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Cryptography with chaotic mixing. (English) Zbl 1139.94005

The authors suggest using the chaotic map \(H=r_p^{-1}\circ T\circ r_p\) defined on the real interval \([0,10^p)\), with \(r_p(x)=\sqrt[p](x)\) (so that \(r_p^{-1}(x)=x^p\)), and \(T(x)=10x \bmod 10\) is the shift map. The secret key is a random real number \(x_0\in[0,10^p)\). A message to be encrypted is a sequence \(m_1,m_2,\dots\in\{0,\dots,10^p-1\}\), and the encrypted message is a list of natural numbers: \(c_1\) is the first number such that \(H^{c_1}(x_0)\in [m_1/10^p,(m_1+1)/10^p)\), \(c_2\) is the first number such that \(H^{c_2}(x_0)\in [m_2/10^p,(m_2+1)/10^p)\), etc. Note that the running time is large if \(p\) is not small enough. Some variants are also discussed. The authors argue, based on the chaotic nature of \(H\), that the system is secure against a brute-force attack which tries to guess \(x_0\) up to a small error. They refer the reader to other papers for cryptanalyses using known plaintext. To what extent is their analysis correct in the real-world case, where random real numbers cannot be chosen nor stored by computers with finite memory? This is not addressed in the paper.

MSC:

94A60 Cryptography
37D45 Strange attractors, chaotic dynamics of systems with hyperbolic behavior
68P25 Data encryption (aspects in computer science)
65P20 Numerical chaos
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References:

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