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Shorter Lattice-Based Group Signatures via “Almost Free” Encryption and Other Optimizations

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Advances in Cryptology – ASIACRYPT 2021 (ASIACRYPT 2021)

Part of the book series: Lecture Notes in Computer Science ((LNSC,volume 13093))

Abstract

We present an improved lattice-based group signature scheme whose parameter sizes and running times are independent of the group size. The signature length in our scheme is around 200KB, which is approximately a 3X reduction over the previously most compact such scheme, based on any quantum-safe assumption, of del Pino et al. (CCS 2018). The improvement comes via several optimizations of some basic cryptographic components that make up group signature schemes, and we think that they will find other applications in privacy-based lattice cryptography.

Supported by the SNSF ERC Transfer Grant CRETP2-166734 FELICITY and the EU H2020 ERC Project 101002845 PLAZA.

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Notes

  1. 1.

    The message \(\mu \) enters the signature as an input to a hash function that is used to convert the interactive proof into a non-interactive one via the Fiat-Shamir transform.

  2. 2.

    While it’s insecure for \(S=\mathcal {R}_q\), it’s unclear whether the size of S actually affects the real security of the scheme or it’s just an artefact of the proof.

  3. 3.

    Observe that we cannot use \(m_{bin}\) as our identity because the set of polynomials with 0/1 NTT coefficients is not closed under subtraction – hence this conversion is necessary.

  4. 4.

    Sometimes to save on computation time, the vector \(\boldsymbol{A}_0\) and \(\boldsymbol{b}_1\) can contain some polynomials that are just 0 or 1 (see [BDL+18]), but in our case we will need them to be uniformly random.

  5. 5.

    In principle, d does not need to be a power-of-2, but then we could not work with the very convenient polynomial rings \(\mathbb {Z}[X]/(X^d+1)\). We think that the slight saving in the public key size is not worth the extra hassle of working aver different rings, and so we only consider power-of-2 d.

  6. 6.

    Here, the inner product is over \(\mathbb {Z}\), i.e. \(\langle \boldsymbol{z},\boldsymbol{v} \rangle = \langle \vec {z},\vec {v} \rangle \) where vectors \(\vec {z},\vec {v}\) are polynomial coefficients of \(\boldsymbol{z}\) and \(\boldsymbol{v}\) respectively.

  7. 7.

    I.e that the group manager can decrypt it and recover the identity m.

  8. 8.

    Equation (31) holds because g’s first d/l coefficients are set to be 0.

  9. 9.

    That is to say \(\bar{\mathbf {y}}_4 \ne \bar{\mathbf {y}}_4'\).

  10. 10.

    Challenges \((c,\phi )\) are in a heavy row when the success probability of the prover conditionned on the first challenges to be these \(c,\phi \) is at least \(\epsilon /2\). We refer to [OO98] for further detail.

  11. 11.

    Otherwise, \(\bar{\mathbf {r}} \bar{e}' - \bar{\mathbf {r}}' \bar{e}\) is a solution for \(\mathsf {MSIS}\) for \(\mathbf {A}_0\) of norm at most \(8 \omega ^2 \sigma ' \sqrt{2 (\kappa + \lambda + \alpha + 5) d}\).

  12. 12.

    Recall that in Fig. 5 we run four rejection algorithms. However, for efficiency purposes we can merge the ones for \(\boldsymbol{z}_1, \boldsymbol{z}_2, \boldsymbol{z}_3\) since they follow the same standard deviation \(\sigma \).

References

  1. Agrawal, S., Boneh, D., Boyen, X.: Efficient lattice (H)IBE in the standard model. In: Gilbert, H. (ed.) EUROCRYPT 2010. LNCS, vol. 6110, pp. 553–572. Springer, Heidelberg (2010). https://doi.org/10.1007/978-3-642-13190-5_28

    Chapter  MATH  Google Scholar 

  2. 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

    Chapter  Google Scholar 

  3. Banaszczyk, W.: New bounds in some transference theorems in the geometry of numbers. Math. Ann. 296(1), 625–635 (1993)

    Article  MathSciNet  Google Scholar 

  4. Boschini, C., Camenisch, J., Neven, G.: Floppy-sized group signatures from lattices. In: Preneel, B., Vercauteren, F. (eds.) ACNS 2018. LNCS, vol. 10892, pp. 163–182. Springer, Cham (2018). https://doi.org/10.1007/978-3-319-93387-0_9

    Chapter  Google Scholar 

  5. 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

    Chapter  Google Scholar 

  6. 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

    Chapter  Google Scholar 

  7. Camenisch, J., Stadler, M.: Efficient group signature schemes for large groups. In: Kaliski, B.S. (ed.) CRYPTO 1997. LNCS, vol. 1294, pp. 410–424. Springer, Heidelberg (1997). https://doi.org/10.1007/BFb0052252

    Chapter  Google Scholar 

  8. Ducas, L., Durmus, A., Lepoint, T., Lyubashevsky, V.: Lattice signatures and bimodal gaussians. In CRYPTO 1, 40–56 (2013)

    MathSciNet  MATH  Google Scholar 

  9. Ducas, L., Kiltz, E., Lepoint, T., Lyubashevsky, V., Schwabe, P., Seiler, G., Stehlé, D.: Crystals-dilithium: a lattice-based digital signature scheme. IACR Trans. Cryptogr. Hardw. Embed. Syst. 2018(1), 238–268 (2018)

    Article  Google Scholar 

  10. 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)

    Google Scholar 

  11. 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

    Chapter  Google Scholar 

  12. 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

    Chapter  Google Scholar 

  13. 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

    Chapter  MATH  Google Scholar 

  14. 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)

    Google Scholar 

  15. 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

    Chapter  Google Scholar 

  16. Gordon, S.D., Katz, J., Vaikuntanathan, V.: A group signature scheme from lattice assumptions. In: Abe, M. (ed.) ASIACRYPT 2010. LNCS, vol. 6477, pp. 395–412. Springer, Heidelberg (2010). https://doi.org/10.1007/978-3-642-17373-8_23

    Chapter  Google Scholar 

  17. Gentry, C., Sahai, A., Waters, B.: Homomorphic encryption from learning with errors: conceptually-simpler, asymptotically-faster, attribute-based. In: Canetti, R., Garay, J.A. (eds.) CRYPTO 2013. LNCS, vol. 8042, pp. 75–92. Springer, Heidelberg (2013). https://doi.org/10.1007/978-3-642-40041-4_5

    Chapter  Google Scholar 

  18. Libert, B., Ling, S., Nguyen, K., Wang, H.: Zero-knowledge arguments for lattice-based accumulators: logarithmic-size ring signatures and group signatures without trapdoors. In: Fischlin, M., Coron, J.-S. (eds.) EUROCRYPT 2016. LNCS, vol. 9666, pp. 1–31. Springer, Heidelberg (2016). https://doi.org/10.1007/978-3-662-49896-5_1

    Chapter  Google Scholar 

  19. Lyubashevsky, V., Neven, G.: One-shot verifiable encryption from lattices. In: Coron, J.-S., Nielsen, J.B. (eds.) EUROCRYPT 2017. LNCS, vol. 10210, pp. 293–323. Springer, Cham (2017). https://doi.org/10.1007/978-3-319-56620-7_11

    Chapter  Google Scholar 

  20. Lyubashevsky, V., Nguyen, N.K., Seiler, G.: Practical lattice-based zero-knowledge proofs for integer relations. In: CCS, pp. 1051–1070. ACM (2020)

    Google Scholar 

  21. 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

    Chapter  Google Scholar 

  22. Lyubashevsky, V., Nguyen, N.K., Seiler, G.: SMILE: set membership from ideal lattices with applications to ring signatures and confidential transactions. In: Malkin, T., Peikert, C. (eds.) CRYPTO 2021. LNCS, vol. 12826, pp. 611–640. Springer, Cham (2021). https://doi.org/10.1007/978-3-030-84245-1_21

    Chapter  Google Scholar 

  23. Langlois, A., Stehlé, D.: Worst-case to average-case reductions for module lattices. Des. Codes Crypt. 75(3), 565–599 (2014). https://doi.org/10.1007/s10623-014-9938-4

    Article  MathSciNet  MATH  Google Scholar 

  24. 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

    Chapter  Google Scholar 

  25. 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

    Chapter  Google Scholar 

  26. Micciancio, D., Peikert, C.: Trapdoors for lattices: simpler, tighter, faster, smaller. In: Pointcheval, D., Johansson, T. (eds.) EUROCRYPT 2012. LNCS, vol. 7237, pp. 700–718. Springer, Heidelberg (2012). https://doi.org/10.1007/978-3-642-29011-4_41

    Chapter  Google Scholar 

  27. Ohta, K., Okamoto, T.: On concrete security treatment of signatures derived from identification. In: Krawczyk, H. (ed.) CRYPTO 1998. LNCS, vol. 1462, pp. 354–369. Springer, Heidelberg (1998). https://doi.org/10.1007/BFb0055741

    Chapter  Google Scholar 

  28. Regev, O.: On lattices, learning with errors, random linear codes, and cryptography. J. ACM 56(6), 1–40 (2009)

    Article  MathSciNet  Google Scholar 

  29. 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

    Chapter  Google Scholar 

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Authors and Affiliations

  1. IBM Research Europe, Zurich, Switzerland

    Vadim Lyubashevsky, Ngoc Khanh Nguyen, Maxime Plancon & Gregor Seiler

  2. ETH Zurich, Zurich, Switzerland

    Ngoc Khanh Nguyen, Maxime Plancon & Gregor Seiler

Authors
  1. Vadim Lyubashevsky
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  2. Ngoc Khanh Nguyen
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  3. Maxime Plancon
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  4. Gregor Seiler
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Corresponding authors

Correspondence to Ngoc Khanh Nguyen , Maxime Plancon or Gregor Seiler .

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Editors and Affiliations

  1. NTT Corporation, Tokyo, Japan

    Mehdi Tibouchi

  2. Nanyang Technological University, Singapore, Singapore

    Huaxiong Wang

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Lyubashevsky, V., Nguyen, N.K., Plancon, M., Seiler, G. (2021). Shorter Lattice-Based Group Signatures via “Almost Free” Encryption and Other Optimizations. 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_8

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