Advertisement

Journal of Cryptology

, Volume 26, Issue 1, pp 39–74 | Cite as

More Constructions of Lossy and Correlation-Secure Trapdoor Functions

  • David Mandell Freeman
  • Oded Goldreich
  • Eike Kiltz
  • Alon RosenEmail author
  • Gil Segev
Article

Abstract

We propose new and improved instantiations of lossy trapdoor functions (Peikert and Waters in STOC’08, pp. 187–196, 2008), and correlation-secure trapdoor functions (Rosen and Segev in TCC’09, LNCS, vol. 5444, pp. 419–436, 2009). Our constructions widen the set of number-theoretic assumptions upon which these primitives can be based, and are summarized as follows:
  • Lossy trapdoor functions based on the quadratic residuosity assumption. Our construction relies on modular squaring, and whereas previous such constructions were based on seemingly stronger assumptions, we present the first construction that is based solely on the quadratic residuosity assumption. We also present a generalization to higher-order power residues.

  • Lossy trapdoor functions based on the composite residuosity assumption. Our construction guarantees essentially any required amount of lossiness, where at the same time the functions are more efficient than the matrix-based approach of Peikert and Waters.

  • Lossy trapdoor functions based on the d-Linear assumption. Our construction both simplifies the DDH-based construction of Peikert and Waters and admits a generalization to the whole family of d-Linear assumptions without any loss of efficiency.

  • Correlation-secure trapdoor functions related to the hardness of syndrome decoding.

Key words

Public-key encryption Lossy trapdoor functions Correlation-secure trapdoor functions 

Preview

Unable to display preview. Download preview PDF.

Unable to display preview. Download preview PDF.

References

  1. [1]
    M. Bellare, Z. Brakerski, M. Naor, T. Ristenpart, G. Segev, H. Shacham, S. Yilek, Hedged public-key encryption: How to protect against bad randomness, in Advances in Cryptology—ASIACRYPT 2009. LNCS, vol. 5912 (Springer, Berlin, 2009), pp. 232–249 CrossRefGoogle Scholar
  2. [2]
    M. Bellare, D. Hofheinz, S. Yilek, Possibility and impossibility results for encryption and commitment secure under selective opening, in Advances in Cryptology—EUROCRYPT 2009. LNCS, vol. 5479 (Springer, Berlin, 2009), pp. 1–35 CrossRefGoogle Scholar
  3. [3]
    D.J. Bernstein, List decoding for binary goppa codes, in International Workshop on Coding and Cryptology—IWCC 2011. LNCS, vol. 6639 (Springer, Berlin, 2011), pp. 62–80 Google Scholar
  4. [4]
    M. Blum, P. Feldman, S. Micali, Non-interactive zero-knowledge and its applications, in Proceedings of the 20th Annual ACM Symposium on Theory of Computing (1988), pp. 103–112 Google Scholar
  5. [5]
    A. Boldyreva, S. Fehr, A. O’Neill, On notions of security for deterministic encryption, and efficient constructions without random oracles, in Advances in Cryptology—CRYPTO 2008. LNCS, vol. 5157 (Springer, Berlin, 2008), pp. 335–359 CrossRefGoogle Scholar
  6. [6]
    D. Boneh, J. Horwitz, Weak trapdoors from the rth-power-residue symbol. Unpublished manuscript (2002) Google Scholar
  7. [7]
    D. Boneh, X. Boyen, H. Shacham, Short group signatures, in Advances in Cryptology—CRYPTO 2004. LNCS, vol. 3152 (Springer, Berlin, 2004), pp. 41–55 CrossRefGoogle Scholar
  8. [8]
    D. Boneh, S. Halevi, M. Hamburg, R. Ostrovsky, Circular-secure encryption from decision Diffie-Hellman, in Advances in Cryptology—CRYPTO 2008. LNCS, vol. 5157 (Springer, Berlin, 2008), pp. 108–125 CrossRefGoogle Scholar
  9. [9]
    D. Boneh, K. Rubin, A. Silverberg, Finding composite order ordinary elliptic curves using the Cocks-Pinch method. J. Number Theory 131, 832–841 (2011) MathSciNetzbMATHCrossRefGoogle Scholar
  10. [10]
    C. Cachin, S. Micali, M. Stadler, Computationally private information retrieval with polylogarithmic communication, in Advances in Cryptology—EUROCRYPT 1999. LNCS, vol. 1592 (Springer, Berlin, 1999), pp. 402–414 Google Scholar
  11. [11]
    H. Cohen, A Course in Computational Algebraic Number Theory, Graduate Texts in Mathematics, vol. 138 (Springer, Berlin, 1993) zbMATHGoogle Scholar
  12. [12]
    D. Coppersmith, Small solutions to polynomial equations, and low exponent RSA vulnerabilities. J. Cryptol. 10(4), 233–260 (1997) MathSciNetzbMATHCrossRefGoogle Scholar
  13. [13]
    R. Cramer, V. Shoup, Universal hash proofs and a paradigm for adaptive chosen ciphertext secure public-key encryption, in Advances in Cryptology—EUROCRYPT 2002 (2002), pp. 45–64 CrossRefGoogle Scholar
  14. [14]
    I. Damgård, M. Jurik, A generalisation, a simplification and some applications of Paillier’s probabilistic public-key system, in Public Key Cryptography—PKC 2001. LNCS, vol. 1992 (Springer, Berlin, 2001), pp. 119–136. Full version (with additional co-author J.B. Nielsen) available at http://www.daimi.au.dk/~ivan/GenPaillier_finaljour.ps CrossRefGoogle Scholar
  15. [15]
    I. Damgård, J.B. Nielsen, Perfect hiding and perfect binding universally composable commitment schemes with constant expansion factor, in Advances in Cryptology—CRYPTO 2002. LNCS, vol. 2442 (Springer, Berlin, 2002), pp. 581–596 CrossRefGoogle Scholar
  16. [16]
    I. Damgård, J.B. Nielsen, Universally composable efficient multiparty computation from threshold homomorphic encryption, in Advances in Cryptology—CRYPTO 2003. LNCS, vol. 2729 (Springer, Berlin, 2003), pp. 247–264 CrossRefGoogle Scholar
  17. [17]
    R. Dowsley, J. Müller-Quade, A.C.A. Nascimento, A CCA2 secure public key encryption scheme based on the McEliece assumptions in the standard model, in Topics in Cryptology—CT-RSA 2009. LNCS, vol. 5473 (Springer, Berlin, 2009), pp. 240–251 CrossRefGoogle Scholar
  18. [18]
    J.-B. Fischer, J. Stern, An efficient pseudo-random generator provably as secure as syndrome decoding, in Advances in Cryptology—EUROCRYPT 1996. LNCS, vol. 1070 (Springer, Berlin, 1996), pp. 245–255 Google Scholar
  19. [19]
    O. Goldreich, Foundations of Cryptography II: Basic Applications (Cambridge University Press, Cambridge, 2004) zbMATHGoogle Scholar
  20. [20]
    S. Goldwasser, V. Vaikuntanathan, New constructions of correlation-secure trapdoor functions and CCA-secure encryption schemes. Manuscript (2008) Google Scholar
  21. [21]
    V.D. Goppa, A new class of linear correcting codes. Probl. Inf. Transm. 6(3), 207–212 (1970) MathSciNetGoogle Scholar
  22. [22]
    V.D. Goppa, Rational representation of codes and (L,g)-codes. Probl. Inf. Transm. 7(3), 223–229 (1971) MathSciNetGoogle Scholar
  23. [23]
    B. Hemenway, R. Ostrovsky, Lossy trapdoor functions from smooth homomorphic hash proof systems. Electronic Colloquium on Computational Complexity, Report TR09-127 (2009) Google Scholar
  24. [24]
    D. Hofheinz, E. Kiltz, Secure hybrid encryption from weakened key encapsulation, in Advances in Cryptology—CRYPTO 2007. LNCS, vol. 4622 (Springer, Berlin, 2007), pp. 553–571 CrossRefGoogle Scholar
  25. [25]
    J. Horwitz, Applications of Cayley graphs, bilinearity, and higher-order residues to cryptology. Ph.D. thesis, Stanford University (2004). Available at http://math.scu.edu/~jhorwitz/pubs/horwitz-phd.pdf
  26. [26]
    K. Ireland, M. Rosen, A Classical Introduction to Modern Number Theory, 2nd edn. Graduate Texts in Mathematics, vol. 84 (Springer, New York, 1990) zbMATHGoogle Scholar
  27. [27]
    E. Kiltz, A. O’Neill, A. Smith, Instantiability of RSA-OAEP under chosen-plaintext attack, in Advances in Cryptology—CRYPTO 2010. LNCS, vol. 6223 (Springer, Berlin, 2010), pp. 295–313 CrossRefGoogle Scholar
  28. [28]
    F.J. MacWilliams, N.J.A. Sloane, The Theory of Error-Correcting Codes (North-Holland, Amsterdam, 1983) Google Scholar
  29. [29]
    R.J. McEliece, A public-key cryptosystem based on algebraic coding theory. DSN Prog. Rep., Jet Prop. Lab., pp. 114–116, Jan 1978 Google Scholar
  30. [30]
    P. Mol, S. Yilek, Chosen-ciphertext security from slightly lossy trapdoor functions, in Public Key Cryptography—PKC 2010. LNCS, vol. 6056 (Springer, Berlin, 2010), pp. 296–377. Full version available at http://eprint.iacr.org/2009/524 CrossRefGoogle Scholar
  31. [31]
    M. Naor, G. Segev, Public-key cryptosystems resilient to key leakage, in Advances in Cryptology—CRYPTO 2009. LNCS, vol. 5677 (Springer, Berlin, 2009), pp. 18–35. Full version available at http://eprint.iacr.org/2009/105 CrossRefGoogle Scholar
  32. [32]
    J. Neukirch, Algebraic Number Theory, Grundlehren der Mathematischen Wissenschaften [Fundamental Principles of Mathematical Sciences], vol. 322 (Springer, Berlin, 1999). Translated from the German by N. Schappacher zbMATHGoogle Scholar
  33. [33]
    H. Niederreiter, Knapsack-type cryptosystems and algebraic coding theory. Probl. Control Inf. Theory 15, 159–166 (1986) MathSciNetzbMATHGoogle Scholar
  34. [34]
    R. Nishimaki, E. Fujisaki, K. Tanaka, Efficient non-interactive universally composable string-commitment schemes, in Provable Security—ProvSec’09. LNCS, vol. 5848 (Springer, Berlin, 2009), pp. 3–18 CrossRefGoogle Scholar
  35. [35]
    P. Paillier, Public-key cryptosystems based on composite degree residuosity classes, in Advances in Cryptology—EUROCRYPT 1999. LNCS, vol. 1592 (Springer, Berlin, 1999), pp. 223–238 Google Scholar
  36. [36]
    C. Peikert, Public-key cryptosystems from the worst-case shortest vector problem, in 41st ACM Symposium on Theory of Computing (2009), pp. 333–342 CrossRefGoogle Scholar
  37. [37]
    C. Peikert, B. Waters, Lossy trapdoor functions and their applications, in 40th ACM Symposium on Theory of Computing (2008), pp. 187–196. Full version available at http://eprint.iacr.org/2007/279 Google Scholar
  38. [38]
    M. Rabin, Digitalized signatures and public-key functions as intractable as factorization. Technical Report MIT/LCS/TR-212, MIT Laboratory for Computer Science (1979) Google Scholar
  39. [39]
    A. Rosen, G. Segev, Chosen-ciphertext security via correlated products, in Theory of Cryptography Conference—TCC 2009. LNCS, vol. 5444 (Springer, Berlin, 2009), pp. 419–436 Google Scholar
  40. [40]
    H. Shacham, A Cramer-Shoup encryption scheme from the Linear assumption and from progressively weaker Linear variants. Cryptology ePrint Archive, Report 2007/074 (2007). Available at http://eprint.iacr.org/2007/074
  41. [41]
    D. Squirrel, Computing reciprocity symbols in number fields. Undergraduate thesis, Reed College (1997) Google Scholar

Copyright information

© International Association for Cryptologic Research 2011

Authors and Affiliations

  • David Mandell Freeman
    • 1
  • Oded Goldreich
    • 2
  • Eike Kiltz
    • 3
  • Alon Rosen
    • 4
    Email author
  • Gil Segev
    • 5
  1. 1.Stanford UniversityStanfordUSA
  2. 2.Weizmann Institute of ScienceRehovotIsrael
  3. 3.Ruhr-Universität BochumBochumGermany
  4. 4.IDC HerzliyaHerzliyaIsrael
  5. 5.Microsoft ResearchMountain ViewUSA

Personalised recommendations