A basic framework for the cryptanalysis of digital chaos-based cryptography (original) (raw)
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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.
Cryptanalytic methods in chaotic cryptosystems
6 pages, 6 figures.-- Communication presented at the 5th World Multiconference on Systemics, Cybernetics and Informatics and 7th International Conference on Information System Analysis and Synthesis (SCI/ISAS 2001, Orlando, Florida, Jul 22-25, 2001). In recent years, telecommunications networks have undergone an explosive growth. As a consequence, there has been a strong demand of information protection mechanisms. Many cryptosystems based on chaos have been proposed, although little or no critical analysis has been made about the security and cryptographic robustness of these algorithms. In this paper we present our tools to examine some of these algorithms from a cryptographic perspective, showing many vulnerabilities that can be exploited to successfully break them. We conclude that most of the chaotic cryptosystems are very insecure and cumbersome, thus, unreliable and impractical for real applications. Peer reviewed
Some hints for the design of digital chaos-based cryptosystems: lessons learned from cryptanalysis
Arxiv preprint arXiv:0812.0765, 2008
In this work we comment some conclusions derived from the analysis of recent proposals in the field of chaos-based cryptography. These observations remark a number of major problems detected in some of those schemes under examination. Therefore, this paper is a list of what to avoid and to pay special attention to when considering chaos as source of new strategies to conceal and protect information.
Central and Eastern European eDem and eGov Days, 2018
The development of information society, which has led to an impressive increase in the volume of information, mainly economic, circulated in computer networks, accelerated the development and mostly the use of modern cryptography tools. In the last years, researchers have pointed out that there is a possible similarity between chaos and cryptography, many of the properties of chaotic dynamic systems having correlation among the cryptographic systems that are based on computational methods. Studies carried out on chaotic dynamic systems usage in digital crypto-systems have determined the occurrence of similar to classic techniques, but also of some specific techniques and methods that have been analyzed and evaluated. The attempts to develop new encryption аlgorithms based on chaos theory have evolved gradually from simple solutions, which suppose the iteration of a dinаmic system to obtain binary sequence used for text masking, to methods that imply coupled dinаmic systems and hybrid techniques that would combine the chaos advantages with classical methods. In this article there are presented 3 encryption algorithms based on chaos theory: RC4, Fractal Encryption and Cellular Automata, implemented in a system of encryption and operation mode analysis for each algorithm separately.
Encryption using Deterministic Chaos
The concepts of randomness, unpredictability, complexity and entropy form the basis of modern cryptography and a cryptosystem can be interpreted as the design of a key-dependent bijective transformation that is unpredictable to an observer for a given computational resource. For any cryptosystem, including a Pseudo-Random Number Generator (PRNG), encryption algorithm or a key exchange scheme, for example, a cryptanalyst has access to the time series of a dynamic system and knows the PRNG function (the algorithm that is assumed to be based on some iterative process) which is taken to be in the public domain by virtue of the Kerchhoff-Shannon principal, i.e. the enemy knows the system. However, the time series is not a compact subset of a trajectory (intermediate states are hidden) and the iteration function is taken to include a 'secret parameter'-the 'key'. We can think of the sample as being 'random', 'unpredictable' and 'complex'. What do these properties mean mathematically and how do they relate to chaos? This paper focuses on answers to this question, links these properties to chaotic dynamics and consider the issues associated with designing pseudo-random number generators based on chaotic systems. The theoretical backound associated with using chaos for encryption is introduced with regard to randomness and complexity. A complexity and information theortic approach is considered based on a study of the complexity and entropy measures associated with chaotic systems. A study of pseudorandomness is then given which provides the foundations for the numerical methods that need to be realed for the practical implementation of data encryption. We study cryptographic systems using finite-state approximations to chaos or 'pseudochaos' and develop an approach based on the concept of multialgorithmic cryptography that exploits the properties of pseudochaotic algorithms.
Lessons Learnt from the Cryptanalysis of Chaos-Based Ciphers
Chaos-Based Cryptography, 2011
The idea of using chaotic transformations in cryptography is explicit in the foundational papers of Shannon on secrecy systems (e.g., [96]). Although the word “chaos” was not minted till the 1970s [71], Shannon clearly refers to this very concept when he proposes the construction of secure ciphers by means of measure-preserving, mixing maps which depend ‘sensitively’ on their parameters. The implementation of Shannon’s intuitions had to wait till the development of Chaos Theory in the 1980s. Indeed, it was around 1990 when the first chaos-based ciphers were proposed (e.g., [78], [46]). Moreover, in 1990 chaos synchronization [91] entered the scene and shortly thereafter, the first applications to secure communications followed [56, 37]. The idea is remarkably simple: mask the message with a chaotic signal and use synchronization at the receiver to filter out the chaotic signal. The realization though had to overcome the desynchronization induced by the message itself. After this initial stage, the number of proposals which exploited the properties of chaotic maps for cryptographical purposes, grew in a spectacular way.
Cryptanalysis of a secure communication scheme combining chaos and noise
This paper studies the security of a secure communication scheme based on two discrete-time intermittently-chaotic systems synchronized via a common random driving signal. Some security defects of the secure communication scheme are revealed: 1) the key space can be remarkably reduced; 2) the decryption is insensitive to the mismatch of the secret key; 3) the key-generation process used in this secure communication scheme is insecure against known/chosen-plaintext attacks. The first two defects mean that the secure communication scheme is not secure enough against brute-force attacks, and the third means that an attacker can easily break the cryptosystem by approximately estimating the secret key once he has a chance to access a fragment of the generated keystream. A direct result of the cryptanalysis given in this paper is the unsuitability of intermittent chaos in the design of secure chaotic cryptosystems.
New approach to chaotic encryption
We describe a computational procedure to encrypt a message, provided that the transmitter and the receiver dispose of identical, but otherwise not synchronized, chaotic dynamical systems. The technique is based upon the fact that the symbolic dynamics of these two systems can be used in order to sequentially construct data blocks which reproduce those from the input file. q