Numerical methods for differential equations as encryption key

Abstract

Several cryptographic systems rely on the generation of pseudo-random sequences to conceal messages, chaotic attractors with intrinsic characteristics such as high sensitivity to initial conditions and the randomness of generated signals, have become able to replace their pseudo-random generator. In a crypto-chaotic system, one chooses one of the temporal functions resulting from the dynamic law characterizing the attractor so that it delivers stream-keys. The attractor plays the role of the generator (Ali-Pacha Generator) of the random data and by applying to it a sampling and then a quantification and finally an encoding, it is xored to the data. Thus it satisfies the condition of the sensitivity to the initial conditions and then it is added to the data to be secured. The majority of the chaotic attractors are defined by a system of differential equations (DE). A differential equation is an equality relation linking a function y with one or even several of its derivatives. When the differential equation is completed by an initial condition, that is, by the knowledge of the image of a particular real, we say that we have to solve a Cauchy problem. They do not have analytical solutions except approximations by numerical methods. Indeed, there are several types of differential equations. Each type requires a particular resolution method. The question then arises, “How to choose the numerical method that best approximates from a cryptographic point of view the attractor that delivers a chaotic stream (streamkeys), which will be added to the data to be secured”. Obtained results show that all of the different numerical analysis methods approximate globally the same differential equation. However, they do not give the same results as locally for all the points. This difference can be exploited by cryptography system. In other words, the name of the numeric method can be a parameter of the encryption key. So, the result found in this work poses another condition for the extraction of the data coming from a coordinate of a given attractor which are the same for the encryption and the decryption. Therefore, we must use the same numerical analysis technique to find solutions of the differential equations of the attractor for encryption and for decryption. In other words, the name of the numeric analysis method can be a parameter of the encryption key.