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Wiretapped Commitment over Binary Channels

Anuj Kumar Yadav, Manideep Mamindlapally, Amitalok J. Budkuley

TL;DR

The effect of collusion (or lack of it) between the eavesdropper Eve and either Alice or Bob is explored and results on the wiretapped commitment capacity under the so-called Y-private regime and the 2-private regime are presented.

Abstract

We propose the problem of wiretapped commitment, where two parties, say committer Alice and receiver Bob, engage in a commitment protocol using a noisy channel as a resource, in the presence of an eavesdropper, say Eve. Noisy versions of Alice's transmission over the wiretap channel are received at both Bob and Eve. We seek to determine the maximum commitment throughput in the presence of an eavesdropper, i.e., wiretapped commitment capacity, where in addition to the standard security requirements for two-party commitment, one seeks to ensure that Eve doesn't learn about the commit string. A key interest in this work is to explore the effect of collusion (or lack of it) between the eavesdropper Eve and either Alice or Bob. Toward the same, we present results on the wiretapped commitment capacity under the so-called 1-private regime (when Alice or Bob cannot collude with Eve) and the 2-private regime (when Alice or Bob may possibly collude with Eve).

Wiretapped Commitment over Binary Channels

TL;DR

The effect of collusion (or lack of it) between the eavesdropper Eve and either Alice or Bob is explored and results on the wiretapped commitment capacity under the so-called Y-private regime and the 2-private regime are presented.

Abstract

We propose the problem of wiretapped commitment, where two parties, say committer Alice and receiver Bob, engage in a commitment protocol using a noisy channel as a resource, in the presence of an eavesdropper, say Eve. Noisy versions of Alice's transmission over the wiretap channel are received at both Bob and Eve. We seek to determine the maximum commitment throughput in the presence of an eavesdropper, i.e., wiretapped commitment capacity, where in addition to the standard security requirements for two-party commitment, one seeks to ensure that Eve doesn't learn about the commit string. A key interest in this work is to explore the effect of collusion (or lack of it) between the eavesdropper Eve and either Alice or Bob. Toward the same, we present results on the wiretapped commitment capacity under the so-called 1-private regime (when Alice or Bob cannot collude with Eve) and the 2-private regime (when Alice or Bob may possibly collude with Eve).
Paper Structure (15 sections, 9 theorems, 35 equations, 1 figure)

This paper contains 15 sections, 9 theorems, 35 equations, 1 figure.

Table of Contents

  1. Introduction
  2. Notation and Preliminaries:
  3. System Model and Problem Description
  4. Our Main Results
  5. Proofs
  6. $1$-privacy converse analysis for the wiretap channel $W_{Y,Z|X}$
  7. $2$-privacy converse analysis for the wiretap channel $W_{Y,Z|X}$.
  8. Achievability for the BS-BC$({p,q})$ wiretap channel under $1-$privacy.
  9. Achievability for I-BS-BC$(p,q)$ wiretap channel under $2-$privacy.
  10. Concluding Remarks and Discussion
  11. Acknowledgements
  12. Proof of Lemma and Lemma
  13. Proof of Lemma and Lemmma
  14. Proof of Claim
  15. Proof of Lemma

Key Result

Theorem 1

Consider a channel $W_{Y,Z|X}\in$ BS-BC$(p,q)$, where $0<p,q<1/2.$ Then, the commitment capacity $\mathbb{C}_1$ of such a BS-BC($p,q$) under $1-$privacy is

Figures (1)

  • Figure 1: The problem setup: commitment over a wiretap channel

Theorems & Definitions (31)

  • Definition 1: $\xi$-Universal hash functions
  • Definition 2: Strong randomness extractors
  • Definition 3: Wiretap channel wyner-inc1978, elgamal-kim
  • Definition 4: BS-BC wiretap channels elgamal-kim
  • Remark 1
  • Definition 5: I-BS-BC wiretap channel elgamal-kim
  • Definition 6: D-BS-BC wiretap channel elgamal-kim
  • Definition 7: Commitment protocol
  • Definition 8: $\epsilon-$sound
  • Definition 9: $\epsilon-1-$concealing
  • ...and 21 more