

We introduce and study a new type of DDH-like assumptions based on groups of prime order q. Whereas standard DDH is based on encoding elements of \(\mathbb{F}_{q}\) “in the exponent” of elements in the group, we ask what happens if instead we put in the exponent elements of the extension ring \(R_f= \mathbb{F}_{q}[X]/(f)\) where f is a degree-d polynomial. The decision problem that follows naturally reduces to the case where f is irreducible. This variant is called the d-DDH problem, where 1-DDH is standard DDH. We show in the generic group model that d-DDH is harder than DDH for d > 1 and that we obtain, in fact, an infinite hierarchy of progressively weaker assumptions whose complexities lie “between” DDH and CDH. This leads to a large number of new schemes because virtually all known DDH-based constructions can very easily be upgraded to be based on d-DDH. We use the same construction and security proof but get better security and moreover, the amortized complexity (e.g, computation per encrypted bit) is the same as when using DDH. We also show that d-DDH, just like DDH, is easy in bilinear groups. We therefore suggest a different type of assumption, the d-vector DDH problems (d-VDDH), which are based on f(X) = X d , but with a twist to avoid problems with reducible polynomials. We show in the generic group model that d-VDDH is hard in bilinear groups and that the problems become harder with increasing d. We show that hardness of d-VDDH implies CCA-secure encryption, efficient Naor-Reingold style pseudorandom functions, and auxiliary input secure encryption. This can be seen as an alternative to the known family of k-LIN assumptions.

Keywords

Encryption Scheme Random Oracle Extension Ring Pseudorandom Function Auxiliary Input 
These keywords were added by machine and not by the authors. This process is experimental and the keywords may be updated as the learning algorithm improves.

References

  1. [BB04]
    Boneh, D., Boyen, X.: Efficient Selective-ID Secure Identity-Based Encryption Without Random Oracles. In: Cachin, C., Camenisch, J.L. (eds.) EUROCRYPT 2004. LNCS, vol. 3027, pp. 223–238. Springer, Heidelberg (2004)CrossRefGoogle Scholar
  2. [BBS04]
    Boneh, D., Boyen, X., Shacham, H.: Short Group Signatures. In: Franklin, M. (ed.) CRYPTO 2004. LNCS, vol. 3152, pp. 41–55. Springer, Heidelberg (2004)Google Scholar
  3. [BHHO08]
    Boneh, D., Halevi, S., Hamburg, M., Ostrovsky, R.: Circular-Secure Encryption from Decision Diffie-Hellman. In: Wagner, D. (ed.) CRYPTO 2008. LNCS, vol. 5157, pp. 108–125. Springer, Heidelberg (2008)Google Scholar
  4. [BMR10]
    Boneh, D., Montgomery, H.W., Raghunathan, A.: Algebraic pseudorandom functions with improved efficiency from the augmented cascade. In: Al-Shaer, E., Keromytis, A.D., Shmatikov, V. (eds.) ACM Conference on Computer and Communications Security, pp. 131–140. ACM (2010)Google Scholar
  5. [CDK+11]
    Cramer, R., Damgaard, I., Kiltz, E., Zakarias, S., Zottarel, A.: Ddh-like assumptions based on extension rings. Cryptology ePrint Archive, Report 2011/280 (2011), http://eprint.iacr.org/
  6. [CS98]
    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)Google Scholar
  7. [DGK+10]
    Dodis, Y., Goldwasser, S., Kalai, Y.T., Peikert, C., Vaikuntanathan, V.: Public-Key Encryption Schemes with Auxiliary Inputs. In: Micciancio, D. (ed.) TCC 2010. LNCS, vol. 5978, pp. 361–381. Springer, Heidelberg (2010)CrossRefGoogle Scholar
  8. [DH76]
    Diffie, W., Hellman, M.E.: New Directions in Cryptography. IEEE Transactions on Information Theory IT-22(6), 644–654 (1976)MathSciNetCrossRefGoogle Scholar
  9. [DY05]
    Dodis, Y., Yampolskiy, A.: A Verifiable Random Function with Short Proofs and Keys. In: Vaudenay, S. (ed.) PKC 2005. LNCS, vol. 3386, pp. 416–431. Springer, Heidelberg (2005)CrossRefGoogle Scholar
  10. [Gam84]
    El Gamal, T.: A Public Key Cryptosystem and a Signature Scheme Based on Discrete Logarithms. In: Blakely, G.R., Chaum, D. (eds.) CRYPTO 1984. LNCS, vol. 196, pp. 10–18. Springer, Heidelberg (1985)CrossRefGoogle Scholar
  11. [HK07]
    Hofheinz, D., Kiltz, E.: Secure Hybrid Encryption from Weakened Key Encapsulation. In: Menezes, A. (ed.) CRYPTO 2007. LNCS, vol. 4622, pp. 553–571. Springer, Heidelberg (2007)CrossRefGoogle Scholar
  12. [HYZX08]
    Huang, H., Yang, B., Zhu, S., Xiao, G.: Generalized ElGamal Public Key Cryptosystem Based on a New Diffie-Hellman Problem. In: Baek, J., Bao, F., Chen, K., Lai, X. (eds.) ProvSec 2008. LNCS, vol. 5324, pp. 1–21. Springer, Heidelberg (2008)CrossRefGoogle Scholar
  13. [Kil07]
    Kiltz, E.: Chosen-Ciphertext Secure Key-Encapsulation Based on Gap Hashed Diffie-Hellman. In: Okamoto, T., Wang, X. (eds.) PKC 2007. LNCS, vol. 4450, pp. 282–297. Springer, Heidelberg (2007)CrossRefGoogle Scholar
  14. [NR97]
    Naor, M., Reingold, O.: Number-theoretic constructions of efficient pseudo-random functions. In: FOCS, pp. 458–467 (1997)Google Scholar
  15. [Pip76]
    Pippenger, N.: On the evaluation of powers and related problems (preliminary version). In: FOCS, pp. 258–263 (1976)Google Scholar
  16. [Sch80]
    Schwartz, J.T.: Fast probabilistic algorithms for verification of polynomial identities. J. ACM 27, 701–717 (1980)MATHCrossRefGoogle Scholar
  17. [Sha07]
    Shacham, H.: A Cramer-Shoup encryption scheme from the Linear Assumption and from progressively weaker Linear variants. Cryptology ePrint Archive, Report 2007/074 (February 2007), http://eprint.iacr.org/
  18. [Zip79]
    Zippel, R.: Probabilistic Algorithms for Sparse Polynomials. In: Ng, K.W. (ed.) EUROSAM 1979 and ISSAC 1979. LNCS, vol. 72, pp. 216–226. Springer, Heidelberg (1979)CrossRefGoogle Scholar

Copyright information

© International Association for Cryptologic Research 2012

Authors and Affiliations

  • Ronald Cramer
    • 1
  • Ivan Damgård
    • 2
  • Eike Kiltz
    • 3
  • Sarah Zakarias
    • 2
  • Angela Zottarel
    • 2
  1. 1.CWI and Leiden UniversityThe Netherlands
  2. 2.Aarhus UniversityDenmark
  3. 3.RU BochumGermany

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