arXiv subjects

Geoffroy Caillat-Grenier

Publications and source records attributed to Geoffroy Caillat-Grenier.

3 recordsLinked to original sources

arXiv2026

Random Schreier graphs as expanders

Expander graphs, due to their mixing properties, are useful in many algorithms and combinatorial constructions. One can produce an expander graph with high probability by taking a random graph (e.g., the union of $d$ random bijections for a bipartite graph of degree $d$). This construction is much simpler than all known explicit constructions of expanders and gives graphs with good mixing properties (small second largest eigenvalue) with high probability. However, from the practical viewpoint, it uses too many random bits. The natural idea is to restrict the class of the bijections that we use. For example, if both sides are vector spaces $\mathbb{F}_q^k$ over a finite field $\mathbb{F}_q$, we may consider only \emph{linear} bijections, making the number of random bits polynomial in $k$ (and not $q^k$). In this paper we provide some experimental data that shows that this approach conserves the mixing properties (the second eigenvalue) for several types of graphs (undirected regular and biregular bipartite graphs). We also prove some upper bounds for the second eigenvalue (though they are quite weak compared with the experimental results) and a lower bound for a special class of graphs. Finally, we discuss the possibility to decrease the number of random bits further by using Toeplitz matrices; our experiments show that this change makes the mixing properties only marginally worse while the number of random bits decreases significantly.

math.CO↗

arXiv2025

Common information in well-mixing graphs and applications to information-theoretic cryptography

We study the connection between mixing properties for bipartite graphs and materialization of the mutual information in one-shot settings. We show that mixing properties of a graph imply impossibility to extract the mutual information shared by the ends of an edge randomly sampled in the graph. We apply these impossibility results to some questions motivated by information-theoretic cryptography. In particular, we show that communication complexity of a secret key agreement in one-shot setting is inherently uneven: for some inputs, almost all communication complexity inevitably falls on only one party.

cs.IT↗

arXiv2023

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

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.

cs.IT↗