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

• URL: http://arxiv.org/abs/2607.28757v1
• Date: Thu, 30 Jul 2026 18:24:52 GMT
• Title: Partial Derandomization for Leakage-Resilient Shamir's Secret Sharing over Composite Order Fields
• Abstract summary: Previously, Maji et al. showed that random evaluation places yield Shamir's secret sharing over the composite order field $mathbbF_pd$.<n>We present a partial derandomization of evaluation places, improving upon Maji et al. result for a restricted regime of parameters.<n>Our construction is valid for the regime $n = O(d/log_p d)$, and any reconstruction threshold $k ge 2$; in fact, the scheme attains perfect security (statistical distance exactly zero) against single-block leakage
• Score: 0.16244541005112745
• License: http://creativecommons.org/licenses/by/4.0/
• 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 $\mathbb{F}_{p^d}$ 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 $\mathbb{F}_{p^d}$ 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 $x_j = Φ^j(x_0)$ of a simple fixed rational function $Φ$, where the initial point $x_0 \in \mathbb{F}_{p^d}^*$ is randomly chosen. The randomness in the evaluation places thus drops from $nd \log p$ bits to $d\log p$ bits. Our construction is valid for the regime $n = O(d/\log_p d)$, and any reconstruction threshold $k \ge 2$; in fact, the scheme attains perfect security (statistical distance exactly zero) against single-block leakage. Our technique is a partial fraction nondegeneracy argument that exploits the distinct poles of the rational iterates.

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