Abstract
Threshold Signatures allow n parties to share the ability of issuing digital signatures so that any coalition of size at least \(t+1\) can sign, whereas groups of t or fewer players cannot. The currently known class-group-based threshold ECDSA constructions are either inefficient (requiring parallel-repetition of the underlying zero knowledge proof with small challenge space) or requiring rather non-standard low order assumption. In this paper, we present efficient threshold ECDSA protocols from encryption schemes based on class groups with neither assuming the low order assumption nor parallel repeating the underlying zero knowledge proof, yielding a significant efficiency improvement in the key generation over previous constructions.
Along the way we introduce a new notion of promise \(\varSigma \)-protocol that satisfies only a weaker soundness called promise extractability. An accepting promise \(\varSigma \)-proof for statements related to class-group-based encryptions does not establish the truth of the statement but provides security guarantees (promise extractability) that are sufficient for our applications. We also show how to simulate homomorphic operations on a (possibly invalid) class-group-based encryption whose correctness has been proven via our promise \(\varSigma \)-protocol. We believe that these techniques are of independent interest and applicable to other scenarios where efficient zero knowledge proofs for statements related to class-group is required.
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Notes
- 1.
In our construction of the two-party protocol, \(\mathcal {P}_2\) does homomorphic operations only on \((c_{key,1},c_{key,2})\) (and not on \((C_{key,1},C_{key,2})\)) and sends back the resulting \(\textsf {CL}\) ciphertext.
- 2.
These encoding schemes \(\mathsf{DL.Code}\) and \(\mathsf{CL.Code}\) may vary with applications, and may be randomized.
- 3.
It is easy to verify that the straight-line extractor for protocol \(\varSigma _\mathrm{prom}^1\), similar to the one for protocol \(\varSigma _\mathrm{prom}^{2}\), does not require the knowledge of \(\textsf {sk}_0\) (which does not exist in protocol \(\varSigma _\mathrm{prom}^{1}\)).
References
Boneh, D., Bünz, B., Fisch, B.: A survey of two verifiable delay functions. Cryptology ePrint Archive, Report 2018/712 (2018). https://eprint.iacr.org/2018/712
Boneh, D., Gennaro, R., Goldfeder, S.: Using Level-1 homomorphic encryption to improve threshold DSA signatures for bitcoin wallet security. In: Lange, T., Dunkelman, O. (eds.) LATINCRYPT 2017. LNCS, vol. 11368, pp. 352–377. Springer, Cham (2019). https://doi.org/10.1007/978-3-030-25283-0_19
Belabas, K., Kleinjung, T., Sanso, A., Wesolowski, B.: A note on the low order assumption in class group of an imaginary quadratic number fields. Cryptology ePrint Archive, Report 2020/1310 (2020). https://eprint.iacr.org/2020/1310
Castagnos, G., Catalano, D., Laguillaumie, F., Savasta, F., Tucker, I.: Two-party ECDSA from hash proof systems and efficient instantiations. In: Boldyreva, A., Micciancio, D. (eds.) CRYPTO 2019. LNCS, vol. 11694, pp. 191–221. Springer, Cham (2019). https://doi.org/10.1007/978-3-030-26954-8_7
Castagnos, G., Catalano, D., Laguillaumie, F., Savasta, F., Tucker, I.: Bandwidth-efficient threshold EC-DSA. In: Kiayias, A., Kohlweiss, M., Wallden, P., Zikas, V. (eds.) PKC 2020. LNCS, vol. 12111, pp. 266–296. Springer, Cham (2020). https://doi.org/10.1007/978-3-030-45388-6_10
Castagnos, G., Laguillaumie, F.: Linearly homomorphic encryption from \(\sf DDH\). In: Nyberg, K. (ed.) CT-RSA 2015. LNCS, vol. 9048, pp. 487–505. Springer, Cham (2015). https://doi.org/10.1007/978-3-319-16715-2_26
Castagnos, G., Laguillaumie, F., Tucker, I.: Practical fully secure unrestricted inner product functional encryption modulo p. In: Peyrin, T., Galbraith, S. (eds.) ASIACRYPT 2018. LNCS, vol. 11273, pp. 733–764. Springer, Cham (2018). https://doi.org/10.1007/978-3-030-03329-3_25
Cohen, H.: A Course in Computational Algebraic Number Theory. Springer, Heidelberg (2000). https://doi.org/10.1007/978-3-662-02945-9
Desmedt, Y.: Society and group oriented cryptography: a new concept. In: Pomerance, C. (ed.) CRYPTO 1987. LNCS, vol. 293, pp. 120–127. Springer, Heidelberg (1988). https://doi.org/10.1007/3-540-48184-2_8
Doerner, J., Kondi, Y., Lee, E., Shelat, A.: Secure two-party threshold ECDSA from ECDSA assumptions. In: 2018 IEEE Symposium on Security and Privacy (SP), pp. 980–997 (2018)
Doerner, J., Kondi, Y., Lee, E., Shelat, A.: Threshold ECDSA from ECDSA assumptions: the multiparty case. In: 2019 IEEE Symposium on Security and Privacy (SP), pp. 1051–1066 (2019)
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
Gennaro, R., Goldfeder, S.: Fast multiparty threshold ECDSA with fast trustless setup. In: Proceedings of the 2018 ACM SIGSAC Conference on Computer and Communications Security, CCS 2018, pp. 1179–1194. Association for Computing Machinery, New York (2018)
Gennaro, R., Goldfeder, S.: One round threshold ECDSA with identifiable abort. Cryptology ePrint Archive, Report 2020/540 (2020). https://eprint.iacr.org/2020/540
Gennaro, R., Goldfeder, S., Narayanan, A.: Threshold-optimal DSA/ECDSA signatures and an application to bitcoin wallet security. In: Manulis, M., Sadeghi, A.-R., Schneider, S. (eds.) ACNS 2016. LNCS, vol. 9696, pp. 156–174. Springer, Cham (2016). https://doi.org/10.1007/978-3-319-39555-5_9
Gennaro, R., Jarecki, S., Krawczyk, H., Rabin, T.: Robust threshold DSS signatures. In: Maurer, U. (ed.) EUROCRYPT 1996. LNCS, vol. 1070, pp. 354–371. Springer, Heidelberg (1996). https://doi.org/10.1007/3-540-68339-9_31
Girault, M., Poupard, G., Stern, J.: On the fly authentication and signature schemes based on groups of unknown order. J. Cryptol. 19, 463–487 (2006)
Lindell, Y.: Fast secure two-party ECDSA signing. In: Katz, J., Shacham, H. (eds.) CRYPTO 2017. LNCS, vol. 10402, pp. 613–644. Springer, Cham (2017). https://doi.org/10.1007/978-3-319-63715-0_21
Lindell, Y., Nof, A.: Fast secure multiparty ECDSA with practical distributed key generation and applications to cryptocurrency custody. In: Proceedings of the 2018 ACM SIGSAC Conference on Computer and Communications Security, CCS 2018, pp. 1837–1854. Association for Computing Machinery, New York (2018)
MacKenzie, P., Reiter, M.K.: Two-party generation of DSA signatures. In: Kilian, J. (ed.) CRYPTO 2001. LNCS, vol. 2139, pp. 137–154. Springer, Heidelberg (2001). https://doi.org/10.1007/3-540-44647-8_8
Pointcheval, D., Stern, J.: Security proofs for signature schemes. In: Maurer, U. (ed.) EUROCRYPT 1996. LNCS, vol. 1070, pp. 387–398. Springer, Heidelberg (1996). https://doi.org/10.1007/3-540-68339-9_33
Yuen, T.H., Cui, H., Xie, X.: Compact zero-knowledge proofs for threshold ECDSA with trustless setup. In: Garay, J.A. (ed.) PKC 2021. LNCS, vol. 12710, pp. 481–511. Springer, Cham (2021). https://doi.org/10.1007/978-3-030-75245-3_18
Acknowledgments
We would like to thank the anonymous reviewers for their valuable suggestions. We are supported by the National Natural Science Foundation of China (Grant No. 61932019, No. 61772521 and No. 61772522) and the Key Research Program of Frontier Sciences, CAS (Grant No. QYZDB-SSW-SYS035).
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Deng, Y., Ma, S., Zhang, X., Wang, H., Song, X., Xie, X. (2021). Promise \(\varSigma \)-Protocol: How to Construct Efficient Threshold ECDSA from Encryptions Based on Class Groups. In: Tibouchi, M., Wang, H. (eds) Advances in Cryptology – ASIACRYPT 2021. ASIACRYPT 2021. Lecture Notes in Computer Science(), vol 13093. Springer, Cham. https://doi.org/10.1007/978-3-030-92068-5_19
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