Optimal share size in the remaining black-box secret-sharing cases

Determine whether the gap between the lower and upper bounds for share size in black-box threshold secret-sharing schemes over finite commutative groups of unknown order can be closed in the cases not already covered by the known optimal constructions.

Background

The paper reviews black-box threshold secret-sharing schemes whose shares are represented by group elements. Existing results give lower bounds on average share size, while the construction of Cramer, Fehr, and Shelat attains the lower bound in particular cases, such as when the number of receivers is a power of two.

The authors explicitly leave open whether the remaining discrepancy between known lower and upper bounds can be eliminated.

References

This leaves at least two questions open: an obvious one is whether we can close the gap between lower and upper bound for share size in the remaining cases; a second, more practical'' question is whether we can obtain betteramortized'' share size in case we share a tuple of secrets from $G$, rather than a single one.

— On exceptional cliques in matrix rings  (2608.24586 - Boutros et al., 25 Aug 2026) in Section 9 (Exceptional cliques in cryptography)

This leaves at least two questions open: an obvious one is whether we can close the gap between lower and upper bound for share size in the remaining cases; a second, more practical'' question is whether we can obtain betteramortized'' share size in case we share a tuple of secrets from $G$, rather than a single one.

— On exceptional cliques in matrix rings  (2608.24586 - Boutros et al., 25 Aug 2026) in Section 9 (Exceptional cliques in cryptography)