Existence of alternative identical leakage functions for additive secret sharing

Determine whether leakage functions other than the Guruswami–Wootters trace-based functions can exploit the simple additive relation in Shamir secret sharing while using identical leakage functions for all three codewords.

Background

The paper proves that, for the simple addition relation, any repair scheme using identical leakage functions for the three codewords reduces to a repair scheme for the underlying base code. Consequently, identical leakage functions cannot improve on the base-case bound through the specific construction analyzed in the paper.

The authors note that exploiting simple addition would require other leakage functions, but leave unresolved whether any such functions exist. This is a concrete open existence question concerning more realistic attacks in which the same leakage function is reused across shares or computations.

References

In cryptographic terms, for Shamir secret sharing, this means that exploiting the simple additive relation requires leakage functions other than those of, if such functions exist.

— On the Leakage of Massey Secret Sharing Schemes under Linear Computations  (2609.19929 - Aoutouf et al., 17 Sep 2026) in Section 5, “Use of Identical Leakage Functions”