Advantage of the key relay protocol over secure network coding

TL;DR

The paper investigates the relationship between the key relay protocol ($KRP$) and secure network coding ($SNC$). It proves that when SNC is generalized to include free public channels, $SNC$ and $KRP$ are equivalent in the one-shot setting, meaning they have the same security capabilities on a given graph. However, with conventional SNC (no public channels) there exist graphs where $KRP$ achieves strictly stronger security than any SNC scheme, establishing a genuine security gap. To formalize this gap, the authors introduce $KRP$-by-SNC and demonstrate a chain of inclusions Secure SNC ⊆ Secure KRP-by-SNC ⊆ Secure KRP, with strict separation showing that $KRP$ can outperform conventional SNC. The results imply that the $KRP$ is a distinct research direction, bridging quantum key distribution networks and network coding theory, and raise questions about asymptotic behaviors and graph-class boundaries.

Abstract

The key relay protocol (KRP) plays an important role in improving the performance and the security of quantum key distribution (QKD) networks. On the other hand, there is also an existing research field called secure network coding (SNC), which has similar goal and structure. We here analyze differences and similarities between the KRP and SNC rigorously. We found, rather surprisingly, that there is a definite gap in security between the KRP and SNC; that is, certain KRPs achieve better security than any SNC schemes on the same graph. We also found that this gap can be closed if we generalize the notion of SNC by adding free public channels; that is, KRPs are equivalent to SNC schemes augmented with free public channels.

Figures (14)

• Figure 1: Relation of secure network coding (SNC) with and without public channels, and the key relay protocol (KRP). The settings and the goals of SNC and the KRP are summarized in Table \ref{['table:differences_similarity_KRs_SNCs']}. The KRP and SNC with publich channels always achieve the same level of security (Theorem \ref{['thm:SNC_and_KR']}). The security of KRP is better than that of the conventional SNC or SNC without public channels (Theorem \ref{['crl:SNCs_and_KRs_are_not_equivalent']}).
• Figure 2: The simplest example of the KRP. On each edge $e_i$ there is a local key source $LKS_{e_i}$ which distributes a random bit $r_{e_i}\in_{\rm R}\{0,1\}$ to both ends. Each node can also use public channels freely. User pair $u^1,u^2$ wishes to share a relayed key $k=(k^1,k^2)$. To this end, the midpoint $v$ announces $\Delta r=r_{e_1}+r_{e_2}$, and then user $u^1$ and $u^2$ each calculate $k^1=r_{e_1}$ and $k^2=r_{e_2}+\Delta r$.
• Figure 3: Somewhat complex examples of the KRP. (a) Serialization of Fig. \ref{['fig:key_relay_example1']}. Nodes $v_i$ each announce $\Delta r_i=r_i+r_{i+1}$, and then users $u^1$ and $u^2$ calculate relayed keys $k^1=r_{e_1}$ and $k^2=r_n+\sum_{i=1}^{n-1}\Delta r_i$ respectively. (b) A parallelization of Fig. \ref{['fig:key_relay_example1']}. Nodes $v_i$ each announce $\Delta r_i=r_{e_{i1}}+r_{e_{i2}}$, and then users $u^1,u^2$ each calculate $k^1=r_{e_{11}}+r_{e_{21}}$, $k^2=\sum_{i=1,2}(r_{e_{2i}}+\Delta r_i)$. Note that the relayed key $k=(k^1,k^2)$ remains secret here even if someone takes over an edge set $E_i=\{e_{i1},e_{i2}\}$ ($i=1$ or 2) and leaks local keys $r_{e_{i1}},r_{e_{i2}}$. In this sense we regard this construction more secure than that of Fig. \ref{['fig:key_relay_example1']}.
• Figure 4: (a) Behavior of local key source $LKS_e$ in the absence of the adversary, on edge $e$ having end nodes $v,w$, (b) public channel $PC_e$ on the same edge.
• Figure 5: Behavior of secret channel $SC_e$ in the absence of the adversary.

Theorems & Definitions (28)

• Definition 1: Local key sources and public channels
• Definition 2
• Definition 3: Security of the KRP
• Lemma 1
• Definition 4: Secret channels
• Definition 5: SNC with public channels
• Definition 6: Security of SNC with public channels
• Theorem 1: The security of KRP $=$ The security of SNC with public channels
• Theorem 2: The security of KRP $\ne$ The security of SNC without public channels (conventional SNC)
• Corollary 1: The security of SNC with public channels $\ne$ The security of SNC without public channels (conventional SNC)