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Trapdoor Functions from the Computational Diffie-Hellman Assumption

  • Sanjam Garg
  • Mohammad Hajiabadi
Conference paper
Part of the Lecture Notes in Computer Science book series (LNCS, volume 10992)

Abstract

Trapdoor functions (TDFs) are a fundamental primitive in cryptography. Yet, the current set of assumptions known to imply TDFs is surprisingly limited, when compared to public-key encryption. We present a new general approach for constructing TDFs. Specifically, we give a generic construction of TDFs from any Chameleon Encryption (Döttling and Garg [CRYPTO’17]) satisfying a novel property which we call recyclability. By showing how to adapt current Computational Diffie-Hellman (CDH) based constructions of chameleon encryption to yield recyclability, we obtain the first construction of TDFs with security proved under the CDH assumption. While TDFs from the Decisional Diffie-Hellman (DDH) assumption were previously known, the possibility of basing them on CDH had remained open for more than 30 years.

Keywords

Trapdoor functions Computational Diffie-Hellman assumption Hash encryption 

Notes

Acknowledgments

We would like to thank the anonymous reviewers for their useful comments, and thank Mohammad Mahmoody and Adam O’Neill for useful discussions.

References

  1. 1.
    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).  https://doi.org/10.1007/978-3-540-74143-5_30CrossRefGoogle Scholar
  2. 2.
    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).  https://doi.org/10.1007/978-3-540-85174-5_20CrossRefGoogle Scholar
  3. 3.
    Bellare, M., Halevi, S., Sahai, A., Vadhan, S.P.: Many-to-one trapdoor functions and their relation to public-key cryptosystems. In: Krawczyk, H. (ed.) CRYPTO 1998. LNCS, vol. 1462, pp. 283–298. Springer, Heidelberg (1998).  https://doi.org/10.1007/BFb0055735CrossRefGoogle Scholar
  4. 4.
    Blum, M., Feldman, P., Micali, S.: Proving security against chosen ciphertext attacks. In: Goldwasser, S. (ed.) CRYPTO 1988. LNCS, vol. 403, pp. 256–268. Springer, New York (1990).  https://doi.org/10.1007/0-387-34799-2_20CrossRefGoogle Scholar
  5. 5.
    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).  https://doi.org/10.1007/978-3-540-85174-5_19CrossRefGoogle Scholar
  6. 6.
    Boneh, D., Boyen, X.: Secure identity based encryption without random oracles. In: Franklin, M. (ed.) CRYPTO 2004. LNCS, vol. 3152, pp. 443–459. Springer, Heidelberg (2004).  https://doi.org/10.1007/978-3-540-28628-8_27CrossRefGoogle Scholar
  7. 7.
    Brakerski, Z., Lombardi, A., Segev, G., Vaikuntanathan, V.: Anonymous IBE, leakage resilience and circular security from new assumptions. In: Nielsen, J.B., Rijmen, V. (eds.) EUROCRYPT 2018, Part I. LNCS, vol. 10820, pp. 535–564. Springer, Cham (2018).  https://doi.org/10.1007/978-3-319-78381-9_20CrossRefGoogle Scholar
  8. 8.
    Canetti, R., Halevi, S., Katz, J.: Chosen-ciphertext security from identity-based encryption. In: Cachin, C., Camenisch, J.L. (eds.) EUROCRYPT 2004. LNCS, vol. 3027, pp. 207–222. Springer, Heidelberg (2004).  https://doi.org/10.1007/978-3-540-24676-3_13CrossRefGoogle Scholar
  9. 9.
    Cash, D., Kiltz, E., Shoup, V.: The twin Diffie-Hellman problem and applications. In: Smart, N.P. (ed.) EUROCRYPT 2008. LNCS, vol. 4965, pp. 127–145. Springer, Heidelberg (2008).  https://doi.org/10.1007/978-3-540-78967-3_8CrossRefGoogle Scholar
  10. 10.
    Cho, C., Döttling, N., Garg, S., Gupta, D., Miao, P., Polychroniadou, A.: Laconic oblivious transfer and its applications. In: Katz, J., Shacham, H. (eds.) CRYPTO 2017, Part II. LNCS, vol. 10402, pp. 33–65. Springer, Cham (2017).  https://doi.org/10.1007/978-3-319-63715-0_2CrossRefGoogle Scholar
  11. 11.
    Diffie, W., Hellman, M.E.: New directions in cryptography. IEEE Trans. Inf. Theory 22(6), 644–654 (1976)MathSciNetCrossRefGoogle Scholar
  12. 12.
    Döttling, N., Garg, S.: From selective IBE to full IBE and selective HIBE. In: Kalai, Y., Reyzin, L. (eds.) TCC 2017, Part I. LNCS, vol. 10677, pp. 372–408. Springer, Cham (2017).  https://doi.org/10.1007/978-3-319-70500-2_13CrossRefGoogle Scholar
  13. 13.
    Döttling, N., Garg, S.: Identity-based encryption from the Diffie-Hellman assumption. In: Katz, J., Shacham, H. (eds.) CRYPTO 2017, Part I. LNCS, vol. 10401, pp. 537–569. Springer, Cham (2017).  https://doi.org/10.1007/978-3-319-63688-7_18CrossRefGoogle Scholar
  14. 14.
    Döttling, N., Garg, S., Hajiabadi, M., Masny, D.: New constructions of identity-based and key-dependent message secure encryption schemes. In: Abdalla, M., Dahab, R. (eds.) PKC 2018, Part I. LNCS, vol. 10769, pp. 3–31. Springer, Cham (2018).  https://doi.org/10.1007/978-3-319-76578-5_1CrossRefMATHGoogle Scholar
  15. 15.
    ElGamal, T.: A public key cryptosystem and a signature scheme based on discrete logarithms. In: Blakley, G.R., Chaum, D. (eds.) CRYPTO 1984. LNCS, vol. 196, pp. 10–18. Springer, Heidelberg (1985).  https://doi.org/10.1007/3-540-39568-7_2CrossRefGoogle Scholar
  16. 16.
    Feige, U., Lapidot, D., Shamir, A.: Multiple non-interactive zero knowledge proofs based on a single random string (extended abstract). In: 31st FOCS, St. Louis, Missouri, 22–24 October 1990, pp. 308–317. IEEE Computer Society Press (1990)Google Scholar
  17. 17.
    Freeman, D.M., Goldreich, O., Kiltz, E., Rosen, A., Segev, G.: More constructions of lossy and correlation-secure trapdoor functions. In: Nguyen, P.Q., Pointcheval, D. (eds.) PKC 2010. LNCS, vol. 6056, pp. 279–295. Springer, Heidelberg (2010).  https://doi.org/10.1007/978-3-642-13013-7_17CrossRefGoogle Scholar
  18. 18.
    Gertner, Y., Malkin, T., Reingold, O.: On the impossibility of basing trapdoor functions on trapdoor predicates. In: 42nd FOCS, Las Vegas, NV, USA, 14–17 October 2001, pp. 126–135. IEEE Computer Society Press (2001)Google Scholar
  19. 19.
    Goldreich, O., Levin, L.A.: A hard-core predicate for all one-way functions. In: 21st ACM STOC, Seattle, WA, USA, 15–17 May 1989, pp. 25–32. ACM Press (1989)Google Scholar
  20. 20.
    Goldwasser, S., Micali, S.: Probabilistic encryption and how to play mental poker keeping secret all partial information. In: 14th ACM STOC, San Francisco, CA, USA, 5–7 May 1982, pp. 365–377. ACM Press (1982)Google Scholar
  21. 21.
    Hajiabadi, M., Kapron, B.M.: Reproducible circularly-secure bit encryption: applications and realizations. In: Gennaro, R., Robshaw, M.J.B. (eds.) CRYPTO 2015, Part I. LNCS, vol. 9215, pp. 224–243. Springer, Heidelberg (2015).  https://doi.org/10.1007/978-3-662-47989-6_11CrossRefGoogle Scholar
  22. 22.
    Haralambiev, K., Jager, T., Kiltz, E., Shoup, V.: Simple and efficient public-key encryption from computational Diffie-Hellman in the standard model. In: Nguyen, P.Q., Pointcheval, D. (eds.) PKC 2010. LNCS, vol. 6056, pp. 1–18. Springer, Heidelberg (2010).  https://doi.org/10.1007/978-3-642-13013-7_1CrossRefMATHGoogle Scholar
  23. 23.
    Kiltz, E., Mohassel, P., O’Neill, A.: Adaptive trapdoor functions and chosen-ciphertext security. In: Gilbert, H. (ed.) EUROCRYPT 2010. LNCS, vol. 6110, pp. 673–692. Springer, Heidelberg (2010).  https://doi.org/10.1007/978-3-642-13190-5_34CrossRefGoogle Scholar
  24. 24.
    Myers, S., Shelat, A.: Bit encryption is complete. In: 50th FOCS, Atlanta, GA, USA, 25–27 October 2009, pp. 607–616. IEEE Computer Society Press (2009)Google Scholar
  25. 25.
    Naor, M., Yung, M: Public-key cryptosystems provably secure against chosen ciphertext attacks. In: 22nd ACM STOC, Baltimore, MD, USA, 14–16 May 1990, pp. 427–437. ACM Press (1990)Google Scholar
  26. 26.
    Peikert, C.: Public-key cryptosystems from the worst-case shortest vector problem: extended abstract. In: Mitzenmacher, M. (ed.) 41st ACM STOC, Bethesda, MD, USA, 31 May–2 June 2009, pp. 333–342. ACM Press (2009)Google Scholar
  27. 27.
    Peikert, C., Waters, B.: Lossy trapdoor functions and their applications. In: Ladner, R.E., Dwork, C. (eds.) 40th ACM STOC, Victoria, British Columbia, Canada, 17–20 May 2008, pp. 187–196. ACM Press (2008)Google Scholar
  28. 28.
    Rivest, R.L., Shamir, A., Adleman, L.M.: A method for obtaining digital signature and public-key cryptosystems. Commun. Assoc. Comput. Mach. 21(2), 120–126 (1978)MathSciNetMATHGoogle Scholar
  29. 29.
    Rosen, A., Segev, G.: Chosen-ciphertext security via correlated products. In: Reingold, O. (ed.) TCC 2009. LNCS, vol. 5444, pp. 419–436. Springer, Heidelberg (2009).  https://doi.org/10.1007/978-3-642-00457-5_25CrossRefGoogle Scholar
  30. 30.
    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).  https://doi.org/10.1007/978-3-642-14623-7_17CrossRefGoogle Scholar
  31. 31.
    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).  https://doi.org/10.1007/978-3-642-29011-4_16CrossRefGoogle Scholar
  32. 32.
    Yao, A.C.-C: Theory and applications of trapdoor functions (extended abstract). In: 23rd FOCS, pp. 80–91. IEEE Computer Society Press, Chicago, 3–5 November 1982Google Scholar

Copyright information

© International Association for Cryptologic Research 2018

Authors and Affiliations

  1. 1.University of CaliforniaBerkeleyUSA
  2. 2.University of VirginiaCharlottesvilleUSA

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