Ratio Attack on G+G Convoluted Gaussian Signature
Chik How Tan, Theo Fanuela Prabowo, Wei Guo Foo
TL;DR
The paper addresses the security of the G+G convoluted Gaussian signature (DPS23) by introducing a ratio-attack that exploits correlations among signatures to recover the secret key. It shows that the ratio of two correlated signature coordinates follows a (truncated) Cauchy distribution, and derives a relation $\mathbb{E}\left(\frac{Z_{i,j}}{Z_{0,0}}\right)=s_{i,j}\alpha_*$ with $\alpha_*=(\sigma_{W_0}^2-\sigma_u^2)/\sigma_{Z_{0,0}}^2$, enabling key recovery from $N$ samples with $N=O(1/p_*^2)$. A SageMath proof-of-concept using scaled-down parameters demonstrates complete secret-key recovery, and the authors discuss DPS23e Security claims and practical parameter issues, noting that secure parameterization remains challenging due to covariance-positivity and forgery risks. Overall, the work highlights vulnerable parameter regimes and informs secure parameter design for lattice-based signatures relying on G+G constructions, with a concrete method to test resilience via ratio statistics.
Abstract
A lattice-based signature, called G+G convoluted Gaussian signature, was proposed in ASIACRYPT 2023 and was proved secure in the quantum random oracle model. In this paper, we propose a ratio attack on the G+G convoluted Gaussian signature to recover the secret key and comment on the revised eprint paper. The attack exploits the fact, proved in this paper, that the secret key can be obtained from the expected value of the ratio of signatures which follows a truncated Cauchy distribution. Moreover, we also compute the number of signatures required to successfully recover the secret key. Furthermore, we simulate the ratio attack in Sagemath with a few different parameters as a proof-of-concept of the ratio attack. In addition, although the revised signature in the revised eprint paper is secure against the ratio attack, we found that either a valid signature cannot be produced or a signature can be forged easily for their given parameters in the eprint.