Chaos-Based Public-Key Cryptography

In this chapter we give an overview and the state of the art in the field of Chaos-based cryptography. The public key cryptosystems based on Chebyshev polynomials enjoy some nice chaotic properties, which makes them suitable for use in both encryption and digital signature. The cryptosystem can work either on real or integer numbers. The cryptosystem that works on real numbers is not secure and permits to recover the corresponding plaintext from a given ciphertext. In addition, it also allows forgeries if the cryptosystem is used for signing messages. On the other hand, ElGamal-like and RSA-like algorithms when using Chebyshev polynomials on integer numbers are secure as the aforementioned encryption algorithms. The chaos-based cryptography is discussed from a point of view which we believe is closer to the spirit of both cryptography and chaos theory than the way the subject has been treated recently by many researchers.

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

[2]  L. Kocarev,et al.  Chaos-based random number generators. Part II: practical realization , 2001 .

[3]  Andrea Gerosa,et al.  A fully integrated chaotic system for the generation of truly random numbers , 2002 .

[4]  Nikolai F. Rulkov,et al.  Chaotic pulse position modulation: a robust method of communicating with chaos , 2000, IEEE Communications Letters.

[5]  Franz Pichler,et al.  Computer Aided Systems Theory — EUROCAST '95 , 1995, Lecture Notes in Computer Science.

[6]  Xiaofeng Liao,et al.  Using Chebyshev chaotic map to construct infinite length hash chains , 2004, 2004 International Conference on Communications, Circuits and Systems (IEEE Cat. No.04EX914).

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

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

[9]  Edmund Taylor Whittaker,et al.  A Course of Modern Analysis , 2021 .

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

[11]  Taher ElGamal,et al.  A public key cyryptosystem and signature scheme based on discrete logarithms , 1985 .

[12]  Ljupco Kocarev,et al.  Public-key encryption based on Chebyshev maps , 2003, Proceedings of the 2003 International Symposium on Circuits and Systems, 2003. ISCAS '03..

[13]  Ljupco Kocarev,et al.  Differential and linear probabilities of a block-encryption cipher , 2003 .

[14]  T. Kohda,et al.  Statistics of chaotic binary sequences , 1997, IEEE Trans. Inf. Theory.

[15]  E. W. Ng.  Symbolic and Algebraic Computation , 1979, Lecture Notes in Computer Science.

[16]  Daniel D. Wheeler,et al.  Supercomputer Investigations of a Chaotic Encryption Algorithm , 1991, Cryptologia.

[17]  Tohru Kohda,et al.  Jacobian elliptic Chebyshev rational maps , 2001 .

[18]  J. Alvin Connelly,et al.  A noise-based IC random number generator for applications in cryptography , 2000 .

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

[20]  Franz Pichler,et al.  Finite Dimensional Generalized Baker Dynamical Systems for Cryptographic Applications , 1995, EUROCAST.

[21]  Whitfield Diffie,et al.  New Directions in Cryptography , 1976, IEEE Trans. Inf. Theory.

[22]  Alfredo De Santis,et al.  Security of public-key cryptosystems based on Chebyshev polynomials , 2004, IEEE Transactions on Circuits and Systems I: Regular Papers.

[23]  Ken Umeno,et al.  Exactly Solvable Chaos and Addition Theorems of Elliptic Functions (Applied Mathematics of Discrete Integrable Systems) , 1997, chao-dyn/9704007.

[24]  L. Kocarev,et al.  Pseudorandom bits generated by chaotic maps , 2003 .

[25]  Helmut Hasse,et al.  Number Theory , 2020, An Introduction to Probabilistic Number Theory.

[26]  E. L. Wachspress.  Evaluating elliptic functions and their inverses , 2000 .

[27]  Harvey Cohn,et al.  A second course in number theory , 1962 .

[28]  Ken Umeno,et al.  METHOD OF CONSTRUCTING EXACTLY SOLVABLE CHAOS , 1996, chao-dyn/9610009.

[29]  Donald Ervin Knuth,et al.  The Art of Computer Programming , 1968 .

[30]  Irene A. Stegun,et al.  Handbook of Mathematical Functions. , 1966 .

[31]  T. J. Rivlin.  Chebyshev polynomials : from approximation theory to algebra and number theory , 1990 .

[32]  Ljupco Kocarev,et al.  Public-Key Encryption Based on Chebyshev Polynomials , 2005 .

[33]  Michael Peter Kennedy,et al.  FM-DCSK: a novel method for chaotic communications , 1998, ISCAS '98. Proceedings of the 1998 IEEE International Symposium on Circuits and Systems (Cat. No.98CH36187).

[34]  L. Kocarev,et al.  Chaos-based random number generators-part I: analysis [cryptography] , 2001 .

[35]  T. Elgamal.  A public key cryptosystem and a signature scheme based on discrete logarithms , 1984, CRYPTO 1984.

[36]  Donald E. Knuth,et al.  The art of computer programming, volume 3: (2nd ed.) sorting and searching , 1998 .

[37]  Gonzalo Álvarez,et al.  Some Basic Cryptographic Requirements for Chaos-Based Cryptosystems , 2003, Int. J. Bifurc. Chaos.

[38]  T. J. Rivlin.  The Chebyshev polynomials , 1974 .

[39]  Daniel D. Wheeler,et al.  Problems with Chaotic Cryptosystems , 1989, Cryptologia.

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

[41]  Claude E. Shannon,et al.  Communication theory of secrecy systems , 1949, Bell Syst. Tech. J..

[42]  S. D. Dhodapkar,et al.  CHAOS BASED CRYPTOGRAPHY : A NEW APPROACH TO SECURE COMMUNICATIONS , 2005 .

[43]  Richard J. Fateman,et al.  Lookup tables, recurrences and complexity , 1989, ISSAC '89.

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

[45]  I. Percival,et al.  Arithmetical properties of strongly chaotic motions , 1987 .