Paper 2026/2202

Fully Anonymous Perfect Threshold Secret Sharing: Explicit One-Bit Construction and Rate Amplification

Abstract

Secret sharing distributes a secret among participants so that only sufficiently large groups can recover it. Fully anonymous secret sharing also allows recovery from share values alone, without participant identities. Below the reconstruction threshold, the shares must look exactly like independent uniform random values, revealing neither the secret nor their holders' identities. Con, Li, and Mazor established that such schemes with error-free recovery exist for every threshold, leaving an efficient general construction open. We give efficient constructions for every threshold 2 ≤ t ≤ n among n participants. For an L-bit secret, each share has L + O(t² log² n) bits. Sharing runs in expected polynomial time, and reconstruction runs in deterministic polynomial time. Our approach combines an efficient one-bit construction with a general transformation that uses it to protect a short message needed for anonymous reconstruction of a long secret. The additional share length has an upper bound depending only on the number of participants and the threshold. For a fixed participant count and threshold, the rate approaches one as secrets grow longer, while anonymity remains exact and recovery never errs.