Paper 2021/202
Subtractive Sets over Cyclotomic Rings: Limits of Schnorrlike Arguments over Lattices
Martin R. Albrecht and Russell W. F. Lai
Abstract
We study when (dual) Vandermonde systems of the form ${V}_T^{{(\intercal)}} \cdot \vec{z} = s\cdot \vec{w}$ admit a solution $\vec{z}$ over a ring $\mathcal{R}$, where ${V}_T$ is the Vandermonde matrix defined by a set $T$ and where the "slack" $s$ is a measure of the quality of solutions. To this end, we propose the notion of $(s,t)$subtractive sets over a ring $\mathcal{R}$, with the property that if $S$ is $(s,t)$subtractive then the above (dual) Vandermonde systems defined by any $t$subset $T \subseteq S$ are solvable over $\mathcal{R}$. The challenge is then to find large sets $S$ while minimising (the norm of) $s$ when given a ring $\mathcal{R}$. By constructing families of $(s,t)$subtractive sets $S$ of size $n = $ poly over cyclotomic rings $\mathcal{R} = \mathbb{Z}[\zeta_{p^\ell}]$ for prime $p$, we construct Schnorrlike latticebased proofs of knowledge for the SIS relation ${A} \cdot \vec{x} = s \cdot \vec{y} \bmod q$ with $O(1/n)$ knowledge error, and $s = 1$ in case $p = $ poly. Our technique slots naturally into the lattice Bulletproof framework from Crypto'20, producing latticebased succinct arguments for NP with better parameters. We then give matching impossibility results constraining $n$ relative to $s$, which suggest that our Bulletproofcompatible protocols are optimal unless fundamentally new techniques are discovered. Noting that the knowledge error of lattice Bulletproofs is \(\Omega(\log k/n)\) for witnesses in \(\mathcal{R}^k\) and subtractive set size \(n\), our result represents a barrier to practically efficient latticebased succinct arguments in the Bulletproof framework. Beyond these main results, the concept of $(s,t)$subtractive sets bridges groupbased threshold cryptography to lattice settings, which we demonstrate by relating it to distributed pseudorandom functions.
Metadata
 Available format(s)
 Category
 Cryptographic protocols
 Publication info
 A minor revision of an IACR publication in CRYPTO 2021
 Keywords
 LatticeBased CryptographyLatticebased Zero Knowledge
 Contact author(s)

martin albrecht @ royalholloway ac uk
russell lai @ cs fau de  History
 20210614: last of 2 revisions
 20210301: received
 See all versions
 Short URL
 https://ia.cr/2021/202
 License

CC BY
BibTeX
@misc{cryptoeprint:2021/202, author = {Martin R. Albrecht and Russell W. F. Lai}, title = {Subtractive Sets over Cyclotomic Rings: Limits of Schnorrlike Arguments over Lattices}, howpublished = {Cryptology ePrint Archive, Paper 2021/202}, year = {2021}, note = {\url{https://eprint.iacr.org/2021/202}}, url = {https://eprint.iacr.org/2021/202} }