Abstract
Structure-Preserving Signatures (SPSs) are an important tool for the design of modular cryptographic protocols. It has been proven that such schemes in the most efficient Type-3 bilinear group setting have a lower bound of 3-element signatures, which must include elements from both base groups, and a verification overhead of at least 2 Pairing-Product Equations (PPEs). Very recently, Ghadafi (ESORICS 2017) showed that by restricting the message space to the set of Diffie-Hellman pairs (which does not hinder applicability of the schemes), some of the existing lower bounds for the single message case can be circumvented. However, the case of signing multiple messages, which is required for many applications, was left as an open problem since the techniques used for signing single messages do not seem to lend themselves to the multi-message setting. In this work we investigate this setting and answer the question in the affirmative. We construct schemes that sign vectors of messages and which yield shorter signatures than optimal schemes for vectors of unilateral messages. More precisely, we construct 2 fully randomiazble schemes that sign vectors of Diffie-Hellman pairs yielding signatures consisting of only 2 elements regardless of the size of the vector signed. We also construct a unilateral scheme that signs a pair of messages yielding signatures consisting of 3 elements from the shorter base group. All of our schemes require a single PPE for verification (not counting the cost of verifying the well-formedness of the messages). Thus, all of our schemes compare favourably to all existing schemes with respect to signature size and verification overhead. Even when considering single messages, our first 2 schemes compare favourably to the best existing schemes in many aspects including the verification overhead and the key size.
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Notes
- 1.
We remark that the scheme remains secure even if the sign oracle only gets \(\tilde{N}_j\) as long as the final forgery is on a well-formed message \((M^*,\tilde{N}^*)\in \widehat{\mathbb {GH}}\).
References
Abe, M., Chase, M., David, B., Kohlweiss, M., Nishimaki, R., Ohkubo, M.: Constant-size structure-preserving signatures: generic constructions and simple assumptions. In: Wang, X., Sako, K. (eds.) ASIACRYPT 2012. LNCS, vol. 7658, pp. 4–24. Springer, Heidelberg (2012). https://doi.org/10.1007/978-3-642-34961-4_3
Abe, M., David, B., Kohlweiss, M., Nishimaki, R., Ohkubo, M.: Tagged one-time signatures: tight security and optimal tag size. In: Kurosawa, K., Hanaoka, G. (eds.) PKC 2013. LNCS, vol. 7778, pp. 312–331. Springer, Heidelberg (2013). https://doi.org/10.1007/978-3-642-36362-7_20
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). https://doi.org/10.1007/978-3-642-14623-7_12
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). https://doi.org/10.1007/978-3-642-22792-9_37
Abe, M., Groth, J., Ohkubo, M.: Separating short structure-preserving signatures from non-interactive assumptions. In: Lee, D.H., Wang, X. (eds.) ASIACRYPT 2011. LNCS, vol. 7073, pp. 628–646. Springer, Heidelberg (2011). https://doi.org/10.1007/978-3-642-25385-0_34
Abe, M., Groth, J., Ohkubo, M., Tibouchi, M.: Unified, minimal and selectively randomizable structure-preserving signatures. In: Lindell, Y. (ed.) TCC 2014. LNCS, vol. 8349, pp. 688–712. Springer, Heidelberg (2014). https://doi.org/10.1007/978-3-642-54242-8_29
Abe, M., Groth, J., Ohkubo, M., Tibouchi, M.: Structure-preserving signatures from type II pairings. In: Garay, J.A., Gennaro, R. (eds.) CRYPTO 2014. LNCS, vol. 8616, pp. 390–407. Springer, Heidelberg (2014). https://doi.org/10.1007/978-3-662-44371-2_22
Abe, M., Kohlweiss, M., Ohkubo, M., Tibouchi, M.: Fully structure-preserving signatures and shrinking commitments. In: Oswald, E., Fischlin, M. (eds.) EUROCRYPT 2015. LNCS, vol. 9057, pp. 35–65. Springer, Heidelberg (2015). https://doi.org/10.1007/978-3-662-46803-6_2
Attrapadung, N., Libert, B., Peters, T.: Computing on authenticated data: new privacy definitions and constructions. In: Wang, X., Sako, K. (eds.) ASIACRYPT 2012. LNCS, vol. 7658, pp. 367–385. Springer, Heidelberg (2012). https://doi.org/10.1007/978-3-642-34961-4_23
Baldimtsi, F., Chase, M., Fuchsbauer, G., Kohlweiss, M.: Anonymous transferable E-Cash. In: Katz, J. (ed.) PKC 2015. LNCS, vol. 9020, pp. 101–124. Springer, Heidelberg (2015). https://doi.org/10.1007/978-3-662-46447-2_5
Barthe, G., Fagerholm, E., Fiore, D., Scedrov, A., Schmidt, B., Tibouchi, M.: Strongly-Optimal structure preserving signatures from Type II pairings: synthesis and lower bounds. In: Katz, J. (ed.) PKC 2015. LNCS, vol. 9020, pp. 355–376. Springer, Heidelberg (2015). https://doi.org/10.1007/978-3-662-46447-2_16
Bellare, M., Garay, J.A., Rabin, T.: Fast batch verification for modular exponentiation and digital signatures. In: Nyberg, K. (ed.) EUROCRYPT 1998. LNCS, vol. 1403, pp. 236–250. Springer, Heidelberg (1998). https://doi.org/10.1007/BFb0054130
Bernhard, D., Fuchsbauer, G., Ghadafi, E.: Efficient signatures of knowledge and DAA in the standard model. In: Jacobson, M., Locasto, M., Mohassel, P., Safavi-Naini, R. (eds.) ACNS 2013. LNCS, vol. 7954, pp. 518–533. Springer, Heidelberg (2013). https://doi.org/10.1007/978-3-642-38980-1_33
Brickell, E., Camenisch, J., Chen, L.: Direct anonymous attestation. In: CCS 2004, ACM, pp. 132–145 (2004)
Camenisch, J., Dubovitskaya, M., Haralambiev, K.: Efficient structure-preserving signature scheme from standard assumptions. In: Visconti, I., De Prisco, R. (eds.) SCN 2012. LNCS, vol. 7485, pp. 76–94. Springer, Heidelberg (2012). https://doi.org/10.1007/978-3-642-32928-9_5
Camenisch, J., Dubovitskaya, M., Haralambiev, K., Kohlweiss, M.: Composable and modular anonymous credentials: definitions and practical constructions. In: Iwata, T., Cheon, J.H. (eds.) ASIACRYPT 2015. LNCS, vol. 9453, pp. 262–288. Springer, Heidelberg (2015). https://doi.org/10.1007/978-3-662-48800-3_11
Camenisch, J., Hohenberger, S., Pedersen, M.Ø.: Batch verification of short signatures. J. Cryptology 25(4), 723–747 (2012)
Chase, M., Kohlweiss, M.: A New hash-and-sign approach and structure-preserving signatures from DLIN. In: Visconti, I., De Prisco, R. (eds.) SCN 2012. LNCS, vol. 7485, pp. 131–148. Springer, Heidelberg (2012). https://doi.org/10.1007/978-3-642-32928-9_8
Chatterjee, S., Menezes, A.: Type 2 structure-preserving signature schemes revisited. In: Iwata, T., Cheon, J.H. (eds.) ASIACRYPT 2015. LNCS, vol. 9452, pp. 286–310. Springer, Heidelberg (2015). https://doi.org/10.1007/978-3-662-48797-6_13
ElGamal, T.: A public key cryptosystem and a signature scheme based on discrete logarithms. IEEE Trans. Inf. Theor. 31(4), 469–472 (1985)
El Kaafarani, A., Ghadafi, E., Khader, D.: Decentralized traceable attribute-based signatures. In: Benaloh, J. (ed.) CT-RSA 2014. LNCS, vol. 8366, pp. 327–348. Springer, Cham (2014). https://doi.org/10.1007/978-3-319-04852-9_17
Fiat, A., Shamir, A.: How To prove yourself: practical solutions to identification and signature problems. In: Odlyzko, A.M. (ed.) CRYPTO 1986. LNCS, vol. 263, pp. 186–194. Springer, Heidelberg (1987). https://doi.org/10.1007/3-540-47721-7_12
Fuchsbauer, G.: Automorphic signatures in bilinear groups and an application to round-optimal blind signatures. In: Cryptology ePrint Archive, Report 2009/320
Fuchsbauer, G.: Commuting Signatures and verifiable encryption. In: Paterson, K.G. (ed.) EUROCRYPT 2011. LNCS, vol. 6632, pp. 224–245. Springer, Heidelberg (2011). https://doi.org/10.1007/978-3-642-20465-4_14
Fuchsbauer, G., Hanser, C., Slamanig, D.: Practical round-optimal blind signatures in the standard model. In: Gennaro, R., Robshaw, M. (eds.) CRYPTO 2015. LNCS, vol. 9216, pp. 233–253. Springer, Heidelberg (2015). https://doi.org/10.1007/978-3-662-48000-7_12
Galbraith, S., Paterson, K., Smart, N.P.: Pairings for cryptographers. Discrete Appl. Math. 156, 3113–3121 (2008)
Ghadafi, E.: Formalizing group blind signatures and practical constructions without random oracles. In: Boyd, C., Simpson, L. (eds.) ACISP 2013. LNCS, vol. 7959, pp. 330–346. Springer, Heidelberg (2013). https://doi.org/10.1007/978-3-642-39059-3_23
Ghadafi, E.: Short structure-preserving signatures. In: Sako, K. (ed.) CT-RSA 2016. LNCS, vol. 9610, pp. 305–321. Springer, Cham (2016). https://doi.org/10.1007/978-3-319-29485-8_18
Ghadafi, E.: More efficient structure-preserving signatures - or: bypassing the Type-III lower bounds. In: Foley, S.N., Gollmann, D., Snekkenes, E. (eds.) ESORICS 2017. LNCS, vol. 10493, pp. 43–61. Springer, Cham (2017). https://doi.org/10.1007/978-3-319-66399-9_3
Ghadafi, E., Smart, N.P., Warinschi, B.: Groth–Sahai proofs revisited. In: Nguyen, P.Q., Pointcheval, D. (eds.) PKC 2010. LNCS, vol. 6056, pp. 177–192. Springer, Heidelberg (2010). https://doi.org/10.1007/978-3-642-13013-7_11
Green, M., Hohenberger, S.: Universally composable adaptive oblivious transfer. In: Pieprzyk, J. (ed.) ASIACRYPT 2008. LNCS, vol. 5350, pp. 179–197. Springer, Heidelberg (2008). https://doi.org/10.1007/978-3-540-89255-7_12
Groth, J.: Simulation-Sound NIZK proofs for a practical language and constant size group signatures. In: Lai, X., Chen, K. (eds.) ASIACRYPT 2006. LNCS, vol. 4284, pp. 444–459. Springer, Heidelberg (2006). https://doi.org/10.1007/11935230_29
Groth, J.: Efficient fully structure-preserving signatures for large messages. In: Iwata, T., Cheon, J.H. (eds.) ASIACRYPT 2015. LNCS, vol. 9452, pp. 239–259. Springer, Heidelberg (2015). https://doi.org/10.1007/978-3-662-48797-6_11
Groth, J., Sahai, A.: Efficient non-interactive proof systems for bilinear groups. SIAM J. Comput. 41(5), 1193–1232 (2012)
Hofheinz, D., Jager, T.: Tightly secure signatures and public-key encryption. In: Safavi-Naini, R., Canetti, R. (eds.) CRYPTO 2012. LNCS, vol. 7417, pp. 590–607. Springer, Heidelberg (2012). https://doi.org/10.1007/978-3-642-32009-5_35
Jutla, C.S., Roy, A.: Improved structure preserving signatures under standard bilinear assumptions. In: Fehr, S. (ed.) PKC 2017. LNCS, vol. 10175, pp. 183–209. Springer, Heidelberg (2017). https://doi.org/10.1007/978-3-662-54388-7_7
Kiltz, E., Pan, J., Wee, H.: Structure-preserving signatures from standard assumptions, revisited. In: Gennaro, R., Robshaw, M. (eds.) CRYPTO 2015. LNCS, vol. 9216, pp. 275–295. Springer, Heidelberg (2015). https://doi.org/10.1007/978-3-662-48000-7_14
Libert, B., Peters, T., Yung, M.: Short group signatures via structure-preserving signatures: standard model security from simple assumptions. In: Gennaro, R., Robshaw, M. (eds.) CRYPTO 2015. LNCS, vol. 9216, pp. 296–316. Springer, Heidelberg (2015). https://doi.org/10.1007/978-3-662-48000-7_15
Maji, H.K., Prabhakaran, M., Rosulek, M.: Attribute-based signatures. In: Kiayias, A. (ed.) CT-RSA 2011. LNCS, vol. 6558, pp. 376–392. Springer, Heidelberg (2011). https://doi.org/10.1007/978-3-642-19074-2_24
Maurer, U.: Abstract models of computation in cryptography. In: Smart, N.P. (ed.) Cryptography and Coding 2005. LNCS, vol. 3796, pp. 1–12. Springer, Heidelberg (2005). https://doi.org/10.1007/11586821_1
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). https://doi.org/10.1007/3-540-69053-0_18
Wang, Y., Zhang, Z., Matsuda, T., Hanaoka, G., Tanaka, K.: How to obtain fully structure-preserving (automorphic) signatures from structure-preserving ones. In: Cheon, J.H., Takagi, T. (eds.) ASIACRYPT 2016. LNCS, vol. 10032, pp. 465–495. Springer, Heidelberg (2016). https://doi.org/10.1007/978-3-662-53890-6_16
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We thank anonymous IMA International Conference on Cryptography and Coding reviewers for valuable comments.
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Ghadafi, E. (2017). How Low Can You Go? Short Structure-Preserving Signatures for Diffie-Hellman Vectors. In: O'Neill, M. (eds) Cryptography and Coding. IMACC 2017. Lecture Notes in Computer Science(), vol 10655. Springer, Cham. https://doi.org/10.1007/978-3-319-71045-7_10
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