Logo
ScienceStack
Table of Contents
Fetching ...
Sign In

Network function computation with vector linear target function and security function

Min Xu, Qian Chen, Gennian Ge

TL;DR

This work studies secure network function computation where both the target function $f$ and the security function $g$ are vector-linear over a directed acyclic graph, defining the secure computing capacity $\widehat{\mathcal{C}}(\mathcal{N},f,g,r)$. It derives two general information-theoretic upper bounds valid for arbitrary networks and vector-linear $f,g$, and provides a sum-target–oriented lower-bound construction by transforming non-secure codes into secure ones, along with a field-size condition for vector-linear security. For three-layer $(U,V,\alpha)$-tree networks, it characterizes the required properties of the global encoding matrices and presents explicit linear secure network codes via branch decomposition, showing how per-branch designs can achieve capacity in some cases. Overall, the results deepen understanding of capacity-security trade-offs in secure network function computation and offer concrete coding strategies for secure aggregation-like distributed computation.

Abstract

In this paper, we study the problem of securely computing a function over a network, where both the target function and the security function are vector linear. The network is modeled as a directed acyclic graph. A sink node wishes to compute a function of messages generated by multiple distributed sources, while an eavesdropper can access exactly one wiretap set from a given collection. The eavesdropper must be prevented from obtaining any information about a specified security function of the source messages. The secure computing capacity is the maximum average number of times that the target function can be securely computed with zero error at the sink node with the given collection of wiretap sets and security function for one use of the network. We establish two upper bounds on this capacity, which hold for arbitrary network topologies and for any vector linear target and security functions. These bounds generalize existing results and also lead to a new upper bound when the target function is the sum over a finite field. For the lower bound, when the target function is the sum, we extend an existing method, which transforms a non-secure network code into a secure one, to the case where the security function is vector linear. Furthermore, for a particular class of networks and a vector linear target function, we characterize the required properties of the global encoding matrix to construct a secure vector linear network code.

Network function computation with vector linear target function and security function

TL;DR

This work studies secure network function computation where both the target function and the security function are vector-linear over a directed acyclic graph, defining the secure computing capacity . It derives two general information-theoretic upper bounds valid for arbitrary networks and vector-linear , and provides a sum-target–oriented lower-bound construction by transforming non-secure codes into secure ones, along with a field-size condition for vector-linear security. For three-layer -tree networks, it characterizes the required properties of the global encoding matrices and presents explicit linear secure network codes via branch decomposition, showing how per-branch designs can achieve capacity in some cases. Overall, the results deepen understanding of capacity-security trade-offs in secure network function computation and offer concrete coding strategies for secure aggregation-like distributed computation.

Abstract

In this paper, we study the problem of securely computing a function over a network, where both the target function and the security function are vector linear. The network is modeled as a directed acyclic graph. A sink node wishes to compute a function of messages generated by multiple distributed sources, while an eavesdropper can access exactly one wiretap set from a given collection. The eavesdropper must be prevented from obtaining any information about a specified security function of the source messages. The secure computing capacity is the maximum average number of times that the target function can be securely computed with zero error at the sink node with the given collection of wiretap sets and security function for one use of the network. We establish two upper bounds on this capacity, which hold for arbitrary network topologies and for any vector linear target and security functions. These bounds generalize existing results and also lead to a new upper bound when the target function is the sum over a finite field. For the lower bound, when the target function is the sum, we extend an existing method, which transforms a non-secure network code into a secure one, to the case where the security function is vector linear. Furthermore, for a particular class of networks and a vector linear target function, we characterize the required properties of the global encoding matrix to construct a secure vector linear network code.
Paper Structure (13 sections, 21 theorems, 166 equations, 11 figures)

This paper contains 13 sections, 21 theorems, 166 equations, 11 figures.

Table of Contents

  1. Introduction
  2. System model and related results
  3. The model of secure network function computation
  4. Network function computation for linear functions
  5. Secure network coding
  6. Upper bounds for secure network computing capacity
  7. Linear Secure Network Codes: Sum Target Function and Vector Linear Security
  8. Linear secure network codes
  9. Secure network coding for sum function
  10. Secure network coding for $(U,V,\alpha)$-trees
  11. Linear secure network codes for branches
  12. Linear secure network codes for $(U,V,\alpha)$-tree
  13. Conclusion

Key Result

Theorem 1

Consider a model of secure network function computation $(\mathcal{N},f,g,r)$, where $f$ is an algebraic sum function over a finite field and the $g$ is the identity function. Then, where $C_{\min}=\min\{|C|:C\in\Lambda(\mathcal{N})\}$.

Figures (11)

  • Figure 1: A three-layer network with $3$ source nodes.
  • Figure 2: A secure network code for $(\mathcal{N},f,g,r)$ with $f=m_1+m_2+m_3,g=m_1+m_3$ (linear functions over $\mathbb{F}_3$) and $r=2$.
  • Figure 3: A secure network code for $(\mathcal{N},f,g,r)$ with $f=m_1+2m_2+3m_3,g=m_1+m_2+m_3$ (linear functions over $\mathbb{F}_5$) and $r=5$.
  • Figure 4: The network $\mathcal{N}$ has $2$ source nodes with binary source messages and the sink node desires the algebraic sum of the two source nodes.
  • Figure 5: A $3$-layer network $\mathcal{N}$ and a $(2,1)$ non-secure network code.
  • ...and 6 more figures

Theorems & Definitions (31)

  • Theorem 1: 2024Guangsourcesecure
  • Theorem 2: 2025Guangfunctionsecure
  • Theorem 3
  • Theorem 4: bai2023multiple
  • Lemma 1
  • Definition 1
  • Definition 2
  • Theorem 5
  • Remark 1
  • Corollary 1
  • ...and 21 more