Abstract
We put forward a generalization of lossy trapdoor functions (LTFs). Namely, all-but-many lossy trapdoor functions (ABM-LTFs) are LTFs that are parametrized with tags. Each tag can either be injective or lossy, which leads to an invertible or a lossy function. The interesting property of ABM-LTFs is that it is possible to generate an arbitrary number of lossy tags by means of a special trapdoor, while it is not feasible to produce lossy tags without this trapdoor.
Our definition and construction can be seen as generalizations of all-but-one LTFs (due to Peikert and Waters) and all-but-N LTFs (due to Hemenway et al.). However, to achieve ABM-LTFs (and thus a number of lossy tags which is not bounded by any polynomial), we have to employ some new tricks. Concretely, we give two constructions that use “disguised” variants of the Waters, resp. Boneh-Boyen signature schemes to make the generation of lossy tags hard without trapdoor. In a nutshell, lossy tags simply correspond to valid signatures. At the same time, tags are disguised (i.e., suitably blinded) to keep lossy tags indistinguishable from injective tags.
ABM-LTFs are useful in settings in which there are a polynomial number of adversarial challenges (e.g., challenge ciphertexts). Specifically, building on work by Hemenway et al., we show that ABM-LTFs can be used to achieve selective opening security against chosen-ciphertext attacks. One of our ABM-LTF constructions thus yields the first SO-CCA secure encryption scheme with compact ciphertexts (\(\ensuremath{\mathbf{O}} (1)\) group elements) whose efficiency does not depend on the number of challenges. Our second ABM-LTF construction yields an IND-CCA (and in fact SO-CCA) secure encryption scheme whose security reduction is independent of the number of challenges and decryption queries.
Chapter PDF
Similar content being viewed by others
References
Abe, M., Gennaro, R., Kurosawa, K.: Tag-KEM/DEM: A new framework for hybrid encryption. Journal of Cryptology 21(1), 97–130 (2008)
Applebaum, B., Ishai, Y., Kushilevitz, E.: How to garble arithmetic circuits. In: 52nd FOCS. IEEE Computer Society Press (2011)
Ateniese, G., de Medeiros, B.: Identity-Based Chameleon Hash and Applications. In: Juels, A. (ed.) FC 2004. LNCS, vol. 3110, pp. 164–180. Springer, Heidelberg (2004)
Bellare, M., Boldyreva, A., Micali, S.: Public-Key Encryption in a Multi-user Setting: Security Proofs and Improvements. In: Preneel, B. (ed.) EUROCRYPT 2000. LNCS, vol. 1807, pp. 259–274. Springer, Heidelberg (2000)
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)
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)
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)
Boneh, D., Boyen, X.: Short signatures without random oracles and the SDH assumption in bilinear groups. Journal of Cryptology 21(2), 149–177 (2008)
Boneh, D., Shen, E., Waters, B.: Strongly Unforgeable Signatures Based on Computational Diffie-Hellman. In: Yung, M., Dodis, Y., Kiayias, A., Malkin, T. (eds.) PKC 2006. LNCS, vol. 3958, pp. 229–240. Springer, Heidelberg (2006)
Boyen, X., Waters, B.: Shrinking the Keys of Discrete-Log-Type Lossy Trapdoor Functions. In: Zhou, J., Yung, M. (eds.) ACNS 2010. LNCS, vol. 6123, pp. 35–52. Springer, Heidelberg (2010)
Canetti, R., Feige, U., Goldreich, O., Naor, M.: Adaptively secure multi-party computation. In: 28th ACM STOC, pp. 639–648. ACM Press (May 1996)
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)
Chevallier-Mames, B., Joye, M.: A Practical and Tightly Secure Signature Scheme Without Hash Function. In: Abe, M. (ed.) CT-RSA 2007. LNCS, vol. 4377, pp. 339–356. Springer, Heidelberg (2006)
Damgård, I., Jurik, M.: A Generalisation, a Simplification and Some Applications of Paillier’s Probabilistic Public-Key System. In: Kim, K. (ed.) PKC 2001. LNCS, vol. 1992, pp. 119–136. Springer, Heidelberg (2001)
Damgård, I., Nielsen, J.B.: Improved Non-committing Encryption Schemes Based on a General Complexity Assumption. In: Bellare, M. (ed.) CRYPTO 2000. LNCS, vol. 1880, pp. 432–450. Springer, Heidelberg (2000)
Damgård, I., Nielsen, J.B.: Perfect Hiding and Perfect Binding Universally Composable Commitment Schemes with Constant Expansion Factor. In: Yung, M. (ed.) CRYPTO 2002. LNCS, vol. 2442, pp. 581–596. Springer, Heidelberg (2002)
Dolev, D., Dwork, C., Naor, M.: Non-malleable cryptography. In: 23rd ACM STOC, pp. 542–552. ACM Press (May 1991)
Dwork, C., Naor, M., Reingold, O., Stockmeyer, L.J.: Magic functions. In: 40th FOCS, pp. 523–534. IEEE Computer Society Press (October 1999)
Fehr, S., Hofheinz, D., Kiltz, E., Wee, H.: Encryption Schemes Secure against Chosen-Ciphertext Selective Opening Attacks. In: Gilbert, H. (ed.) EUROCRYPT 2010. LNCS, vol. 6110, pp. 381–402. Springer, Heidelberg (2010)
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)
Gennaro, R., Shoup, V.: A note on an encryption scheme of Kurosawa and Desmedt. Cryptology ePrint Archive, Report 2004/194 (2004), http://eprint.iacr.org/
Goldwasser, S., Micali, S.: Probabilistic encryption. Journal of Computer and System Sciences 28(2), 270–299 (1984)
Hemenway, B., Libert, B., Ostrovsky, R., Vergnaud, D.: Lossy Encryption: Constructions from General Assumptions and Efficient Selective Opening Chosen Ciphertext Security. In: Lee, D.H., Wang, X. (eds.) ASIACRYPT 2011. LNCS, vol. 7073, pp. 70–88. Springer, Heidelberg (2011)
Hohenberger, S., Waters, B.: Realizing Hash-and-Sign Signatures under Standard Assumptions. In: Joux, A. (ed.) EUROCRYPT 2009. LNCS, vol. 5479, pp. 333–350. Springer, Heidelberg (2009)
Ishai, Y., Kushilevitz, E.: Perfect Constant-Round Secure Computation via Perfect Randomizing Polynomials. In: Widmayer, P., Triguero, F., Morales, R., Hennessy, M., Eidenbenz, S., Conejo, R. (eds.) ICALP 2002. LNCS, vol. 2380, pp. 244–256. Springer, Heidelberg (2002)
Krawczyk, H., Rabin, T.: Chameleon signatures. In: NDSS 2000. The Internet Society (February 2000)
Nishimaki, R., Fujisaki, E., Tanaka, K.: Efficient Non-interactive Universally Composable String-Commitment Schemes. In: Pieprzyk, J., Zhang, F. (eds.) ProvSec 2009. LNCS, vol. 5848, pp. 3–18. Springer, Heidelberg (2009)
Paillier, P.: Public-Key Cryptosystems Based on Composite Degree Residuosity Classes. In: Stern, J. (ed.) EUROCRYPT 1999. LNCS, vol. 1592, pp. 223–238. Springer, Heidelberg (1999)
Pedersen, T.P.: Non-interactive and Information-Theoretic Secure Verifiable Secret Sharing. In: Feigenbaum, J. (ed.) CRYPTO 1991. LNCS, vol. 576, pp. 129–140. Springer, Heidelberg (1992)
Peikert, C., Waters, B.: Lossy trapdoor functions and their applications. In: Ladner, R.E., Dwork, C. (eds.) 40th ACM STOC, pp. 187–196. ACM Press (May 2008)
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)
Villar, J.: An efficient reduction from DDH to the rank problem (2011)
Waters, B.: Efficient Identity-Based Encryption Without Random Oracles. In: Cramer, R. (ed.) EUROCRYPT 2005. LNCS, vol. 3494, pp. 114–127. Springer, Heidelberg (2005)
Author information
Authors and Affiliations
Editor information
Editors and Affiliations
Rights and permissions
Copyright information
© 2012 International Association for Cryptologic Research
About this paper
Cite this paper
Hofheinz, D. (2012). All-But-Many Lossy Trapdoor Functions. In: Pointcheval, D., Johansson, T. (eds) Advances in Cryptology – EUROCRYPT 2012. EUROCRYPT 2012. Lecture Notes in Computer Science, vol 7237. Springer, Berlin, Heidelberg. https://doi.org/10.1007/978-3-642-29011-4_14
Download citation
DOI: https://doi.org/10.1007/978-3-642-29011-4_14
Publisher Name: Springer, Berlin, Heidelberg
Print ISBN: 978-3-642-29010-7
Online ISBN: 978-3-642-29011-4
eBook Packages: Computer ScienceComputer Science (R0)