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Communication Complexity of the Secret Key Agreement in Algorithmic Information Theory

Emirhan Gürpınar, Andrei Romashchenko

TL;DR

The paper investigates the information-theoretic secret-key agreement problem in algorithmic information theory, focusing on the worst-case communication complexity when private randomness is allowed. It develops a spectral-graph framework, embedding input pairs into bipartite graphs with large spectral gaps and applying the Expander Mixing Lemma to derive information inequalities that bound transcript length and secrecy for secret-key protocols. For two explicit input constructions (lines-points in a finite plane and a discrete Euclidean-distance model), the authors prove a sharp lower bound: any protocol achieving a secret key of size close to the mutual information $I(x:y)=0.5n+O( log n)$ requires about $0.5n$ communication, matching known upper bounds up to logarithmic terms. They also show a contrasting behavior for inputs at fixed Hamming distance, where secret-key size and communication can be traded continuously, with both achievable and lower-bound results establishing linear dependences. These results bridge algorithmic information theory with spectral graph methods and illuminate the role of graph structure in extractability of mutual information in one-shot settings.

Abstract

It is known that the mutual information, in the sense of Kolmogorov complexity, of any pair of strings x and y is equal to the length of the longest shared secret key that two parties can establish via a probabilistic protocol with interaction on a public channel, assuming that the parties hold as their inputs x and y respectively. We determine the worst-case communication complexity of this problem for the setting where the parties can use private sources of random bits. We show that for some x, y the communication complexity of the secret key agreement does not decrease even if the parties have to agree on a secret key whose size is much smaller than the mutual information between x and y. On the other hand, we discuss examples of x, y such that the communication complexity of the protocol declines gradually with the size of the derived secret key. The proof of the main result uses spectral properties of appropriate graphs and the expander mixing lemma, as well as information theoretic techniques.

Communication Complexity of the Secret Key Agreement in Algorithmic Information Theory

TL;DR

The paper investigates the information-theoretic secret-key agreement problem in algorithmic information theory, focusing on the worst-case communication complexity when private randomness is allowed. It develops a spectral-graph framework, embedding input pairs into bipartite graphs with large spectral gaps and applying the Expander Mixing Lemma to derive information inequalities that bound transcript length and secrecy for secret-key protocols. For two explicit input constructions (lines-points in a finite plane and a discrete Euclidean-distance model), the authors prove a sharp lower bound: any protocol achieving a secret key of size close to the mutual information requires about communication, matching known upper bounds up to logarithmic terms. They also show a contrasting behavior for inputs at fixed Hamming distance, where secret-key size and communication can be traded continuously, with both achievable and lower-bound results establishing linear dependences. These results bridge algorithmic information theory with spectral graph methods and illuminate the role of graph structure in extractability of mutual information in one-shot settings.

Abstract

It is known that the mutual information, in the sense of Kolmogorov complexity, of any pair of strings x and y is equal to the length of the longest shared secret key that two parties can establish via a probabilistic protocol with interaction on a public channel, assuming that the parties hold as their inputs x and y respectively. We determine the worst-case communication complexity of this problem for the setting where the parties can use private sources of random bits. We show that for some x, y the communication complexity of the secret key agreement does not decrease even if the parties have to agree on a secret key whose size is much smaller than the mutual information between x and y. On the other hand, we discuss examples of x, y such that the communication complexity of the protocol declines gradually with the size of the derived secret key. The proof of the main result uses spectral properties of appropriate graphs and the expander mixing lemma, as well as information theoretic techniques.

Paper Structure

This paper contains 16 sections, 21 theorems, 103 equations, 8 figures.

Table of Contents

  1. Introduction
  2. Preliminaries
  3. Kolmogorov Complexity.
  4. Communication Complexity.
  5. The Amount of Randomness in Communication Protocols.
  6. From Information-Theoretic Properties to Combinatorics of Graphs
  7. Randomized Communication Protocols in the Information-Theoretic Framework.
  8. Bounds with the Spectral Method
  9. Information Inequalities from the Graph Spectrum
  10. Information Inequalities for a Secret Key Agreement
  11. Proof of Theorem \ref{['th:main']}
  12. Pairs with a Fixed Hamming Distance
  13. Conclusion
  14. Acknowledgements
  15. Information-theoretic security in the language of Kolmogorov complexity.
  16. ...and 1 more sections

Key Result

Theorem 1

(a) There is a secret key agreement protocol that, for every $n$-bit strings $x$ and $y$, allows Alice and Bob to compute with high probability a shared secret key $z$ of length equal to the mutual information of $x$ and $y$ (up to an $O (\log n)$ additive term). (b) No protocol can produce a longer

Figures (8)

  • Figure 1: Complexity profile for a triple $x,y,z$. On this diagram it is easy to observe several standard equations: $\bullet$$C(x) = C(x|y,z) + I(x:y|z) + I(x:z|y) + I(x:y:z)$ (the sum of all quantities inside the left circle representing $x$); $\bullet$$I(x:y) = I(x:y|z) + I(x:y:z)$ (the sum of the quantities in the intersection of the left and the right circles representing $x$ and $y$ respectively); $\bullet$$C(x,y) = C(x|y,z) + C(y|x,z) + I(x:y|z) + I(x:z|y) + I(y:z|x) + I(x:y:z)$ (the sum of all quantities inside the union of the left and the right circles); $\bullet$$C(x|y) = C(x|y,z) + I(x:z|y)$ (the sum of the quantities inside the left circle but outside the right one); and so on; all these equations are valid up to $O(\log(|x|+|y|+|z|))$.
  • Figure 2: A diagram for the complexity profile of two strings $x,y$: from the Kolmogorov--Levin theorem we have $C(x) =^{+} C(x|y)+ I(x:y) =^{+} n$, $C(y) =^{+} C(y|x)+ I(x:y) =^{+} n$, and $C(x,y)=^{+} C(x|y)+ C(y|x)+I(x:y)=^{+} 1.5n.$
  • Figure 3: Complexity profile for Alice' and Bob's data and a "natural" communication transcript.
  • Figure 4: Complexity profile for inputs $x,y,$ and the transcript $t$ of a communication protocol with given inputs. Note that $C(t|x,y)$ is negligibly small (we can compute $t$ by simulating the communication protocol) and $I(x:y|t)\mathbin{\le^{+}} I(x:y)$ due to Lemma \ref{['l:triple-info']}.
  • Figure 5: Complexity profile for $x,y,$ and $t':=\langle t_x,t_y\rangle$ from Lemma \ref{['l:simplifying-t']}. Note that $C(t_x)= I(x:t|y)$, $C(t_y)= I(y:t|x)$, and $I(x:y|t')= I(x:y).$
  • ...and 3 more figures

Theorems & Definitions (58)

  • Theorem 1: sketchy version; see RZ1 for a more precise statement
  • Theorem 2
  • Remark 1
  • Proposition 1
  • Lemma 1
  • proof
  • Remark 2
  • Remark 3
  • Example 1: discrete plane
  • Example 2: discrete Euclidean distance
  • ...and 48 more