Framework for the analysis and design of encryption strategies based on discrete-time chaotic dynamical systems

Resumen

Since 1990s chaotic dynamical systems have been widely used to design new strategies to encrypt information. Indeed, the dependency to initial conditions and control parameters, along with the ergodicity of their temporal evolution allow the establishment of chaos as the base of new cryptosystems, i.e., of new schemes of confusion and diffusion of information. However, an optimum design in the context of chaos-based cryptography demands a thorough knowledge not only of the foundations of cryptography, but also of the dynamics and inner structure of chaos. Therefore, any proposal to use chaos in the context of cryptography must respect a series of design rules, in order to avoid the reconstruction of the dynamics of the underlying chaotic system, and to determine an optimum use of the virtues of the chaotic dynamics. Although it is possible to use chaos to design analog cryptosystems based on synchronization techniques, this Thesis is focused on the application of chaotic maps, i.e., chaotic dynamical systems defined in discrete time to cryptography. In this sense, a set of mathematical tools are defined to establish the adequacy of a chaotic map as the base of a cryptosystem, and the requirements that an encryption architecture must satisfy to avoid the dynamical reconstruction of the underlying chaotic map. More precisely, this Thesis provides an extension and systematization of the results derived from the cryptanalysis of chaos-based cryptosystems. The above goal comprises three different stages: 1.- Definition of a set of mathematical tools that allow the selection of the adequate configurations of a dynamical system to implement strategies of confusion and diffusion of information. 2.- Study of the most popular chaotic maps in the field of chaos-based cryptography to determine whether these maps can be used to design new cryptosystems without incurring in security problems. 3.- Summary and conclusions of the first two stages. The aim is to define a set of rules or recommendations as a guide for the design of chaos-based cryptosystems. Recalling the first stage, its main purpose is the search of procedures to infer or estimate the initial conditions and/or the control parameters from the orbits of a chaotic map. Different scenarios are considered depending on whether complete orbits are accesible or it is only possible to work with sampled or discretized versions of the orbits. In all scenarios the goal consist in building bijective functions with respect to the initial conditions and/or the control parameters. The requirements to build these bijective functions are clarified, along with the procedures to guide the estimation of the initial conditions and/or the control parameters. In order to test the set of mathematical tools and the estimation methods, the logistic map and its associated topological conjugate maps are thoroughly studied, since these maps are the most widely used in the design of new digital chaotic cryptosystems. Specially relevant is the study of the symbolic dynamics and order patterns of unimodal maps. The study of this family of chaotic maps leads to a series of very useful results to define a set of recommendations for both the evaluation of the security of chaos-based cryptosystems and the design of encryption schemes based on chaos.