Determine the correct rate threshold for one-bit local leakage resilience

Determine the smallest rate threshold R^* such that every Shamir secret-sharing scheme over a prime field with rate at least R^* is one-bit locally leakage resilient, and establish whether the conjectured threshold can be an arbitrarily small positive constant.

Background

The paper studies one-bit local leakage resilience (LLR) for Shamir secret sharing over prime fields. Its main LLR result proves resilience for rates slightly below one half, thereby breaking the previously identified one-half barrier, but it does not determine the optimal threshold. The unresolved problem is to identify the smallest rate at which one-bit LLR holds for all Shamir schemes and to resolve the cited conjecture that this threshold may be any arbitrarily small positive constant. The authors explicitly note that substantial progress remains necessary beyond their result.

References

What is the ``correct'' rate threshold for LLR for Shamir schemes? That is, what is the smallest $R*$ so that any Shamir scheme with rate at least $R*$ is one-bit LLR? A conjecture of posits that $R*$ can be an arbitrarily small constant; we have shown that $R*$ can be less than $1/2$, but there is a long way to go.

— Improved Local Leakage Resilience of Shamir Secret Sharing and Worst-Case Optimal Polynomial Intersection  (2610.12357 - Sun et al., 8 Oct 2026) in Section ‘Open Problems’

The full BDIR conjecture remains open, in particular at threshold rate 1/3.

— Local Leakage Resilience of Shamir Secret Sharing: Below Half Threshold Rate  (2610.17711 - Nguyen, 13 Oct 2026) in Section 1, subsection “Our contributions,” paragraph “Below half”

At smaller thresholds, Nielsen and Simkin give an inefficient one-bit leakage attack when k is of order $n /\log p$, which is n/\log n when p=poly(n). Closing this gap remains an open problem.

— Local Leakage Resilience of Shamir Secret Sharing: Below Half Threshold Rate  (2610.17711 - Nguyen, 13 Oct 2026) in Section 1, subsection “Our contributions,” paragraph “Below half”

The intermediate range is open.

— Local Leakage Resilience of Shamir Secret Sharing: Below Half Threshold Rate  (2610.17711 - Nguyen, 13 Oct 2026) in Section 4, subsection “Field size for square-root thresholds,” paragraph “A contrasting attack at square-root thresholds”