Abstract
We explore a bitwise modification in Ajtai’s one-way function. Our main contribution is to define the higher-bit approximate inhomogeneous short integer solution (ISIS) problem and prove its reduction to the ISIS problem. In this new instance, our main idea is to discard low-weighted bits to gain compactness.
As an application, we construct a bitwise version of a hash-and-sign signature in the random oracle model whose security relies on the (Ring)-LWE and (Ring)-ISIS assumptions. Our scheme is built from the hash-and-sign digital signature scheme based on the relaxed notion of approximate trapdoors introduced by Chen, Genise and Mukherjee (2019). Their work can be interpreted as a bitwise optimization of the work of Micciancio and Peikert (2012). We extend this idea and apply our technique to the scheme by discarding low-weighted bits in the public key. Our modification brings improvement in the public key size but also in the signature size when used in the right setting.
However, constructions based on the higher-bit approximate ISIS save memory space at the expense of loosening security. Parameters must be set in regards with this trade-off.
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
References
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
Ajtai, M.: Generating hard instances of lattice problems (extended abstract). In: STOC 1996. ACM, New York (1996). https://doi.org/10.1145/237814.237838
Ajtai, M.: Generating hard instances of the short basis problem. In: Wiedermann, J., van Emde Boas, P., Nielsen, M. (eds.) ICALP 1999. LNCS, vol. 1644, pp. 1–9. Springer, Heidelberg (1999). https://doi.org/10.1007/3-540-48523-6_1
Alkim, E., Barreto, P.S.L.M., Bindel, N., Krämer, J., Longa, P., Ricardini, J.E.: The lattice-based digital signature scheme qTESLA. In: Conti, M., Zhou, J., Casalicchio, E., Spognardi, A. (eds.) ACNS 2020. LNCS, vol. 12146, pp. 441–460. Springer, Cham (2020). https://doi.org/10.1007/978-3-030-57808-4_22
Alwen, J., Peikert, C.: Generating shorter bases for hard random lattices. Theor. Comp. Sys. 48(3), 535–553 (2010). https://doi.org/10.1007/s00224-010-9278-3
Brakerski, Z., Langlois, A., Peikert, C., Regev, O., Stehlé, D.: Classical hardness of learning with errors. In: STOC 2013. ACM, New York (2013). https://doi.org/10.1145/2488608.2488680
Cash, D., Hofheinz, D., Kiltz, E., Peikert, C.: Bonsai trees, or how to delegate a lattice basis. J. Cryptol. 25(4), 601–639 (2011). https://doi.org/10.1007/s00145-011-9105-2
Chen, Y.: Private communication (2021)
Chen, Y., Genise, N., Mukherjee, P.: Approximate trapdoors for lattices and smaller hash-and-sign signatures. In: Galbraith, S.D., Moriai, S. (eds.) ASIACRYPT 2019. LNCS, vol. 11923, pp. 3–32. Springer, Cham (2019). https://doi.org/10.1007/978-3-030-34618-8_1
Ducas, L., et al.: CRYSTALS-Dilithium: a lattice-based digital signature scheme. IACR Trans. Cryptographic Hardware Embedded Syst. (1), 238–268 (2018). https://doi.org/10.13154/tches.v2018.i1.238-268
Fouque, P.-A., e.a.: Falcon: fast-Fourier lattice-based compact signatures over NTRU (2018). https://falcon-sign.info/
Gentry, C.: Fully homomorphic encryption using ideal lattices. In: STOC 2009. ACM (2009). https://doi.org/10.1145/1536414.1536440
Gentry, C., Peikert, C., Vaikuntanathan, V.: Trapdoors for hard lattices and new cryptographic constructions. In: STOC 2008. ACM (2008). https://doi.org/10.1145/1374376.1374407
Hoffstein, J., Pipher, J., Silverman, J.H.: NTRU: a ring-based public key cryptosystem. In: Buhler, J.P. (ed.) ANTS 1998. LNCS, vol. 1423, pp. 267–288. Springer, Heidelberg (1998). https://doi.org/10.1007/BFb0054868
Lindner, R., Peikert, C.: Better key sizes (and attacks) for LWE-based encryption. In: Kiayias, A. (ed.) CT-RSA 2011. LNCS, vol. 6558, pp. 319–339. Springer, Heidelberg (2011). https://doi.org/10.1007/978-3-642-19074-2_21
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., Micciancio, D.: Asymptotically efficient lattice-based digital signatures. In: Canetti, R. (ed.) TCC 2008. LNCS, vol. 4948, pp. 37–54. Springer, Heidelberg (2008). https://doi.org/10.1007/978-3-540-78524-8_3
Lyubashevsky, V., Peikert, C., Regev, O.: On ideal lattices and learning with errors over rings. In: Gilbert, H. (ed.) EUROCRYPT 2010. LNCS, vol. 6110, pp. 1–23. Springer, Heidelberg (2010). https://doi.org/10.1007/978-3-642-13190-5_1
Micciancio, D., Regev, O.: Worst-case to average-case reductions based on Gaussian measures. In: 45th IEEE Symposium on FOCS, pp. 372–381 (2004). https://doi.org/10.1109/FOCS.2004.72
Micciancio, D.: Generalized compact knapsacks, cyclic lattices, and efficient one-way functions from worst-case complexity assumptions. In: FOCS 2002. IEEE (2002). https://doi.org/10.1109/SFCS.2002.1181960
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
Moody, D., et al.: Status report on the second round of the NIST post-quantum cryptography standardization process (2020). https://doi.org/10.6028/NIST.IR.8309
Peikert, C.: An efficient and parallel Gaussian sampler for lattices. In: Rabin, T. (ed.) CRYPTO 2010. LNCS, vol. 6223, pp. 80–97. Springer, Heidelberg (2010). https://doi.org/10.1007/978-3-642-14623-7_5
Regev, O.: On lattices, learning with errors, random linear codes, and cryptography. J. ACM 56(6) (2009). https://doi.org/10.1145/1568318.1568324
Shor, P.W.: Algorithms for quantum computation: discrete logarithms and factoring. In: SFCS 1994. IEEE Computer Society (1994). https://doi.org/10.1109/SFCS.1994.365700
Acknowledgments
We would like to thank Yilei Chen, Nicholas Genise and Pratyay Mukherjee for kindly sharing with us their implementation of Hash-and-Sign signature based on F-trapdoors. We are especially grateful to Yilei Chen for his invaluable advice to our work.
Author information
Authors and Affiliations
Corresponding author
Editor information
Editors and Affiliations
Rights and permissions
Copyright information
© 2021 Springer Nature Switzerland AG
About this paper
Cite this paper
Le Dévéhat, A., Shizuya, H., Hasegawa, S. (2021). On the Higher-Bit Version of Approximate Inhomogeneous Short Integer Solution Problem. In: Conti, M., Stevens, M., Krenn, S. (eds) Cryptology and Network Security. CANS 2021. Lecture Notes in Computer Science(), vol 13099. Springer, Cham. https://doi.org/10.1007/978-3-030-92548-2_14
Download citation
DOI: https://doi.org/10.1007/978-3-030-92548-2_14
Published:
Publisher Name: Springer, Cham
Print ISBN: 978-3-030-92547-5
Online ISBN: 978-3-030-92548-2
eBook Packages: Computer ScienceComputer Science (R0)