Leveled Lattice-Based Linearly Homomorphic Signature Scheme in the Standard Model for Network Coding
Linearly homomorphic signature scheme is an important cryptographic primitive which can be used to against the pollution attacks in network coding. To achieve the security protection for network coding even in quantum environment, an efficient lattice-based linearly homomorphic signature scheme in the standard model is proposed in this paper. Unlike the known lattice-based scheme in the standard model, in our construction, lattice-based delegation algorithm is not needed to achieve the standard security. Hence, all the messages are signed over the same lattice in the proposed scheme. Hence, the public key of the proposed scheme only consists as a group of vectors compared with that a group of public and random matrices are necessary in known construction used lattice-based delegation tool. As a result, the public key size of the proposed scheme is shorter than that of the known lattice-based schemes (standard model). Moreover, the proposed scheme also shares advantage about the signature length. Based on the hardness of the standard short integer solution problem, we prove that the proposed scheme is adaptively unforgeable against the type 1 and type 2 adversaries in the standard model. We also shown that the proposed scheme satisfies the weakly context hiding property.
KeywordsLinearly homomorphic signature Standard model Lattice Short integer solution Pre-image sampling function
This work was supported in part by the National Natural Science Foundation of China under Grant 61803228, Project of Shandong Province Higher Education Science and Technology Program under grant J18KA361.
- 2.Arita, S., Kozaki, S.: A homomorphic signature scheme for quadratic polynomials, in Smart Computing (SMARTCOMP). In: 2017 IEEE International Conference on, IEEE, pp. 1–6 (2017)Google Scholar
- 3.Alwen, J., Peikert, C.: Generating shorter bases for hard random lattices. In: Proceedings of 26th International Symposium on Theoretical Aspects of Computer Science, vol. 09001, Freiburg, Germany, pp. 75–86 (2009)Google Scholar
- 4.Boneh, D., Freeman, D.M., Katz, J., et al.: Singing a linear subspace: signature schemes for network coding. In: Proceedings of PKC 2009, LNCS 5443, pp. 68–87. Springer-Verlag, Berlin (2009)Google Scholar
- 5.Boneh, D., Freeman, D.M.: Linearly homomorphic signatures over binary fields and new tools for lattice-based signatures. In: Catalano, D., Fazio, N., Gennaro, R., Nicolosi, A. (eds.) PKC 2011. LNCS, vol. 6571, pp. 1–16. Springer, Heidelberg (2011). https://doi.org/10.1007/978-3-642-19379-8_1CrossRefGoogle Scholar
- 13.Gentry, C., Peikert, C., Vaikuntanathan, V.: Trapdoors for hard lattices and new cryptographic constructions. In: Proceedings of the 40th Annual ACM Symposium on Theory of Computing STOC 2008, British Columbia, Canada, pp. 197–206 (2008)Google Scholar
- 14.Gorbunov, S., Vaikuntanathan, V., Wichs, D.: (Leveled) fully homomorphic signatures from lattices. In: Proceedings of STOC, pp. 469–477 (2015)Google Scholar
- 16.Liu, H.W., Cao, W.M.: Public proof of cloud storage from lattice assumption. Chin. J. Electron. 23(1), 186–190 (2014)Google Scholar
- 18.Micciancio, D., Regev, O.: Worst-case to average-case reductions based on gaussian measures. In: Proceedings of 45th Annual IEEE Symposium on Foundations of Computer Science (FOCS), Rome, Italy, pp. 372–381 (2004)Google Scholar
- 20.Boyen, X., Fan, X., Shi, E.: Adaptively secure fully homomorphic signatures based on lattices. IACR Cryptology ePrint Archive, 916 (2014)Google Scholar