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Secure Network Function Computation for Linear Functions, Part II: Target-Function Security

Yang Bai, Xuan Guang, Raymond W. Yeung

TL;DR

This work advances secure network function computation by analyzing target-function security in linear settings over arbitrary networks. It establishes a topology-dependent upper bound on the secure computing capacity, proves algebraic equivalences between target-function and source security, and provides a constructive linear coding scheme that achieves a guaranteed lower bound up to $C_{\min}-r$. The authors also compare target-function security with the prior source-security model, showing the broader capacity and flexibility of the former, including improved field-size requirements via primary minimum cuts. Overall, the paper advances theory and code construction for securely computing linear functions with zero-error guarantees in networks under strong security constraints, with implications for secure distributed computation and network coding.

Abstract

In this Part II of a two-part paper, we put forward secure network function computation, where in a directed acyclic network, a sink node is required to compute a target function of which the inputs are generated as source messages at multiple source nodes, while a wiretapper, who can access any one but not more than one wiretap set in a given collection of wiretap sets, is not allowed to obtain any information about a security function of the source messages. In Part I of the two-part paper, we have investigated securely computing linear functions with the wiretapper who can eavesdrop any edge subset up to a certain size r, referred to as the security level, where the security function is the identity function. The notion of this security is called source security. In the current paper, we consider another interesting model which is the same as the above one except that the security function is identical to the target function, i.e., we need to protect the information on the target function from being leaked to the wiretapper. The notion of this security is called target-function security. We first prove a non-trivial upper bound on the secure computing capacity, which is applicable to arbitrary network topologies and arbitrary security levels. In particular, when the security level r is equal to 0, the upper bound reduces to the computing capacity without security consideration. Further, from an algebraic point of view, we prove two equivalent conditions for target-function security and source security for the existence of the corresponding linear function-computing secure network codes. With them, for any linear function over a given finite field, we develop a code construction of linear secure network codes for target-function security and thus obtain a lower bound on the secure computing capacity; and also generalize the code construction developed in Part I for source security.

Secure Network Function Computation for Linear Functions, Part II: Target-Function Security

TL;DR

This work advances secure network function computation by analyzing target-function security in linear settings over arbitrary networks. It establishes a topology-dependent upper bound on the secure computing capacity, proves algebraic equivalences between target-function and source security, and provides a constructive linear coding scheme that achieves a guaranteed lower bound up to . The authors also compare target-function security with the prior source-security model, showing the broader capacity and flexibility of the former, including improved field-size requirements via primary minimum cuts. Overall, the paper advances theory and code construction for securely computing linear functions with zero-error guarantees in networks under strong security constraints, with implications for secure distributed computation and network coding.

Abstract

In this Part II of a two-part paper, we put forward secure network function computation, where in a directed acyclic network, a sink node is required to compute a target function of which the inputs are generated as source messages at multiple source nodes, while a wiretapper, who can access any one but not more than one wiretap set in a given collection of wiretap sets, is not allowed to obtain any information about a security function of the source messages. In Part I of the two-part paper, we have investigated securely computing linear functions with the wiretapper who can eavesdrop any edge subset up to a certain size r, referred to as the security level, where the security function is the identity function. The notion of this security is called source security. In the current paper, we consider another interesting model which is the same as the above one except that the security function is identical to the target function, i.e., we need to protect the information on the target function from being leaked to the wiretapper. The notion of this security is called target-function security. We first prove a non-trivial upper bound on the secure computing capacity, which is applicable to arbitrary network topologies and arbitrary security levels. In particular, when the security level r is equal to 0, the upper bound reduces to the computing capacity without security consideration. Further, from an algebraic point of view, we prove two equivalent conditions for target-function security and source security for the existence of the corresponding linear function-computing secure network codes. With them, for any linear function over a given finite field, we develop a code construction of linear secure network codes for target-function security and thus obtain a lower bound on the secure computing capacity; and also generalize the code construction developed in Part I for source security.

Paper Structure

This paper contains 19 sections, 11 theorems, 168 equations, 3 figures.

Table of Contents

  1. Introduction
  2. Preliminaries
  3. A Model for Secure Network Function Computation
  4. Function-Computing Secure Network Code for $(\mathcal{N}, f, \mathcal{W}, \zeta)$
  5. Secure Network Function Computation for Linear Functions
  6. Upper Bound on the Secure Computing Capacity
  7. The Upper Bound
  8. A Comparison of the Upper Bounds for the Models $\langle \mathcal{N}, f, r \rangle$ and $(\mathcal{N}, f, r)$
  9. Linear (Function-Computing) Secure Network Coding
  10. Construction of Linear Secure Network Codes for Target-Function Security
  11. Code Construction
  12. The Existence of $\vec{b}_1, \vec{b}_2, \cdots, \vec{b}_{R-r}$ and the Required Field Size of the Code Construction
  13. An Example
  14. Comparison of the Code Constructions for the Models $\langle \mathcal{N}, f, r \rangle$ and $(\mathcal{N}, f, r)$
  15. A Generalized Code Construction for the Source-Security Model $(\mathcal{N}, f, r)$
  16. ...and 4 more sections

Key Result

Theorem 1

Consider the secure model $\langle \mathcal{N}, f, r \rangle$, where the target function $f$ is an algebraic sum over a finite field $\mathbb{F}_q$. Then,

Figures (3)

  • Figure 1: A toy example to show the unnecessariness of randomness.
  • Figure 2: An $\mathbb{F}_3$-valued $(2, 1)$ linear network code for the model $(\mathcal{N}, f)$.
  • Figure 3: An $\mathbb{F}_3$-valued $(1,1)$ linear secure network code for the secure model $\langle \mathcal{N}, f, r\rangle$.

Theorems & Definitions (13)

  • Example 1
  • Theorem 1
  • Corollary 2
  • Corollary 3
  • Corollary 4
  • Theorem 5: PartI
  • Theorem 6
  • Theorem 7
  • Theorem 8
  • Theorem 9
  • ...and 3 more