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Applied mathematics, Information Science
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 background associated with using chaos for encryption is introduced with regard to randomness and complexity. A complexity and information theoretic 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.
Blackledge, J., Ptitsyn, N.: Encryption using Deterministic Chaos. ISAST Transactions on Electronics and Signal Processing, vol. 4, issue 1, pp. 6-17. 2010.