Full threshold change range of threshold changeable secret sharing

A threshold changeable secret sharing (TCSS) scheme is designed for changing the initial threshold pair of the privacy threshold and reconstruction threshold to a given threshold pair after the dealer distributes shares to participants, while a universal threshold changeable secret sharing (uTCSS) scheme is threshold changeable to multiple new threshold pairs. We focus on the threshold changeability in a dealer-free scenario with an outside adversary and the absence of secure channels among participants. There are some known threshold change regimes that are realized by (optimal) TCSS schemes or (optimal) uTCSS schemes. In this work, by combining the frequently used two methods in previous constructions: folding shares of a given secret sharing scheme and packing shares of multiple secret sharing schemes, we construct an optimal TCSS scheme and an optimal uTCSS scheme with a new threshold change regime, respectively. This helps us determine the full threshold change range that can be realized by optimal TCSS schemes and optimal uTCSS schemes, respectively. Moreover, we construct some near optimal TCSS schemes to show that the full threshold change range of TCSS schemes (without requiring optimality) is completely covered by the threshold change regimes of our near optimal TCSS schemes together with the full threshold change range of optimal TCSS schemes.