Additive Access Structures

• Additive Access Structure is a framework that defines authorized and unauthorized participant groups using cumulative, monotone rules in secret-sharing schemes.
• It leverages additive codes, graph multigraph representations, and algebraic criteria to enable secure reconstruction in both static and dynamic models.
• The methodology integrates information-theoretic bounds and combinatorial designs to optimize secret recovery rates and ensure uniform coverage among participants.

An Additive Access Structure governs the rule set for which groups of participants are authorized or unauthorized to reconstruct a secret in secret-sharing schemes where shares are derived via additive protocols, codes, or correlated randomness. This paradigm subsumes both static, code-based schemes and dynamic, time-evolving models in which the access structure grows monotonically by authorizing new subsets at each time step. The characterization and analysis of such structures utilize algebraic constructs, graphical encodings, and information-theoretic bounds.

1. Formal Definition and Algebraic Foundations

Let $\mathcal{L}=\{1,\dots,L\}$ denote participant indices. An Additive Access Structure (AAS) is a sequence $$\bigl(\mathcal{A}_t,\mathcal{U}_t\bigr)_{t\in\mathcal{T}}$$, where $\mathcal{A}_t$ (authorized sets) and $\mathcal{U}_t$ (unauthorized sets) satisfy:

• $$\mathcal{A}_0 = \emptyset \subset \mathcal{A}_1 \subset \cdots \subset \mathcal{A}_{t_{\max}}$$ (authorized sets are cumulatively enlarged),
• $$2^{\mathcal{L}} = \mathcal{U}_0 \supset \mathcal{U}_1 \supset \cdots \supset \mathcal{U}_{t_{\max}}$$ (unauthorized sets shrink monotonically),
• For each $t$, $\mathcal{A}_t$ and $\mathcal{U}_t$ are monotone: if $A\in\mathcal{A}_t$ and $A\subseteq B$, then $B\in\mathcal{A}_t$; similarly for $\mathcal{U}_t$ (Miller et al., 14 Jan 2026).

The additive property typically stems from the underlying structure of share generation—using additive codes, correlated random variables, or combinatorial designs—where the authorized sets are tightly coupled to algebraic or graph-theoretic criteria.

2. Additive Codes and Static Access Structures

Additive codes are pivotal in static (non-evolving) AAS. For $GF(4) = \{0,1,\omega,\bar{\omega}\}$ ($\omega^2 = \bar{\omega} = \omega+1$), an additive code $C \subseteq GF(4)^n$ is a GF(2)-vector space: $|C|=2^k$. Share distribution is realized via a generator matrix $G = [g_0, \dotsc, g_{n-1}]$ with nonzero columns.

The dual code $C^\perp$ is defined using the trace inner product $x \star y = \sum_{i=1}^{n} \text{Tr}(x_i y_i)$ with the trace $\text{Tr}(x)=x+x^2$ mapping $GF(4)\to GF(2)$. The access structure is encoded through cosets:

• $H_i = \{ h \in C^\perp : h_0 = a_i \ne 0 \}$ for $a_i \in \{1, \omega, \bar{\omega}\}$.
• Supports $$T_i = \{ \text{supp}(h) \setminus \{0\} : h \in H_i \}$$ define minimal authorized sets.

Critical property: Recovery of the secret $s \in GF(4)$ requires two linearly independent trace equations from sets in distinct $T_i$, rendering one-step reconstruction impossible (Kim et al., 2017).

3. Time-Evolving Additive Access Structures

In dynamic AAS models, the authorized sets $\mathcal{A}_t$ are updated over discrete time steps. At each $t$, new groups are appended to the current access structure. The dealer—knowing only the present structure—constructs secrets and public messages through random binning functions and adapts parameters $(k_t, \sigma_t)$ quantizing the message and secrecy rates based on conditional entropies and correlated random samples $(Y^n, X_{1:L}^n)$ (Miller et al., 14 Jan 2026).

The secret rate $R_t$ at step $t$ is specified by:

$$R_t = \min_{U\in\mathcal{U}_t} H(Y|X_U) - \max_{A\in\mathcal{A}_t} H(Y|X_A)$$

with reliability and secrecy requirements enforced asymptotically.

When threshold structures are used (i.e., sets of size $\geq u$ authorized, $\leq v$ unauthorized), the capacity simplifies to $R_t = H(X)(u-v)$.

4. Graphical Characterization of Access Structures

Access structures can be encoded via $q$-multigraphs ($q$ prime), whose adjacency matrix $\Gamma$ controls authorization. For participant set $V$ and dealer vertex $d \in V$, the set $B \subseteq V \setminus \{d\}$ is authorized iff there exists $D \in \mathbb{F}_q^B$ such that:

$\Gamma[B,\{d\}] D = 1, \quad \Gamma[B,F] D = 0$

where $F = V \setminus (B \cup \{d\})$. The reconstruction map for classical secrets is additive in the shares, and the access structure is fully determined by the cut-rank criterion:

$$\pi_G(B,d) := \text{rank}(\Gamma[B, V \setminus B]) - \text{rank}(\Gamma[B, V \setminus (B \cup \{d\})]) = 1$$

(Marin et al., 2013).

5. Minimal Qualified Sets and Reconstruction

A minimal qualified group in static additive code schemes is an ordered pair $(A,B)$ with $A \in T_i$, $B \in T_j$, $i \ne j$, such that no proper subset of $A$ or $B$ is authorized under the respective $T_i$ or $T_j$. The total number of such minimal pairs, for example in the hexacode $(6,2^6)$, is $3 \cdot 10 \cdot 10 = 300$.

Reconstruction requires:

1. Participants in $A$ compute $\alpha = \sum_{r\in A} \text{Tr}(\sigma_r h_r)$.
2. Participants in $B$ compute $\beta = \sum_{r\in B} \text{Tr}(\sigma_r h'_r)$.
3. The secret $s$ is recovered via the bijection $s \mapsto (\alpha, \beta) \in GF(2)^2$ (Kim et al., 2017).

6. Design-Theoretic Properties and Uniform Coverage

Support sets of codewords in extremal self-dual additive codes frequently form generalized $t$-designs. A set $S \subseteq GF(4)^n$ of fixed weight $k$ is a generalized $t$-design of type 3 if every subvector of weight $t$ is covered with exact multiplicity $\lambda_t$.

For extremal codes—such as the hexacode, dodecacode—the $\mathcal{T}_i$ have uniform size for each $i$ and all single-point coalitions are uniformly covered. This ensures parameter regularity and uniformity in access degrees and reconstruction probabilities.

7. Probabilistic Bounds and Graph-Based Schemes

Random $q$-multigraphs with $n$ vertices yield threshold secret-sharing schemes with the threshold parameter $k \leq \alpha n$, where $\alpha$ solves $H_{q^2}(1-\alpha) < 1/2$, with $H_{q^2}$ the $q^2$-ary entropy (Marin et al., 2013).

The authorized subsets are precisely those whose inclusion of the dealer’s vertex increases the matrix cut-rank by one. This graphical formalism generalizes to classical and quantum secret-sharing with additive structure.

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Summary Table: Mathematical Criteria for Additive Access Structures

Model	Static/Time-Evolving	Reconstruction Rule
Additive Codes	Static	Two trace equations from duals $H_i$, $H_j$
Correlated AAS	Time-evolving	Typicality decoding, binning functions
Graph Multigraph	Static/Quantum	Cut-rank increment, linear dependency

In both static and growing additive access structures, the authorized sets are delineated via combination of algebraic, graph-theoretic, and information-theoretic conditions, yielding precise reconstruction methodologies and capacity bounds for secret-sharing applications (Kim et al., 2017, Miller et al., 14 Jan 2026, Marin et al., 2013).