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Secure Computation-and-Forward with Linear Codes

Masahito Hayashi, Tadashi Wadayama, Angeles Vazquez-Castro

TL;DR

A code that securely transmits the modulo sum of the messages of two nodes via a multiple access channel is constructed by simple combination of an existing linear code and universal2 hash function.

Abstract

We discuss secure transmission via an untrusted relay when we have a multiple access phase from two nodes to the relay and broadcast phase from the relay to the two nodes. To realize the security, we construct a code that securely transmits the modulo sum of the messages of two nodes via a multiple access channel. In this code, the relay cannot obtain any information for the message of each node, and can decode only the messages of the two nodes. Our code is constructed by simple combination of an existing liner code and universal2 hash function.

Secure Computation-and-Forward with Linear Codes

TL;DR

A code that securely transmits the modulo sum of the messages of two nodes via a multiple access channel is constructed by simple combination of an existing linear code and universal2 hash function.

Abstract

We discuss secure transmission via an untrusted relay when we have a multiple access phase from two nodes to the relay and broadcast phase from the relay to the two nodes. To realize the security, we construct a code that securely transmits the modulo sum of the messages of two nodes via a multiple access channel. In this code, the relay cannot obtain any information for the message of each node, and can decode only the messages of the two nodes. Our code is constructed by simple combination of an existing liner code and universal2 hash function.

Paper Structure

This paper contains 10 sections, 4 theorems, 42 equations, 4 figures, 1 table.

Table of Contents

  1. Introduction
  2. Code Construction
  3. Secrecy Analysis with Transmission Rate
  4. Examples
  5. Random coding with universal$2$ condition
  6. LDPC code
  7. Conclusion
  8. Proof of Lemma \ref{['GUR']}
  9. Proof of Theorems \ref{['TTA']} and \ref{['TTY']}
  10. Proof of \ref{['LODT']}

Key Result

Theorem 1

Given a map $G=g$, using $B_{i,n,s,1}:= 3 q^{s (n-k_n-\bar{k}_n)} e^{s n I_{\frac{1}{1-s}}^{\downarrow} (Y; X_i )[W] }$, we have

Figures (4)

  • Figure 1: MAC phase and broadcast phase.
  • Figure 2: Butterfly network coding.
  • Figure 3: Encoding and decoding process.
  • Figure 4: Achievable rates with BPSK when the variance $N_0$ is $=1$. The base of logarithm is chosen to be $e$. The horizontal axis expresses the intensity $h$. The vertical axis expresses transmission rate. The solid black line express the 2nd type of rate with random coding given as \ref{['H13']}. This value is positive with $h \ge 2.443$ and approaches $\frac{1}{2}\log 2$. The dashed blue line express the 1st type of rate with random coding given as \ref{['H17']}. This value is positive with $h \ge 2.518$ and approaches $\frac{1}{2}\log 2$. The black points express the 3rd type of rate with $(d_l,d_r,L)$ spatial coupling LDPC code with sufficiently large $L$, whose rate is \ref{['H14']}. The blue points express the 1st type of rate with $(d_l,d_r,L)$ spatial coupling LDPC code with sufficiently large $L$, whose rate is \ref{['H18']}. According to these formulas, the value is negative when $h$ is less than a certain threshold. In this case, the secure transmission of $M_1+M_2$ is impossible in these methods.

Theorems & Definitions (4)

  • Theorem 1
  • Theorem 2
  • Lemma 1
  • Lemma 2