Cryptography with Chaos and Shadowing

Abstract In this paper, we present a novel approach to encrypt a message (a text composed by some alphabets) using chaos and shadowing. First, we generate a numerical chaotic orbit based on the logistic map, and use the shadowing algorithm of Smaoui and Kostelich [Smaoui N, Kostelich E. Using chaos to shadow the quadratic map for all time. Int J Comput Math 1998;70:117–29] to show that there exists a finite number of true orbits that shadow the numerical orbit. Then, the finite number of maps generated is used in Baptista’s algorithm [Baptista MS. Cryptography with chaos. Phys Lett A 1998;240:50-4] to encrypt each character of the message. It is shown that the use of chaos and shadowing in the encryption process enhances the security level.

[1]  W. Schwarz,et al.  Chaos and cryptography , 2001 .

[2]  L. Kocarev.  Chaos-based cryptography: a brief overview , 2001 .

[3]  J. Yorke,et al.  Pseudo-orbit shadowing in the family of tent maps , 1988 .

[4]  Z. Kotulski,et al.  APPLICATION OF DISCRETE CHAOTIC DYNAMICAL SYSTEMS IN CRYPTOGRAPHY — DCC METHOD , 1999 .

[5]  L. Kocarev,et al.  Chaos and cryptography: block encryption ciphers based on chaotic maps , 2001 .

[6]  Gonzalo Alvarez,et al.  Chaotic cryptosystems , 1999, Proceedings IEEE 33rd Annual 1999 International Carnahan Conference on Security Technology (Cat. No.99CH36303).

[7]  Xuanqin Mou,et al.  Pseudo-random Bit Generator Based on Couple Chaotic Systems and Its Applications in Stream-Cipher Cryptography , 2001, INDOCRYPT.

[8]  Kwok-Wo Wong,et al.  A chaotic cryptography scheme for generating short ciphertext , 2003 .

[9]  J. Fridrich.  Symmetric Ciphers Based on Two-Dimensional Chaotic Maps , 1998 .

[10]  G. Álvarez,et al.  Cryptanalysis of dynamic look-up table based chaotic cryptosystems , 2003, nlin/0311043.

[11]  Iwao Sasase,et al.  A Secret Key Cryptosystem by Iterating a Chaotic Map , 1991, EUROCRYPT.

[12]  Kwok-wo Wong,et al.  A modified chaotic cryptographic method , 2001 .

[13]  Kevin M. Short,et al.  Signal Extraction from Chaotic Communications , 1997 .

[14]  H. Dedieu,et al.  Some tools for attacking secure communication systems employing chaotic carriers , 1998, ISCAS '98. Proceedings of the 1998 IEEE International Symposium on Circuits and Systems (Cat. No.98CH36187).

[15]  Kwok-Wo Wong,et al.  A new block cipher based on chaotic map and group theory , 2009 .

[16]  Grebogi,et al.  Communicating with chaos. , 1993, Physical review letters.

[17]  G. Álvarez,et al.  Cryptanalysis of an ergodic chaotic cipher , 2003 .

[18]  Kwok-Wo Wong,et al.  A combined chaotic cryptographic and hashing scheme , 2003 .

[19]  James A. Yorke,et al.  Rigorous verification of trajectories for the computer simulation of dynamical systems , 1991 .

[20]  X. Liao,et al.  A novel chaotic encryption scheme based on arithmetic coding , 2008 .

[21]  Carroll,et al.  Synchronization in chaotic systems. , 1990, Physical review letters.

[22]  L. Chua,et al.  CLARIFYING CHAOS: EXAMPLES AND COUNTEREXAMPLES , 1996 .

[23]  A. Kanso,et al.  Logistic chaotic maps for binary numbers generations , 2009 .

[24]  Robert A. J. Matthews,et al.  On the Derivation of a "Chaotic" Encryption Algorithm , 1989, Cryptologia.

[25]  X. Mou,et al.  Performance analysis of Jakimoski–Kocarev attack on a class of chaotic cryptosystems , 2003 .

[26]  R. Bowen.  ω-Limit sets for Axiom A diffeomorphisms , 1975 .

[27]  Anastasios Bezerianos,et al.  A Probabilistic Symmetric Encryption Scheme for Very Fast Secure Communication Based on Chaotic Systems of difference equations , 2001, Int. J. Bifurc. Chaos.

[28]  Grebogi,et al.  Using chaos to direct orbits to targets in systems describable by a one-dimensional map. , 1992, Physical review. A, Atomic, molecular, and optical physics.

[29]  A. Palacios,et al.  Cryptography with cycling chaos , 2002 .

[30]  Roland Schmitz,et al.  Use of chaotic dynamical systems in cryptography , 2001, J. Frankl. Inst..

[31]  E. Alvarez,et al.  New approach to chaotic encryption , 1999 .

[32]  Ljupco Kocarev,et al.  Analysis of some recently proposed chaos-based encryption algo-rithms , 2001 .

[33]  Alfred Menezes,et al.  Handbook of Applied Cryptography , 2018 .

[34]  X. Liao,et al.  A new chaotic cryptosystem , 2006 .

[35]  A. Akhavan,et al.  A novel algorithm for image encryption based on mixture of chaotic maps , 2008 .

[36]  Gonzalo Alvarez,et al.  Keystream cryptanalysis of a chaotic cryptographic method , 2004 .

[37]  Grebogi,et al.  Higher-dimensional targeting. , 1993, Physical review. E, Statistical physics, plasmas, fluids, and related interdisciplinary topics.

[38]  Nejib Smaoui,et al.  Using chaos to shadow the quadratic map for all time , 1998, Int. J. Comput. Math..

[39]  K. Aihara,et al.  Cryptosystems with discretized chaotic maps , 2002 .

[40]  M. Baptista.  Cryptography with chaos , 1998 .

[41]  K. Wong,et al.  A fast chaotic cryptographic scheme with dynamic look-up table , 2002 .

[42]  James A. Yorke,et al.  Is every approximate trajectory of some process near an exact trajectory of a nearby process? , 1988 .

[43]  Celso Grebogi,et al.  Cryptography with chaos at the physical level. , 2004 .

[44]  X. Mou,et al.  Improving security of a chaotic encryption approach , 2001, Physics Letters A.

[45]  Celso Grebogi,et al.  Do numerical orbits of chaotic dynamical processes represent true orbits? , 1987, J. Complex..

[46]  Celso Grebogi,et al.  Numerical orbits of chaotic processes represent true orbits , 1988 .