Paper 2026/2654

When Algebraic Structure Meets Too Many Hints

Abstract

The hint framework of Dachman-Soled, Ducas, Gong, and Rossi (CRYPTO 2020), together with the later too-many-hints analyses from May and Nowakowski (ASIACRYPT 2023) and Cao, Jiang, and Cheng (CRYPTO 2025), gives a classification of secret-dependent hints and LLL-breaking hint thresholds for LWE schemes. These values are now frequently adopted as the default benchmark for side-channel security evaluation. We show that applying them as a security metric is problematic. DDGR categorizes hints based only on their local algebraic form, while the threshold analyses assume uniformly sampled hint vectors. Neither captures how the specific algebraic form of a hint governs the geometry of the resulting sparsified lattice. A hint’s class label and the corresponding uniform threshold therefore cannot, on their own, capture its real cryptanalytic power. We illustrate this mismatch with two forms of structured leakage that naturally appear in ring and module lattice implementations, validated via synthetic experiments on Kyber-512. NTT-domain residues qualify formally as mod-$q$ hints, yet they recover the full Kyber-512 secret with roughly half as many hints as the uniform mod-$q$ bound predicts. Finite-precision FFT samples fall into the weaker approximate-hint class, and still only need $25\%$--$55\%$ of the hint budget required for uniform perfect hints. The gap stems from algebraic structure: these hint families carry additional ring structure that enables tailored lattice recovery strategies, a distinction invisible under the uniform-sampling model. We do not propose new attack optimizations. Our findings indicate that DDGR hint classes and too-many-hints thresholds are syntactic descriptions, not standalone security metrics for real leakage traces. Evaluating a side-channel leak requires considering the algebraic structure of the hint vectors, not just their category and quantity.