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.


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


[1] Pecora, L.M.; Carroll, T.L., Synchronization in chaotic systems, Phys rev lett, 64, 821-824, (1990) · Zbl 0938.37019
[2] Hayes, S.; Grebogy, C.; Ott, E.; Mark, A., Experimental control of chaos for communication, Phys rev lett, 73, 1781-1784, (1994)
[3] Baptista, M.S., Cryptography with chaos, Phys lett A, 240, 50-54, (1998) · Zbl 0936.94013
[4] Wong, W.-K.; Lee, L.-P.; Wong, K.-W., A modified chaotic cryptographic method, Comput phys commun, 138, 234-236, (2001) · Zbl 0987.94033
[5] Jakimoski, G.; Kocarev, L., Analysis of some recently proposed chaos-based encryption algorithms, Phys lett A, 291, 381-384, (2001) · Zbl 0978.68061
[6] Li, S.; Mou, X.; Ji, Z.; Zhang, J.; Cai, Y., Performance analysis of jakimoski – kocarev attack on a class of chaotic cryptosystems, Phys lett A, 307, 22-28, (2003) · Zbl 1006.94018
[7] Weisstein EW. Normal number. From MathWorld-A Wolfram Web Resource. Available from: http://mathworld.wolfram.com/NormalNumber.html, visited in 25 August 2005.
This reference list is based on information provided by the publisher or from digital mathematics libraries. Its items are heuristically matched to zbMATH identifiers and may contain data conversion errors. It attempts to reflect the references listed in the original paper as accurately as possible without claiming the completeness or perfect precision of the matching.