Abstract
Projecting bilinear pairings have frequently been used for designing cryptosystems since they were first derived from composite order bilinear groups. There have been only a few studies on the (im)possibility of projecting bilinear pairings. Groth and Sahai showed that projecting bilinear pairings can be achieved in the prime-order group setting. They constructed both projecting asymmetric bilinear pairings and projecting symmetric bilinear pairings, where a bilinear pairing e is symmetric if it satisfies e(g,h) = e(h,g) for any group elements g and h; otherwise, it is asymmetric.
In this paper, we provide impossibility results on projecting bilinear pairings in a prime-order group setting. More precisely, we specify the lower bounds of
-
the image size of a projecting asymmetric bilinear pairing
-
the image size of a projecting symmetric bilinear pairing
-
the computational cost for a projecting asymmetric bilinear pairing
-
the computational cost for a projecting symmetric bilinear pairing
in a prime-order group setting naturally induced from the k-linear assumption, where the computational cost means the number of generic operations.
Our lower bounds regarding a projecting asymmetric bilinear pairing are tight, i.e., it is impossible to construct a more efficient projecting asymmetric bilinear pairing than the constructions of Groth-Sahai and Freeman. However, our lower bounds regarding a projecting symmetric bilinear pairing differ from Groth and Sahai’s results regarding a symmetric bilinear pairing results; We fill these gaps by constructing projecting symmetric bilinear pairings.
In addition, on the basis of the proposed symmetric bilinear pairings, we construct more efficient instantiations of cryptosystems that essentially use the projecting symmetric bilinear pairings in a modular fashion. Example applications include new instantiations of the Boneh-Goh-Nissim cryptosystem, the Groth-Sahai non-interactive proof system, and Seo-Cheon round optimal blind signatures proven secure under the DLIN assumption. These new instantiations are more efficient than the previous ones, which are also provably secure under the DLIN assumption. These applications are of independent interest.
Chapter PDF
Similar content being viewed by others
Keywords
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
Abe, M., Fuchsbauer, G., Groth, J., Haralambiev, K., Ohkubo, M.: Structure-Preserving Signatures and Commitments to Group Elements. In: Rabin, T. (ed.) CRYPTO 2010. LNCS, vol. 6223, pp. 209–236. Springer, Heidelberg (2010)
Abe, M., Groth, J., Haralambiev, K., Ohkubo, M.: Optimal Structure-Preserving Signatures in Asymmetric Bilinear Groups. In: Rogaway, P. (ed.) CRYPTO 2011. LNCS, vol. 6841, pp. 649–666. Springer, Heidelberg (2011)
Abe, M., Groth, J., Ohkubo, M.: Separating Short Structure-Preserving Signatures from Non-interactive Assumptions. In: Lee, D.H. (ed.) ASIACRYPT 2011. LNCS, vol. 7073, pp. 628–646. Springer, Heidelberg (2011)
Abe, M., Haralambiev, K., Ohkubo, M.: Group to Group Commitments Do Not Shrink. In: Pointcheval, D., Johansson, T. (eds.) EUROCRYPT 2012. LNCS, vol. 7237, pp. 301–317. Springer, Heidelberg (2012)
Adida, B., Wikström, D.: How to Shuffle in Public. In: Vadhan, S.P. (ed.) TCC 2007. LNCS, vol. 4392, pp. 555–574. Springer, Heidelberg (2007)
Belenkiy, M., Camenisch, J., Chase, M., Kohlweiss, M., Lysyanskaya, A., Shacham, H.: Randomizable Proofs and Delegatable Anonymous Credentials. In: Halevi, S. (ed.) CRYPTO 2009. LNCS, vol. 5677, pp. 108–125. Springer, Heidelberg (2009)
Belenkiy, M., Chase, M., Kohlweiss, M., Lysyanskaya, A.: P-signatures and Noninteractive Anonymous Credentials. In: Canetti, R. (ed.) TCC 2008. LNCS, vol. 4948, pp. 356–374. Springer, Heidelberg (2008)
Boneh, D., Boyen, X.: Short Signatures Without Random Oracles. In: Cachin, C., Camenisch, J.L. (eds.) EUROCRYPT 2004. LNCS, vol. 3027, pp. 56–73. Springer, Heidelberg (2004)
Boneh, D., Boyen, X., Shacham, H.: Short Group Signatures. In: Franklin, M. (ed.) CRYPTO 2004. LNCS, vol. 3152, pp. 41–55. Springer, Heidelberg (2004)
Boneh, D., Goh, E.-J., Nissim, K.: Evaluating 2-DNF Formulas on Ciphertexts. In: Kilian, J. (ed.) TCC 2005. LNCS, vol. 3378, pp. 325–341. Springer, Heidelberg (2005)
Camenisch, J., Haralambiev, K., Kohlweiss, M., Lapon, J., Naessens, V.: Structure Preserving CCA Secure Encryption and Applications. In: Lee, D.H. (ed.) ASIACRYPT 2011. LNCS, vol. 7073, pp. 89–106. Springer, Heidelberg (2011)
Cathalo, J., Libert, B., Yung, M.: Group Encryption: Non-interactive Realization in the Standard Model. In: Matsui, M. (ed.) ASIACRYPT 2009. LNCS, vol. 5912, pp. 179–196. Springer, Heidelberg (2009)
Chandran, N., Groth, J., Sahai, A.: Ring Signatures of Sub-linear Size Without Random Oracles. In: Arge, L., Cachin, C., Jurdziński, T., Tarlecki, A. (eds.) ICALP 2007. LNCS, vol. 4596, pp. 423–434. Springer, Heidelberg (2007)
Chase, M., Kohlweiss, M., Lysyanskaya, A., Meiklejohn, S.: Malleable Proof Systems and Applications. In: Pointcheval, D., Johansson, T. (eds.) EUROCRYPT 2012. LNCS, vol. 7237, pp. 281–300. Springer, Heidelberg (2012)
Freeman, D.M.: Converting Pairing-Based Cryptosystems from Composite-Order Groups to Prime-Order Groups. In: Gilbert, H. (ed.) EUROCRYPT 2010. LNCS, vol. 6110, pp. 44–61. Springer, Heidelberg (2010), http://eprint.iacr.org/2009/540
Galbraith, S.D., Paterson, K.G., Smart, N.P.: Pairings for cryptographers. Discrete Applied Mathematics 156, 3113–3121 (2008)
Granger, R., Smart, N.: On computing products of pairings. Cryptology ePrint Archive, Report 2006/172 (2006)
Green, M., Hohenberger, S.: Universally Composable Adaptive Oblivious Transfer. In: Pieprzyk, J. (ed.) ASIACRYPT 2008. LNCS, vol. 5350, pp. 179–197. Springer, Heidelberg (2008)
Groth, J.: Fully Anonymous Group Signatures Without Random Oracles. In: Kurosawa, K. (ed.) ASIACRYPT 2007. LNCS, vol. 4833, pp. 164–180. Springer, Heidelberg (2007)
Groth, J., Lu, S.: A Non-interactive Shuffle with Pairing Based Verifiability. In: Kurosawa, K. (ed.) ASIACRYPT 2007. LNCS, vol. 4833, pp. 51–67. Springer, Heidelberg (2007)
Groth, J., Ostrovsky, R., Sahai, A.: Perfect Non-interactive Zero Knowledge for NP. In: Vaudenay, S. (ed.) EUROCRYPT 2006. LNCS, vol. 4004, pp. 339–358. Springer, Heidelberg (2006)
Groth, J., Sahai, A.: Efficient Non-interactive Proof Systems for Bilinear Groups. In: Smart, N.P. (ed.) EUROCRYPT 2008. LNCS, vol. 4965, pp. 415–432. Springer, Heidelberg (2008)
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)
Lewko, A.: Tools for Simulating Features of Composite Order Bilinear Groups in the Prime Order Setting. In: Pointcheval, D., Johansson, T. (eds.) EUROCRYPT 2012. LNCS, vol. 7237, pp. 318–335. Springer, Heidelberg (2012)
Meiklejohn, S., Shacham, H., Freeman, D.M.: Limitations on Transformations from Composite-Order to Prime-Order Groups: The Case of Round-Optimal Blind Signatures. In: Abe, M. (ed.) ASIACRYPT 2010. LNCS, vol. 6477, pp. 519–538. Springer, Heidelberg (2010)
Okamoto, T., Takashima, K.: Homomorphic Encryption and Signatures from Vector Decomposition. In: Galbraith, S.D., Paterson, K.G. (eds.) Pairing 2008. LNCS, vol. 5209, pp. 57–74. Springer, Heidelberg (2008)
Okamoto, T., Takashima, K.: Hierarchical Predicate Encryption for Inner-Products. In: Matsui, M. (ed.) ASIACRYPT 2009. LNCS, vol. 5912, pp. 214–231. Springer, Heidelberg (2009)
Okamoto, T., Takashima, K.: Fully Secure Functional Encryption with General Relations from the Decisional Linear Assumption. In: Rabin, T. (ed.) CRYPTO 2010. LNCS, vol. 6223, pp. 191–208. Springer, Heidelberg (2010)
Okamoto, T., Takashima, K.: Efficient Attribute-Based Signatures for Non-monotone Predicates in the Standard Model. In: Catalano, D., Fazio, N., Gennaro, R., Nicolosi, A. (eds.) PKC 2011. LNCS, vol. 6571, pp. 35–52. Springer, Heidelberg (2011)
Okamoto, T., Takashima, K.: Adaptively Attribute-Hiding (Hierarchical) Inner Product Encryption. In: Pointcheval, D., Johansson, T. (eds.) EUROCRYPT 2012. LNCS, vol. 7237, pp. 591–608. Springer, Heidelberg (2012)
Ostrovsky, R., Skeith, W.: Private searching on streaming data. Journal of Cryptology 20, 397–430 (2007)
Sang, Y., Shen, H.: Efficient and secure protocols for privacy-preserving set operations. ACM Transactions on Information and Systems Security 13 (2009)
Schwartz, J.: Fast probabilistic algorithms for verification of polynomials identities. Journal of the ACM 27, 701–717 (1980)
Scott, M.: Computing the Tate Pairing. In: Menezes, A. (ed.) CT-RSA 2005. LNCS, vol. 3376, pp. 293–304. Springer, Heidelberg (2005)
Seo, J.H., Cheon, J.H.: Beyond the Limitation of Prime-Order Bilinear Groups, and Round Optimal Blind Signatures. In: Cramer, R. (ed.) TCC 2012. LNCS, vol. 7194, pp. 133–150. Springer, Heidelberg (2012)
Shacham, H.: A cramer-shoup encryption scheme from the linear assumption and from progressively weaker linear variants. Cryptology ePrint Archive, Report 2007/074 (2007), http://eprint.iacr.org/2007/074
Shoup, V.: Lower Bounds for Discrete Logarithms and Related Problems. In: Fumy, W. (ed.) EUROCRYPT 1997. LNCS, vol. 1233, pp. 256–266. Springer, Heidelberg (1997)
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
Seo, J.H. (2012). On the (Im)possibility of Projecting Property in Prime-Order Setting. In: Wang, X., Sako, K. (eds) Advances in Cryptology – ASIACRYPT 2012. ASIACRYPT 2012. Lecture Notes in Computer Science, vol 7658. Springer, Berlin, Heidelberg. https://doi.org/10.1007/978-3-642-34961-4_6
Download citation
DOI: https://doi.org/10.1007/978-3-642-34961-4_6
Publisher Name: Springer, Berlin, Heidelberg
Print ISBN: 978-3-642-34960-7
Online ISBN: 978-3-642-34961-4
eBook Packages: Computer ScienceComputer Science (R0)