Mutual learning in a tree parity machine and its application to cryptography

TL;DR

This paper analytically studies mutual learning in tree parity machines with continuous and discrete weights, revealing phase transitions and synchronization dynamics, and applies these findings to develop a cryptographic key-exchange protocol.

Contribution

It provides a novel analytical framework for understanding mutual learning in tree parity machines and demonstrates their application in cryptography.

Findings

Phase transition from partial to full synchronization in continuous machines.

Finite steps to full synchronization in discrete machines.

Analytical results agree with simulations.

Abstract

Mutual learning of a pair of tree parity machines with continuous and discrete weight vectors is studied analytically. The analysis is based on a mapping procedure that maps the mutual learning in tree parity machines onto mutual learning in noisy perceptrons. The stationary solution of the mutual learning in the case of continuous tree parity machines depends on the learning rate where a phase transition from partial to full synchronization is observed. In the discrete case the learning process is based on a finite increment and a full synchronized state is achieved in a finite number of steps. The synchronization of discrete parity machines is introduced in order to construct an ephemeral key-exchange protocol. The dynamic learning of a third tree parity machine (an attacker) that tries to imitate one of the two machines while the two still update their weight vectors is also analyzed. In particular, the synchronization times of the naive attacker and the flipping attacker recently introduced in [1] are analyzed. All analytical results are found to be in good agreement with simulation results.