Two-Dimensional XOR Secret Sharing Scheme

• The paper introduces a novel two-dimensional XOR-based secret sharing scheme that ensures unconditional, information-theoretic privacy and recovery under a layered failure model.
• The scheme uses nested one-layer XOR encodings to generate a 3×3 share matrix, allowing the secret to be perfectly reconstructed even when one row and one column are lost.
• The lightweight design with only XOR operations and linear complexity makes the approach ideal for resource-constrained and quantum-safe communication networks.

Searching arXiv for the specified paper and closely related context. A two-dimensional XOR-based secret sharing scheme is a secret-sharing construction for layered multipath communication networks in which a secret is encoded into a $3\times 3$ matrix of shares so that a receiver can recover the message despite the loss of any one full row and any one full column, while an adversary that observes any one row and any one column learns nothing about the secret. In "Two-Dimensional XOR-Based Secret Sharing for Layered Multipath Communication" (Chan et al., 29 Sep 2025), the scheme is presented as the first two-dimensional XOR-based secret sharing scheme for layered multipath communication networks. Its security objective is explicitly information-theoretic: successful reconstruction is guaranteed under the stated disruption model, and privacy is unconditional even against an adversary with unbounded computational power, including quantum capabilities (Chan et al., 29 Sep 2025).

1. Problem setting and network model

The scheme is formulated for a layered multipath communication topology with two layers. Layer 1 consists of base-stations with $N_1=3$, indexed by columns $j=1,2,3$. Layer 2 consists of routes with $N_2=3$, indexed by rows $i=1,2,3$. End-to-end communication paths are treated as black-box links uniquely identified by ordered pairs $(i,j)$ (Chan et al., 29 Sep 2025).

The secret is a bitstring $S\in\{0,1\}^{|S|}$, with the explicit assumption that $|S|$ is even. It is split into two equal halves,

$S=(S_1\|\;S_2), \qquad S_1,S_2\in\{0,1\}^{|S|/2}.$

The encoder outputs a $3\times 3$ share matrix

$N_1=3$0

where each subshare $N_1=3$1 is the concatenation of two $N_1=3$2-bit strings. The construction also uses intermediate shares $N_1=3$3 for $N_1=3$4 (Chan et al., 29 Sep 2025).

The design target is specific and operational: the receiver must tolerate the loss of any one full row and one full column, whereas an eavesdropper that sees any one row and one column must obtain no information about $N_1=3$5. This gives the scheme a distinctly layered interpretation: failure and observation are each modeled as occurring once per transmission layer.

2. Security objectives and adversary model

The adversary is information-theoretic and may have unbounded computational resources, explicitly including quantum computation. Two adversarial capabilities are modeled. In the passive case, the adversary eavesdrops on all shares in one row $N_1=3$6 and one column $N_1=3$7. In the active case, the adversary jams or removes all shares in one row and one column (Chan et al., 29 Sep 2025).

The security goals are expressed in terms of entropy and mutual information. For availability, if row $N_1=3$8 and column $N_1=3$9 are lost, the receiver retains

$j=1,2,3$0

and the requirement is

$j=1,2,3$1

This states that the surviving $j=1,2,3$2 submatrix determines the secret with certainty.

For privacy, if the adversary observes

$j=1,2,3$3

the requirement is

$j=1,2,3$4

This is perfect privacy: the observation of a full row together with a full column is statistically independent of the secret (Chan et al., 29 Sep 2025).

These conditions distinguish the construction from standard single-layer XOR secret-sharing settings. The stated comparison is that traditional one-dimensional XOR schemes tolerate only single-path failure, whereas the two-dimensional scheme tolerates one failure per layer simultaneously. The paper also contrasts the construction with encryption-based approaches such as AES, which provide confidentiality but not information-theoretic availability under path failures and rely on computational hardness assumptions (Chan et al., 29 Sep 2025).

3. Encoding construction

Share generation begins by splitting the secret into two halves and sampling eight independent uniform random sequences: $j=1,2,3$5 The construction then applies a one-layer XOR encoder, denoted $j=1,2,3$6, in nested form. First,

$j=1,2,3$7

Then, for each $j=1,2,3$8,

$j=1,2,3$9

The output is the $N_2=3$0 matrix $N_2=3$1 (Chan et al., 29 Sep 2025).

The one-layer encoder $N_2=3$2, for $N_2=3$3, produces three shares $N_2=3$4 according to

$N_2=3$5

The two-dimensional construction is therefore a composition of two one-layer XOR encodings (Chan et al., 29 Sep 2025).

The paper also gives explicit formulas for all entries of the share matrix. Writing each $N_2=3$6 as a concatenation of two $N_2=3$7-bit blocks, the columns are

$N_2=3$8

$N_2=3$9

and

$i=1,2,3$0

These formulas make explicit that every secret-dependent component appears XOR-masked by independent random blocks (Chan et al., 29 Sep 2025).

A plausible implication is that the construction is best understood not as a generic threshold code over a finite field, but as a structured XOR masking system tailored to the geometry of a two-layer path matrix.

4. Reconstruction and availability

Reconstruction assumes that one row $i=1,2,3$1 and one column $i=1,2,3$2 are lost. Let the surviving rows be

$i=1,2,3$3

and the surviving columns be

$i=1,2,3$4

The decoder proceeds in two stages. First, for each surviving column $i=1,2,3$5, it reconstructs the intermediate share $i=1,2,3$6 from the two surviving entries in that column: $i=1,2,3$7 Second, it reconstructs the secret from the two surviving intermediate shares: $i=1,2,3$8 The paper states that this always succeeds, establishing availability (Chan et al., 29 Sep 2025).

The proof sketch relies on a one-layer lemma: any two shares suffice to determine the source, giving zero conditional entropy. Since two surviving entries remain in every surviving column, each such column yields its intermediate $i=1,2,3$9. Then any two intermediate shares yield $(i,j)$0. The resulting argument is compositional: one-layer decodability lifts to two-layer decodability through nesting (Chan et al., 29 Sep 2025).

This structure is significant because the availability guarantee is not merely resilience to arbitrary erasures among nine links. The guarantee is specifically aligned with a layered failure model: one complete row and one complete column may disappear simultaneously, yet the remaining $(i,j)$1 portion still contains enough information for exact recovery.

5. Privacy proof and unconditional security

The privacy claim rests on the independence and uniformity of $(i,j)$2 and on the masking behavior of XOR. The proof sketch states that observing one row $(i,j)$3 plus one column $(i,j)$4 reveals only XORs of $(i,j)$5 or $(i,j)$6 masked by independent $(i,j)$7. By evaluating the mutual information in all nine $(i,j)$8 cases, the paper concludes that

$(i,j)$9

for every row-column observation pattern (Chan et al., 29 Sep 2025).

The argument uses the one-layer availability and privacy lemmas together with composition via the data-processing inequality. In effect, the two-dimensional scheme inherits privacy from the one-layer encoder because each observable expression remains hidden behind fresh randomness, and the layered composition does not introduce exploitable dependence on $S\in\{0,1\}^{|S|}$0 (Chan et al., 29 Sep 2025).

The paper characterizes the resulting security as unconditional security regardless of computational resources available to adversaries. It explicitly contrasts this with encryption-based approaches vulnerable to quantum computing advances. Within the terms given, the construction offers provable security without dependence on computational assumptions, finite-field multiplications, or cryptanalytic hardness conjectures (Chan et al., 29 Sep 2025).

A common misconception in discussions of XOR-based schemes is that XOR implies heuristic or computationally weak protection. The present construction does not adopt that premise. Its security claim is information-theoretic, and the privacy condition is stated as exact mutual-information zero, not as computational indistinguishability.

6. Complexity, comparison, and operational context

The complexity analysis is linear in the message length. Share generation uses $S\in\{0,1\}^{|S|}$1 random blocks of $S\in\{0,1\}^{|S|}$2 bits together with $S\in\{0,1\}^{|S|}$3 block-wise XORs, yielding $S\in\{0,1\}^{|S|}$4 bit-XOR cost. Reconstruction requires at most $S\in\{0,1\}^{|S|}$5 two-share XORs plus one final decode, again giving $S\in\{0,1\}^{|S|}$6 bit-XOR complexity (Chan et al., 29 Sep 2025).

The paper emphasizes that the scheme uses only bitwise XOR operations. It is therefore described as lightweight and suitable for resource-constrained military environments where computational assumptions may fail. The comparison drawn is explicit: encryption-based systems such as AES provide confidentiality but not information-theoretic availability under path failures, whereas the XOR-only construction provides both the stated availability and perfect privacy in the modeled setting (Chan et al., 29 Sep 2025).

The main contrasts presented in the source can be summarized as follows:

Aspect	Two-dimensional XOR-based scheme	Comparison stated in source
Failure tolerance	Tolerates one failure per layer simultaneously	Traditional one-dimensional XOR schemes tolerate only single-path failure
Security basis	Information-theoretic, unconditional	Encryption-based approaches rely on computational hardness
Core operations	Bitwise XOR only	No finite-field multiplies

This suggests that the construction is intended not as a replacement for all cryptographic transport mechanisms, but as a specialized primitive for layered multipath systems in which the dominant requirements are low computational cost, quantum-safe confidentiality, and deterministic recoverability under structured path loss.

7. Generalization and extensions

The paper states that general $S\in\{0,1\}^{|S|}$7 layering follows the same pattern: generate $S\in\{0,1\}^{|S|}$8 random blocks and apply nested $S\in\{0,1\}^{|S|}$9 and $|S|$0 XOR-based encodings (Chan et al., 29 Sep 2025). In the reported $|S|$1 instance, the construction handles up to one failure per layer; by increasing redundancy through $|S|$2 and $|S|$3 schemes, the framework can tolerate multiple failures (Chan et al., 29 Sep 2025).

Practical issues identified for deployment include random-seed synchronization, share-indexing, and loss detection. The source states that these can be handled with small sequence numbers or lightweight MACs, with complementary systems work cited for such mechanisms (Chan et al., 29 Sep 2025). No further protocol details are specified in the provided description.

In research terms, the two-dimensional XOR-based secret sharing scheme occupies the intersection of secret sharing, network resilience, and layered multipath communication. Its distinguishing feature is the alignment of a compositional XOR secret-sharing design with a topology-aware adversary model. The formal objectives are exact and narrow—recovery after one row and one column failure, and privacy against observation of one row and one column—but within that regime the scheme is presented as the first construction of its kind for layered multipath communication networks (Chan et al., 29 Sep 2025).