Abstract
In this work, we construct a short one-out-of-many proof from (module) lattices, allowing one to prove knowledge of a secret associated with one of the public values in a set. The proof system builds on a combination of ideas from the efficient proposals in the discrete logarithm setting by Groth and Kohlweiss (EUROCRYPT ’15) and Bootle et al. (ESORICS ’15), can have logarithmic communication complexity in the set size and does not require a trusted setup.
Our work resolves an open problem mentioned by Libert et al. (EUROCRYPT ’16) of how to efficiently extend the above discrete logarithm proof techniques to the lattice setting. To achieve our result, we introduce new technical tools for design and analysis of algebraic lattice-based zero-knowledge proofs, which may be of independent interest.
Using our proof system as a building block, we design a short ring signature scheme, whose security relies on “post-quantum” lattice assumptions. Even for a very large ring size such as 1 billion, our ring signature size is only 3 MB for 128-bit security level compared to 216 MB in the best existing lattice-based result by Libert et al. (EUROCRYPT ’16).
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Notes
- 1.
There are some constructions of ring signatures that give a constant size signature but require a trusted setup.
- 2.
Our scheme, like [18], is only analyzed in the classical random oracle model (ROM) (rather than quantum ROM). Also, note that the linear-sized ring signature schemes are inherently long for large ring sizes.
- 3.
M-SIS is used usually (e.g. in [12]) to fix the ring dimension d and to avoid the need for a change of it to accommodate new security parameters. It does not have a significant effect on efficiency due to extracted witness norm unlike in our case.
- 4.
As in [3], we define M-SIS in “Hermite normal form”, which is equivalent to M-SIS with completely random \(\varvec{A}\).
- 5.
- 6.
A more detailed table is available in the full version of the manuscript [14].
- 7.
In protocol’s application to a ring signature (and for other applications in general), simulation of aborts is not needed as the protocol is made non-interactive.
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Acknowledgements
The work of Ron Steinfeld and Amin Sakzad was supported in part by ARC grant DP150100285. Ron Steinfeld and Joseph K. Liu were also supported in part by ARC grant DP180102199.
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Esgin, M.F., Steinfeld, R., Sakzad, A., Liu, J.K., Liu, D. (2019). Short Lattice-Based One-out-of-Many Proofs and Applications to Ring Signatures. In: Deng, R., Gauthier-Umaña, V., Ochoa, M., Yung, M. (eds) Applied Cryptography and Network Security. ACNS 2019. Lecture Notes in Computer Science(), vol 11464. Springer, Cham. https://doi.org/10.1007/978-3-030-21568-2_4
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