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Spectral approach to the communication complexity of multi-party key agreement

Geoffroy Caillat-Grenier, Andrei Romashchenko

TL;DR

This work develops a spectral graph–information theory framework to bound multi-party secret key agreement in the information-theoretic setting. By combining expander-based spectral bounds (via the Expander Mixing Lemma) with Kolmogorov complexity and mutual information, the authors show that, for symmetric three-party inputs, any one-round simultaneous-messages protocol must incur near-omniscience communication to achieve a public-key secrecy target. They prove asymptotically tight lower bounds matching the known omniscience protocol for this class, and extend the results to all symmetric profiles through reductions, while also establishing an upper bound via interactive protocols that can outperform the simultaneous-message bound in some cases. The results provide a robust, non-i.i.d., one-shot framework linking spectral graph properties to fundamental limits in cryptographic key agreement, with clear open problems for multi-round protocols and broader participant counts.

Abstract

We propose a linear algebraic method, rooted in the spectral properties of graphs, that can be used to prove lower bounds in communication complexity. Our proof technique effectively marries spectral bounds with information-theoretic inequalities. The key insight is the observation that, in specific settings, even when data sets $X$ and $Y$ are closely correlated and have high mutual information, the owner of $X$ cannot convey a reasonably short message that maintains substantial mutual information with $Y$. In essence, from the perspective of the owner of $Y$, any sufficiently brief message $m=m(X)$ would appear nearly indistinguishable from a random bit sequence. We employ this argument in several problems of communication complexity. Our main result concerns cryptographic protocols. We establish a lower bound for communication complexity of multi-party secret key agreement with unconditional, i.e., information-theoretic security. Specifically, for one-round protocols (simultaneous messages model) of secret key agreement with three participants we obtain an asymptotically tight lower bound. This bound implies optimality of the previously known omniscience communication protocol (this result applies to a non-interactive secret key agreement with three parties and input data sets with an arbitrary symmetric information profile). We consider communication problems in one-shot scenarios when the parties' inputs are not produced by any i.i.d. sources, and there are no ergodicity assumptions on the input data. In this setting, we found it natural to present our results using the framework of Kolmogorov complexity.

Spectral approach to the communication complexity of multi-party key agreement

TL;DR

This work develops a spectral graph–information theory framework to bound multi-party secret key agreement in the information-theoretic setting. By combining expander-based spectral bounds (via the Expander Mixing Lemma) with Kolmogorov complexity and mutual information, the authors show that, for symmetric three-party inputs, any one-round simultaneous-messages protocol must incur near-omniscience communication to achieve a public-key secrecy target. They prove asymptotically tight lower bounds matching the known omniscience protocol for this class, and extend the results to all symmetric profiles through reductions, while also establishing an upper bound via interactive protocols that can outperform the simultaneous-message bound in some cases. The results provide a robust, non-i.i.d., one-shot framework linking spectral graph properties to fundamental limits in cryptographic key agreement, with clear open problems for multi-round protocols and broader participant counts.

Abstract

We propose a linear algebraic method, rooted in the spectral properties of graphs, that can be used to prove lower bounds in communication complexity. Our proof technique effectively marries spectral bounds with information-theoretic inequalities. The key insight is the observation that, in specific settings, even when data sets and are closely correlated and have high mutual information, the owner of cannot convey a reasonably short message that maintains substantial mutual information with . In essence, from the perspective of the owner of , any sufficiently brief message would appear nearly indistinguishable from a random bit sequence. We employ this argument in several problems of communication complexity. Our main result concerns cryptographic protocols. We establish a lower bound for communication complexity of multi-party secret key agreement with unconditional, i.e., information-theoretic security. Specifically, for one-round protocols (simultaneous messages model) of secret key agreement with three participants we obtain an asymptotically tight lower bound. This bound implies optimality of the previously known omniscience communication protocol (this result applies to a non-interactive secret key agreement with three parties and input data sets with an arbitrary symmetric information profile). We consider communication problems in one-shot scenarios when the parties' inputs are not produced by any i.i.d. sources, and there are no ergodicity assumptions on the input data. In this setting, we found it natural to present our results using the framework of Kolmogorov complexity.
Paper Structure (17 sections, 20 theorems, 86 equations, 6 figures)

This paper contains 17 sections, 20 theorems, 86 equations, 6 figures.

Table of Contents

  1. Introduction
  2. Preliminaries and Notation
  3. General notation.
  4. Communication complexity.
  5. Reminder of the spectral graph technique.
  6. Main technical tools and the scheme of the proof
  7. When it is hard to say anything that the interlocutor already knows
  8. Protocols with simultaneous messages : a warm-up example
  9. Secret key agreement: a lower bound for the most crucial profile
  10. Secret key agreement: a lower bound for all symmetric profiles
  11. Upper bound for interactive protocols
  12. Conclusion and open problems
  13. Preliminaries: Kolmogorov complexity in some more detail
  14. Bound of the spectral gap for the tri-expander
  15. Useful information inequalities
  16. ...and 2 more sections

Key Result

Theorem 1

(i) For any numbers $k,\ell\in \mathbb{N}$ and $\epsilon,\delta >0$ there exist a randomized communication protocols $\pi_{k,\ell,\epsilon,\delta}$ such that on every pair of input strings $(x,y)$ (of length at most $n$) satisfyingHere the term ${\mathrm C}(x)$ stands for the plain Kolmogorov comple (for $n = |x|+|y|$), which means that the size of the produced secret key is asymptotically equal t

Figures (6)

  • Figure 1: Diagrams with complexity profiles for Examples \ref{['example:1']}-\ref{['example:line-point']} and Proposition \ref{['p:hypergraph']}.
  • Figure 2: The profile in Theorem \ref{['thm:1']}.
  • Figure 3: Alice holding $a$ and Bob holding $b$ send simultaneous messages to Charlie, who computes $c$.
  • Figure 4: Complexity profile for two lines ($a$ and $b$) and their intersection point $c$ in the plane over $\mathbb{F}_{2^n}$.
  • Figure 5: Complexity profile for a triple $x,y,z$. On this diagram it is easy to observe several standard equations: $\bullet$${\mathrm C}(x) \mathop{\mathrm{\mathrel{\stackrel{\hbox{\normalfont\tiny $\lg$}}{=}}}}\nolimits {\mathrm C}(x\mid y,z) + {\mathrm I}(x:y\mid z) + {\mathrm I}(x:z\mid y) + {\mathrm I}(x:y:z)$$\bullet$${\mathrm C}(x,y) \mathop{\mathrm{\mathrel{\stackrel{\hbox{\normalfont\tiny $\lg$}}{=}}}}\nolimits {\mathrm C}(x\mid y,z) + {\mathrm C}(y\mid x,z) + {\mathrm I}(x:y\mid z) + {\mathrm I}(x:z\mid y) + {\mathrm I}(y:z\mid x) + {\mathrm I}(x:y:z)$$\bullet$${\mathrm C}(x\mid y) \mathop{\mathrm{\mathrel{\stackrel{\hbox{\normalfont\tiny $\lg$}}{=}}}}\nolimits {\mathrm C}(x\mid y,z) + {\mathrm I}(x:z\mid y)$$\bullet$${\mathrm I}(x:y) \mathop{\mathrm{\mathrel{\stackrel{\hbox{\normalfont\tiny $\lg$}}{=}}}}\nolimits {\mathrm I}(x:y\mid z) + {\mathrm I}(x:y:z)$$\bullet$${\mathrm I}(x:yz) \mathop{\mathrm{\mathrel{\stackrel{\hbox{\normalfont\tiny $\lg$}}{=}}}}\nolimits {\mathrm I}(x:y\mid z) + {\mathrm I}(x:z\mid y) + {\mathrm I}(x:y:z)$ and so on; all these equations are valid up to $O(\log(|x|+|y|+|z|))$.
  • ...and 1 more figures

Theorems & Definitions (48)

  • Example 1
  • Remark 1
  • Theorem 1: jacm2019
  • Remark 2
  • Remark 3
  • Theorem 2: symmetric version of jacm2019
  • Remark 4
  • Theorem 3: main result
  • Lemma 1: Expander Mixing Lemma for bipartite graphs, see mixing-lemma
  • Corollary 1
  • ...and 38 more