Paper 2026/1971

Component-Dual Compression and the Exact Characteristic-Two Contribution Region of the Relaxed Non-Fano Port

Abstract

Jafari and Khazaei (Journal of Cryptology, 2021) introduced a kernel-based lower-bound method for linear secret-sharing schemes by fixing one minimal qualified coalition and comparing its participant components with those arising from auxiliary minimal qualified coalitions. These comparisons form a star. We extend the same mechanism from stars to coalition-labelled trees and obtain new characteristic-two inequalities for the relaxed-line non-Fano port $\widehat N$. For this access structure, the tree method is strictly stronger than the star method, yielding facet inequalities not implied by the star inequalities. We also show that the tree method does not determine the full contribution region. To complete the analysis of $\widehat N$, we introduce component-dual compression (CDC). CDC replaces each share by the span of the components selected from minimal reconstructions and realizes the duals of these compressed spaces in a common coordinate system indexed by the original minimal coalitions. This yields the remaining lower-bound inequalities. Together with matching constructions, the star, tree, and CDC bounds determine the complete characteristic-two linear contribution region of $\widehat N$, with maximum and average linear information ratios $5/4$.