Gram Root Decomposition over the Polynomial Ring: Application to Sphericalization of Discrete Gaussian

Hiroki Okada, Hiroki Okada, Tsuyoshi Takagi

2025

Abstract

Efficient construction of lattice-based cryptography is often based on the polynomial ring. Furthermore, many advanced lattice-based cryptosystems require the analysis of the discrete Gaussian under convolutions and linear transformations. In this paper, we present an efficient Gram root decomposition algorithm of the polynomial ring and an application to sphericalization of the discrete Gaussian. Let r be a polynomial of spherical discrete Gaussian coefficients and e be a fixed polynomial. Then, the coefficient vector of r · e is (statistically close to) non-spherical discrete Gaussian whose (scaled) covariance matrix is Ge := EE⊺, where E is composed of rotations of the coefficient vector of e. Given Ge, our algorithm outputs polynomials ζ1,...,ζl s.t. ∑l i=1 Gζi +Ge is a scalar matrix. The objective of this algorithm is similar to the (ring version of) integral Gram root decomposition proposed by Ducas et al. (Eurocrypt 2020). Notably, our algorithm ensures the bounds of the norm of ζi and the minimum eigenvalue of Gζi , whereas Ducas et al.’s algorithm does not ensure such bounds. Owing to the bounds, we can obtain a polynomial (r0e + ∑l i=1 riζi) whose coefficients are spherical discrete Gaussians, where ri are polynomials with discrete Gaussian coefficients; i.e., we can “cancel out” the dependence between the coefficients.

Paper Citation

in Harvard Style

Okada H. and Takagi T. (2025). Gram Root Decomposition over the Polynomial Ring: Application to Sphericalization of Discrete Gaussian. In Proceedings of the 11th International Conference on Information Systems Security and Privacy - Volume 2: ICISSP; ISBN 978-989-758-735-1, SciTePress, pages 306-317. DOI: 10.5220/0013139700003899

in Bibtex Style

@conference{icissp25,
author={Hiroki Okada and Tsuyoshi Takagi},
title={Gram Root Decomposition over the Polynomial Ring: Application to Sphericalization of Discrete Gaussian},
booktitle={Proceedings of the 11th International Conference on Information Systems Security and Privacy - Volume 2: ICISSP},
year={2025},
pages={306-317},
publisher={SciTePress},
organization={INSTICC},
doi={10.5220/0013139700003899},
isbn={978-989-758-735-1},
}

in EndNote Style

TY - CONF

JO - Proceedings of the 11th International Conference on Information Systems Security and Privacy - Volume 2: ICISSP
TI - Gram Root Decomposition over the Polynomial Ring: Application to Sphericalization of Discrete Gaussian
SN - 978-989-758-735-1
AU - Okada H.
AU - Takagi T.
PY - 2025
SP - 306
EP - 317
DO - 10.5220/0013139700003899
PB - SciTePress