Cryptography-Based Chaos via Geometric Undersampling of Ring-Coupled Attractors

We propose a new mechanism for undersampling chaotic numbers obtained by the ring coupling of one-dimensional maps. In the case of 2 coupled maps this mechanism allows the building of a PRNG which passes all NIST Test. This new geometric undersampling is very effective for generating 2 parallel streams of pseudorandom numbers, as we show, computing carefully their properties, up to sequences of 10 consecutives iterates of the ring coupled mapping which provides more than 3.35 x 10 random numbers in very short time. Both 3 and 4 dimension cases can be managed in the same way. In addition we recall a novel method of noise-resisting ciphering. The originality lies in the use of a chaotic pseudo-random number generator: several co-generated sequences can be used at different steps of the ciphering process, since they present the strong property of being uncorrelated. Each letter of the initial alphabet of the plain text is encoded as a subinterval of [-1,1]. The bounds of each interval are defined in function of the known bound of the additive noise. A pseudo-random sequence is used to enhance the complexity of the ciphering. The transmission consists of a substitution technique inside a chaotic carrier, depending on another cogenerated sequence. This novel noise-resisting ciphering method can be used with geometric undersampling when 4 mappings are coupled. CONTENTS Introduction 2 1. Recovering chaotic properties of numerically computed chaotic numbers 2 1.1. Numerical approximation of chaotic numbers 2 1.2. Very long periodic orbits for ultra-weakly coupled tent map 4 1.2.1. 2-coupled symmetric tent map 4 1.2.2. p-coupled symmetric tent map 4 1.2.3. Computation of approximated invariant measure 6 2. The route from chaos to pseudo-randomness via chaotic undersampling 8 2.1. Chaotic under-sampling 8 2.2. Chaotic mixing 9 2.3. Enhanced chaotic under-sampling 10 2.4. A window of emergence of randomness 10 3. Geometric undersampling 10 3.1. Pseudo-random numbers generated by ring coupled mapping 11 3.2. Ring coupling of 2-dimensional symmetric tent map 12 3.2.1. Critical lines 13 3.2.2. Markov partition 14 3.2.3. Exact computation of invariant measure associated to 2 M 16 3.3. Geometric undersampling 17 3.3.1. Algorithm of geometric undersampling 17 3.3.2. Numerical tests 18 4. Noise-resisting ciphering 20 4.1. Ciphering principle 20 4.2. Transmission principle 24 4.3. Decoding principle 24 4.4. Numerical illustration 24 Conclusion 25 1 Université de Nice Sophia-Antipolis, Laboratoire J. A. Dieudonné, UMR CNRS 7351, Parc Valrose, 06108 NICE, Cedex 02, France, email : [email protected]