From combinatorics of graphs to communication complexity in algorithmic information theory

Several connexions between information theory and combinatorics of graphs have been explored in the last decades. Following this path, we use tools from spectral graph theory to show impossibility results in information theoretic cryptography. We place ourselves in the Kolmogorov complexity framework. After a short introduction to algorithmic information theory and secret key agreement, we will study some spectral and combinatorial properties of bipartite graphs explicitly constructed.

Those graphs can represent the inputs for a communication problem. Spectral and combinatorial properties of these objects imply information inequalities that make impossible some communication protocols for secret key agreement. For instance, if the graph representing the inputs of the secret key agreement is a spectral expander, we get the worst case in terms of communication complexity. Moreover, the communication protocol is inherently uneven: for two participants, only one of them needs to send all the needed information. Theses facts are connected to the mutual information unextractability phenomenon.