Abstract
In a set membership proof, the public information consists of a set of elements and a commitment. The prover then produces a zero-knowledge proof showing that the commitment is indeed to some element from the set. This primitive is closely related to concepts like ring signatures and “one-out-of-many” proofs that underlie many anonymity and privacy protocols. The main result of this work is a new succinct lattice-based set membership proof whose size is logarithmic in the size of the set.
We also give a transformation of our set membership proof to a ring signature scheme. The ring signature size is also logarithmic in the size of the public key set and has size \(16\) KB for a set of \(2^5\) elements, and \(22\) KB for a set of size \(2^{25}\). At an approximately 128-bit security level, these outputs are between 1.5\(\times \) and 7\(\times \) smaller than the current state of the art succinct ring signatures of Beullens et al. (Asiacrypt 2020) and Esgin et al. (CCS 2019).
We then show that our ring signature, combined with a few other techniques and optimizations, can be turned into a fairly efficient Monero-like confidential transaction system based on the MatRiCT framework of Esgin et al. (CCS 2019). With our new techniques, we are able to reduce the transaction proof size by factors of about 4X - 10X over the aforementioned work. For example, a transaction with two inputs and two outputs, where each input is hidden among \(2^{15}\) other accounts, requires approximately 30KB in our protocol .
This is a preview of subscription content, log in via an institution to check access.
Access this chapter
Tax calculation will be finalised at checkout
Purchases are for personal use only
Notes
- 1.
One can also obtain ring signatures which are linear (rather than logarithmic) in the size of the public key set by plugging in a lattice-based signature scheme based on a trapdoor function, such as [23], into the generic framework of [24]. Even though for small set sizes (around a dozen), this may be smaller than our solution, it quickly becomes much larger (see Fig. 2).
- 2.
For efficiency, a large portion of \(\boldsymbol{B}_i\) can be the identity matrix (c.f. [3]), but we ignore the form of the public randomness in this paper, as it does not affect any output sizes.
- 3.
Actually the inverse NTT of the vector \(\vec {v}_i\), which is an element of \(\mathcal {R}_q\), will be committed – see Sect. 1.3.
- 4.
Intuitively, if the coefficients of \(\vec {v}_i\) were polynomials of degree \(>0\), then the term \(\langle \vec {v}_1,P'(\vec {v}_{2}\otimes \ldots \otimes \vec {v}_m)\rangle \) in (7) would make very little algebraic sense because there is a multiplication on one side of \(P'\) which involves reduction modulo \(X^4-r_j\), and then there would be a multiplication on the other side which would get reduced modulo different \(X^4-r_{j'}\). But since vectors \(\vec {v}_i\) only have constant terms, the “inner product” with \(\vec {v}_i\) does not involve any modular reduction.
- 5.
If there are less than \(l^m\) users then we simply add the zero vectors as public keys so that the ring has exactly \(l^m\) elements. Then the proof that the prover knows a short preimage to one of the columns implies that they must know a preimage to one of the actual public keys because knowing a preimage for one of the zero columns would constitute a SIS solution.
- 6.
References
Attema, T., Lyubashevsky, V., Seiler, G.: Practical product proofs for lattice commitments. In: Micciancio, D., Ristenpart, T. (eds.) CRYPTO 2020. LNCS, vol. 12171, pp. 470–499. Springer, Cham (2020). https://doi.org/10.1007/978-3-030-56880-1_17
Baum, C., Bootle, J., Cerulli, A., del Pino, R., Groth, J., Lyubashevsky, V.: Sub-linear lattice-based zero-knowledge arguments for arithmetic circuits. In: Shacham, H., Boldyreva, A. (eds.) CRYPTO 2018. LNCS, vol. 10992, pp. 669–699. Springer, Cham (2018). https://doi.org/10.1007/978-3-319-96881-0_23
Baum, C., Damgård, I., Lyubashevsky, V., Oechsner, S., Peikert, C.: More efficient commitments from structured lattice assumptions. In: Catalano, D., De Prisco, R. (eds.) SCN 2018. LNCS, vol. 11035, pp. 368–385. Springer, Cham (2018). https://doi.org/10.1007/978-3-319-98113-0_20
Beullens, W., Katsumata, S., Pintore, F.: Calamari and Falafl: logarithmic (linkable) ring signatures from isogenies and lattices. In: Moriai, S., Wang, H. (eds.) ASIACRYPT 2020. LNCS, vol. 12492, pp. 464–492. Springer, Cham (2020). https://doi.org/10.1007/978-3-030-64834-3_16
Bootle, J., Cerulli, A., Chaidos, P., Ghadafi, E., Groth, J., Petit, C.: Short accountable ring signatures based on DDH. In: Pernul, G., Ryan, P.Y.A., Weippl, E. (eds.) ESORICS 2015. LNCS, vol. 9326, pp. 243–265. Springer, Cham (2015). https://doi.org/10.1007/978-3-319-24174-6_13
Bootle, J., Lyubashevsky, V., Seiler, G.: Algebraic techniques for short(er) exact lattice-based zero-knowledge proofs. In: Boldyreva, A., Micciancio, D. (eds.) CRYPTO 2019. LNCS, vol. 11692, pp. 176–202. Springer, Cham (2019). https://doi.org/10.1007/978-3-030-26948-7_7
del Pino, R., Lyubashevsky, V., Seiler, G.: Lattice-based group signatures and zero-knowledge proofs of automorphism stability. In: ACM Conference on Computer and Communications Security, pp. 574–591. ACM (2018)
Ducas, L., Durmus, A., Lepoint, T., Lyubashevsky, V.: Lattice signatures and bimodal Gaussians. In: Canetti, R., Garay, J.A. (eds.) CRYPTO 2013. LNCS, vol. 8042, pp. 40–56. Springer, Heidelberg (2013). https://doi.org/10.1007/978-3-642-40041-4_3
Ducas, L., et al.: Crystals-dilithium: a lattice-based digital signature scheme. IACR Trans. Cryptogr. Hardw. Embed. Syst. 2018(1), 238–268 (2018)
Esgin, M.F., Nguyen, N.K., Seiler, G.: Practical exact proofs from lattices: new techniques to exploit fully-splitting rings. In: Moriai, S., Wang, H. (eds.) ASIACRYPT 2020. LNCS, vol. 12492, pp. 259–288. Springer, Cham (2020). https://doi.org/10.1007/978-3-030-64834-3_9
Esgin, M.F., Steinfeld, R., Liu, J.K., Liu, D.: Lattice-based zero-knowledge proofs: new techniques for shorter and faster constructions and applications. In: Boldyreva, A., Micciancio, D. (eds.) CRYPTO 2019. LNCS, vol. 11692, pp. 115–146. Springer, Cham (2019). https://doi.org/10.1007/978-3-030-26948-7_5
Esgin, M.F., Steinfeld, R., Sakzad, A., Liu, J.K., Liu, D.: Short lattice-based one-out-of-many proofs and applications to ring signatures. In: Deng, R.H., Gauthier-Umaña, V., Ochoa, M., Yung, M. (eds.) ACNS 2019. LNCS, vol. 11464, pp. 67–88. Springer, Cham (2019). https://doi.org/10.1007/978-3-030-21568-2_4
Esgin, M.F., Zhao, R.K., Steinfeld, R., Liu, J.K., Liu, D.: MatRiCT: efficient, scalable and post-quantum blockchain confidential transactions protocol. In: CCS, pp. 567–584. ACM (2019)
Groth, J., Kohlweiss, M.: One-out-of-many proofs: or how to leak a secret and spend a coin. In: Oswald, E., Fischlin, M. (eds.) EUROCRYPT 2015. LNCS, vol. 9057, pp. 253–280. Springer, Heidelberg (2015). https://doi.org/10.1007/978-3-662-46803-6_9
Lu, X., Au, M.H., Zhang, Z.: Raptor: a practical lattice-based (linkable) ring signature. In: Deng, R.H., Gauthier-Umaña, V., Ochoa, M., Yung, M. (eds.) ACNS 2019. LNCS, vol. 11464, pp. 110–130. Springer, Cham (2019). https://doi.org/10.1007/978-3-030-21568-2_6
Lyubashevsky, V.: Fiat-Shamir with aborts: applications to lattice and factoring-based signatures. In: Matsui, M. (ed.) ASIACRYPT 2009. LNCS, vol. 5912, pp. 598–616. Springer, Heidelberg (2009). https://doi.org/10.1007/978-3-642-10366-7_35
Lyubashevsky, V.: Lattice signatures without trapdoors. In: Pointcheval, D., Johansson, T. (eds.) EUROCRYPT 2012. LNCS, vol. 7237, pp. 738–755. Springer, Heidelberg (2012). https://doi.org/10.1007/978-3-642-29011-4_43
Lyubashevsky, V., Nguyen, N.K., Seiler, G.: Practical lattice-based zero-knowledge proofs for integer relations. In: CCS, pp. 1051–1070. ACM (2020)
Lyubashevsky, V., Nguyen, N.K., Seiler, G.: Shorter lattice-based zero-knowledge proofs via one-time commitments. In: Garay, J.A. (ed.) PKC 2021. LNCS, vol. 12710, pp. 215–241. Springer, Cham (2021). https://doi.org/10.1007/978-3-030-75245-3_9
Lyubashevsky, V., Nguyen, N.K., Seiler, G.: SMILE: set membership from ideal lattices with applications to ring signatures and confidential transactions. Cryptology ePrint Archive, Report 2021/564 (2021). https://eprint.iacr.org/2021/564
Lyubashevsky, V., Seiler, G.: Short, invertible elements in partially splitting cyclotomic rings and applications to lattice-based zero-knowledge proofs. In: Nielsen, J.B., Rijmen, V. (eds.) EUROCRYPT 2018. LNCS, vol. 10820, pp. 204–224. Springer, Cham (2018). https://doi.org/10.1007/978-3-319-78381-9_8
Noether, S.: Ring signature confidential transactions for Monero. IACR Cryptol. ePrint Arch. 2015, 1098 (2015)
Prest, T., et al.: FALCON. Technical report, National Institute of Standards and Technology (2017). https://csrc.nist.gov/projects/post-quantum-cryptography/round-1-submissions
Rivest, R.L., Shamir, A., Tauman, Y.: How to leak a secret. In: Boyd, C. (ed.) ASIACRYPT 2001. LNCS, vol. 2248, pp. 552–565. Springer, Heidelberg (2001). https://doi.org/10.1007/3-540-45682-1_32
Yang, R., Au, M.H., Zhang, Z., Xu, Q., Yu, Z., Whyte, W.: Efficient lattice-based zero-knowledge arguments with standard soundness: construction and applications. In: Boldyreva, A., Micciancio, D. (eds.) CRYPTO 2019. LNCS, vol. 11692, pp. 147–175. Springer, Cham (2019). https://doi.org/10.1007/978-3-030-26948-7_6
Acknowledgements
We would like to thank anonymous reviews for useful feedback. This work was supported by the SNSF ERC Transfer Grant CRETP2-166734 FELICITY.
Author information
Authors and Affiliations
Editor information
Editors and Affiliations
Rights and permissions
Copyright information
© 2021 International Association for Cryptologic Research
About this paper
Cite this paper
Lyubashevsky, V., Nguyen, N.K., Seiler, G. (2021). SMILE: Set Membership from Ideal Lattices with Applications to Ring Signatures and Confidential Transactions. In: Malkin, T., Peikert, C. (eds) Advances in Cryptology – CRYPTO 2021. CRYPTO 2021. Lecture Notes in Computer Science(), vol 12826. Springer, Cham. https://doi.org/10.1007/978-3-030-84245-1_21
Download citation
DOI: https://doi.org/10.1007/978-3-030-84245-1_21
Published:
Publisher Name: Springer, Cham
Print ISBN: 978-3-030-84244-4
Online ISBN: 978-3-030-84245-1
eBook Packages: Computer ScienceComputer Science (R0)
