Information Rates of Minimal Non-Matroid-Related Access Structures

In a secret sharing scheme, shares of a secret are distributed to participants in such a way that only certain predetermined sets of participants are qualified to reconstruct the secret. An access structure on a set of participants specifies which sets are to be qualified. The information rate of an access structure is a bound on how efficient a secret sharing scheme for that access structure can be. Marti-Farre and Padro showed that all access structures with information rate greater than two-thirds are matroid-related, and Stinson showed that four of the minor-minimal, non-matroid-related access structures have information rate exactly two-thirds. By a result of Seymour, there are infinitely many remaining minor-minimal, non-matroid-related access structures. In this paper we find the exact information rates for all such structures.

💡 Research Summary

The paper determines the exact information rate of the infinite family of “king‑and‑n‑pawns” access structures, denoted Γₙ, for every n ≥ 2. An access structure Γ on a participant set P specifies which subsets are qualified to reconstruct a secret. The information rate ρ(Γ) is the supremum of the minimum participant‑wise rate over all secret‑sharing schemes that realize Γ. Prior work showed that any access structure with rate > 2⁄3 must be matroid‑related, and that four minor‑minimal non‑matroid‑related structures have rate exactly 2⁄3. Seymour proved that the only infinite family of minor‑minimal non‑matroid‑related structures consists of the Γₙ’s (together with three small examples). However, the exact rates of Γₙ had not been known.

The authors first formalize secret sharing using random variables: S for the secret and X₁,…,Xₙ for participants. They define a normalized entropy function h(X)=H(X)/H(S). This function satisfies monotonicity, submodularity, and an additional “+‑submodularity” that applies when two qualified sets intersect in an unqualified set. Using these properties they prove two lemmas:

• Lemma 2: h(k p₁…p_{n‑1}) ≥ h(p₁)+(n‑1).
• Lemma 3: h(k p₁)+∑{i=2}^{n}h(p_i) ≥ h(k p₁…p{n‑1})+(n‑2).

Combining the lemmas with the basic submodular inequality h(p₁)+h(k) ≥ h(k p₁) yields h(k)+∑_{i=2}^{n}h(p_i) ≥ 2n‑3. Since the minimum of the h‑values over all participants is at most the average, at least one participant p* satisfies h(p*) ≥ (2n‑3)/(n‑1). Because the information rate of a participant is the reciprocal of h, this gives the universal upper bound ρ(Γₙ) ≤ (n‑1)/(2n‑3).

To match this bound, the paper constructs an explicit secret‑sharing scheme Σ whose rate equals (n‑1)/(2n‑3). Two auxiliary schemes are built first:

1. Σ₁ – a variant of Shamir’s (n, 2n‑1) threshold scheme. The secret s∈𝔽_q is encoded as a random polynomial f of degree ≤ n‑1 with f(0)=s. The king receives the evaluations f(1),…,f(n‑1); pawn p_i receives f(n‑1+i). Each share is uniformly distributed over 𝔽_q, and any qualified set can reconstruct s.

2. Σ₂ – a combination of a (2,2) threshold scheme and an (n,n) threshold scheme (the “decomposition method”). In the (2,2) part the king gets a random r and each pawn receives r+s, making every {king, pawn} qualified. In the (n,n) part the first n‑1 pawns receive independent random values r_i, while the last pawn receives s+∑_{i=1}^{n‑1} r_i, ensuring that the whole set of pawns is qualified. All shares are independent and uniformly distributed.

The final scheme Σ is obtained by taking one instance of Σ₁ and n‑2 independent copies of Σ₂, and giving each participant the concatenation of all his shares. The king thus holds (n‑1)+(n‑2)=2n‑3 field elements; each pawn holds 1+2(n‑2)=2n‑3 field elements as well. The secret is taken to be a vector of n‑1 independent field elements, so H(S)=log q^{n‑1}. Consequently each participant’s entropy is log q^{2n‑3}, yielding an information rate of (n‑1)/(2n‑3). Because this scheme achieves the previously derived upper bound, the authors conclude

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