All-but-One Dual Projective Hashing and Its Applications

  • Zongyang Zhang
  • Yu Chen
  • Sherman S. M. Chow
  • Goichiro Hanaoka
  • Zhenfu Cao
  • Yunlei Zhao
Part of the Lecture Notes in Computer Science book series (LNCS, volume 8479)


Recently, Wee (EUROCRYPT’12) introduced the notion of dual projective hashing as an extension of the Cramer-Shoup projective hashing, with a simple construction of lossy trapdoor functions, and a simple construction of deterministic encryption schemes which is chosen-plaintext-attack secure with respect to hard-to-invert auxiliary input. In this work, we further extend it to the all-but-one setting by introducing the notion of all-but-one dual projective hashing.

  • We provide a simple construction of all-but-one lossy trapdoor functions. Our construction encompasses many known constructions of all-but-one lossy trapdoor functions, as presented by Peikert and Waters (STOC’08), and Freeman et al. (JoC’13). Particularly, we present a new construction of all-but-one lossy trapdoor functions based on the DLIN assumption, which can be viewed as an extension of Freeman et al.’s DDH-based construction to the DLIN setting, and therefore solves an open problem left by Freeman et al.

  • We also provide a general construction of chosen-ciphertext-attack (CCA) secure deterministic encryption schemes in the standard model, under an additional assumption about the projective map. This extends the general approach of designing CCA secure deterministic encryption schemes by Boldyreva, Fehr and O’Neill (CRYPTO’08). In addition, we present a new construction of CCA secure deterministic encryption schemes based on the DLIN assumption.


Smooth projective hashing ABO lossy trapdoor function deterministic encryption CCA security 


  1. 1.
    Abdalla, M., Chevalier, C., Pointcheval, D.: Smooth Projective Hashing for Conditionally Extractable Commitments. In: Halevi, S. (ed.) CRYPTO 2009. LNCS, vol. 5677, pp. 671–689. Springer, Heidelberg (2009)CrossRefGoogle Scholar
  2. 2.
    Bellare, M., Boldyreva, A., O’Neill, A.: Deterministic and Efficiently Searchable Encryption. In: Menezes, A. (ed.) CRYPTO 2007. LNCS, vol. 4622, pp. 535–552. Springer, Heidelberg (2007)CrossRefGoogle Scholar
  3. 3.
    Bellare, M., Brakerski, Z., Naor, M., Ristenpart, T., Segev, G., Shacham, H., Yilek, S.: Hedged Public-Key Encryption: How to Protect against Bad Randomness. In: Matsui, M. (ed.) ASIACRYPT 2009. LNCS, vol. 5912, pp. 232–249. Springer, Heidelberg (2009)CrossRefGoogle Scholar
  4. 4.
    Bellare, M., Fischlin, M., O’Neill, A., Ristenpart, T.: Deterministic Encryption: Definitional Equivalences and Constructions without Random Oracles. In: Wagner, D. (ed.) CRYPTO 2008. LNCS, vol. 5157, pp. 360–378. Springer, Heidelberg (2008)CrossRefGoogle Scholar
  5. 5.
    Bellare, M., Hofheinz, D., Yilek, S.: Possibility and Impossibility Results for Encryption and Commitment Secure under Selective Opening. In: Joux, A. (ed.) EUROCRYPT 2009. LNCS, vol. 5479, pp. 1–35. Springer, Heidelberg (2009)CrossRefGoogle Scholar
  6. 6.
    Blazy, O., Pointcheval, D., Vergnaud, D.: Round-Optimal Privacy-Preserving Protocols with Smooth Projective Hash Functions. In: Cramer, R. (ed.) TCC 2012. LNCS, vol. 7194, pp. 94–111. Springer, Heidelberg (2012)CrossRefGoogle Scholar
  7. 7.
    Boldyreva, A., Fehr, S., O’Neill, A.: On Notions of Security for Deterministic Encryption, and Efficient Constructions without Random Oracles. In: Wagner, D. (ed.) CRYPTO 2008. LNCS, vol. 5157, pp. 335–359. Springer, Heidelberg (2008)CrossRefGoogle Scholar
  8. 8.
    Brakerski, Z., Segev, G.: Better Security for Deterministic Public-Key Encryption: The Auxiliary-Input Setting. In: Rogaway, P. (ed.) CRYPTO 2011. LNCS, vol. 6841, pp. 543–560. Springer, Heidelberg (2011)CrossRefGoogle Scholar
  9. 9.
    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)CrossRefGoogle Scholar
  10. 10.
    Cramer, R., Shoup, V.: Universal Hash Proofs and a Paradigm for Adaptive Chosen Ciphertext Secure Public-Key Encryption. In: Knudsen, L.R. (ed.) EUROCRYPT 2002. LNCS, vol. 2332, pp. 45–64. Springer, Heidelberg (2002)CrossRefGoogle Scholar
  11. 11.
    Damgård, I., Jurik, M., Nielsen, J.B.: A generalization of Paillier’s public-key system with applications to electronic voting. Int. J. Inf. Sec. 9(6), 371–385 (2010)CrossRefGoogle Scholar
  12. 12.
    Dolev, D., Dwork, C., Naor, M.: Nonmalleable Cryptography. SIAM J. Comput. 30(2), 391–437 (2000)CrossRefzbMATHMathSciNetGoogle Scholar
  13. 13.
    Freeman, D.M., Goldreich, O., Kiltz, E., Rosen, A., Segev, G.: More Constructions of Lossy and Correlation-Secure Trapdoor Functions. J. Cryptology 26(1), 39–74 (2013)CrossRefzbMATHMathSciNetGoogle Scholar
  14. 14.
    Fuller, B., O’Neill, A., Reyzin, L.: A Unified Approach to Deterministic Encryption: New Constructions and a Connection to Computational Entropy. In: Cramer, R. (ed.) TCC 2012. LNCS, vol. 7194, pp. 582–599. Springer, Heidelberg (2012)CrossRefGoogle Scholar
  15. 15.
    Gennaro, R., Lindell, Y.: A framework for password-based authenticated key exchange. ACM Trans. Inf. Syst. Secur. 9(2), 181–234 (2006)CrossRefGoogle Scholar
  16. 16.
    Gentry, C., Peikert, C., Vaikuntanathan, V.: Trapdoors for hard lattices and new cryptographic constructions. In: Dwork, C. (ed.) STOC, pp. 197–206. ACM (2008)Google Scholar
  17. 17.
    Halevi, S., Kalai, Y.T.: Smooth Projective Hashing and Two-Message Oblivious Transfer. J. Cryptology 25(1), 158–193 (2012)CrossRefzbMATHMathSciNetGoogle Scholar
  18. 18.
    Joye, M., Libert, B.: Efficient Cryptosystems from 2k-th Power Residue Symbols. In: Johansson, T., Nguyen, P.Q. (eds.) EUROCRYPT 2013. LNCS, vol. 7881, pp. 76–92. Springer, Heidelberg (2013)CrossRefGoogle Scholar
  19. 19.
    Katz, J., Vaikuntanathan, V.: Smooth Projective Hashing and Password-Based Authenticated Key Exchange from Lattices. In: Matsui, M. (ed.) ASIACRYPT 2009. LNCS, vol. 5912, pp. 636–652. Springer, Heidelberg (2009)CrossRefGoogle Scholar
  20. 20.
    Mironov, I., Pandey, O., Reingold, O., Segev, G.: Incremental Deterministic Public-Key Encryption. In: Pointcheval, D., Johansson, T. (eds.) EUROCRYPT 2012. LNCS, vol. 7237, pp. 628–644. Springer, Heidelberg (2012)CrossRefGoogle Scholar
  21. 21.
    Naor, M., Segev, G.: Public-Key Cryptosystems Resilient to Key Leakage. In: Halevi, S. (ed.) CRYPTO 2009. LNCS, vol. 5677, pp. 18–35. Springer, Heidelberg (2009)CrossRefGoogle Scholar
  22. 22.
    Peikert, C., Waters, B.: Lossy Trapdoor Functions and Their Applications. SIAM J. Comput. 40(6), 1803–1844 (2011)CrossRefzbMATHMathSciNetGoogle Scholar
  23. 23.
    Raghunathan, A., Segev, G., Vadhan, S.P.: Deterministic Public-Key Encryption for Adaptively Chosen Plaintext Distributions. In: Johansson, T., Nguyen, P.Q. (eds.) EUROCRYPT 2013. LNCS, vol. 7881, pp. 93–110. Springer, Heidelberg (2013)CrossRefGoogle Scholar
  24. 24.
    Wee, H.: Efficient Chosen-Ciphertext Security via Extractable Hash Proofs. In: Rabin, T. (ed.) CRYPTO 2010. LNCS, vol. 6223, pp. 314–332. Springer, Heidelberg (2010)CrossRefGoogle Scholar
  25. 25.
    Wee, H.: Dual Projective Hashing and Its Applications - Lossy Trapdoor Functions and More. In: Pointcheval, D., Johansson, T. (eds.) EUROCRYPT 2012. LNCS, vol. 7237, pp. 246–262. Springer, Heidelberg (2012)CrossRefGoogle Scholar
  26. 26.
    Wichs, D.: Barriers in cryptography with weak, correlated and leaky sources. In: Kleinberg, R.D. (ed.) ITCS, pp. 111–126. ACM (2013)Google Scholar

Copyright information

© Springer International Publishing Switzerland 2014

Authors and Affiliations

  • Zongyang Zhang
    • 1
    • 4
  • Yu Chen
    • 2
  • Sherman S. M. Chow
    • 3
  • Goichiro Hanaoka
    • 1
  • Zhenfu Cao
    • 4
  • Yunlei Zhao
    • 5
  1. 1.National Institute of Advanced Industrial Science and Technology (AIST)Japan
  2. 2.State Key Laboratory of Information Security (SKLOIS), Institute of Information EngineeringChinese Academy of SciencesChina
  3. 3.Department of Information EngineeringThe Chinese University of Hong KongHong Kong
  4. 4.Department of Computer Science and EngineeringShanghai Jiao Tong UniversityJapan
  5. 5.Software SchoolFudan UniversityChina

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