Partial Derandomization for Leakage-Resilient Shamir’s Secret Sharing over Composite Order Fields

Abstract

We make progress on the question of constructing explicit evaluation places for leakage-resilient Shamir’s secret sharing, over composite order fields. Previously, Maji et al. (EUROCRYPT 2024) showed that random evaluation places yield Shamir’s secret sharing over the composite order field 𝔽_{p^d} that is statistically secure against physical bit leakage. Later, Nguyen (EUROCRYPT 2025) established a dichotomy that linear code-based secret-sharing scheme over the field 𝔽_{p^d} is either statistically secure or completely insecure against such leakage.
Building upon Nguyen’s dichotomy, we present a partial derandomization of evaluation places, improving upon the Maji et al. result for a restricted regime of parameters. We replace the random choice of n independent evaluation places by the iterates x_j = Φ^j(x₀) of a simple fixed rational function Φ, where the initial point x₀ ∈ 𝔽_{p^d}^* is randomly chosen. The randomness in the evaluation places thus drops from nd log p bits to dlog p bits. Our construction is valid for the regime n = O(d/log_p d), and any reconstruction threshold k ≥ 2; in fact, the scheme attains perfect security (statistical distance exactly zero) against single-block leakage.
Building on Nguyen’s dichotomy, our technique is a partial-fraction non-degeneracy argument that exploits the distinct poles of the rational iterates.