Left invertibility, flatness and identifiability of switched linear dynamical systems: a framework for cryptographic applications

The purpose of this paper is to illustrate the potential interest of the control theory framework for cryptographic applications. It is shown that under the properties of left invertibility and flatness, dynamical systems are structurally equivalent to some specific cryptographic primitives called self-synchronizing stream ciphers. After having motivated the interest of considering hybrid systems for such ciphers, the development is particularized for the special class of switched linear systems. We also show that identifiability is a necessary condition for security, describe an identification procedure as a possible attack and assess its complexity. Hybrid systems have inspired a great deal of research from both control theory and theoretical computer science ([HSCC(2008)]). They provide a strong theoretical foundation which combines discrete-event and continuoustime systems in a manner that can capture software logic and physical dynamics in a unified modeling framework. The most well-known area of applicability of hybrid systems are naturally modeling, analysis and control design of embedded systems. From a theoretical point of view, stability, identifiability, controller or observer design are challenging problems widely studied in the literature ([Shorten et al.(2007), Bemporad et al.(2000), Juloski et al.(2005), Balluchi et al.(2002), Babaali and Egerstedt(2004)]). In this paper, left invertibility, flatness and identifiability of discretetime switched linear systems are investigated. The hybrid aspect is really taken into account insofar as the minimum ”dwell-time” assumption is relaxed or in other words, since the switching rule is not restricted to the case when the modes are active during a sufficient large time. The conditions characterizing left invertibility ([Sain and Massey(1969)]) and flatness 1 ha l-0 04 45 92 9, v er si on 1 3 M ar 2 01 0 Author manuscript, published in "International Journal of Control 83, 1 (2010) 145-153" DOI : 10.1080/00207170903104190