Parameter Selection in Public Key Cryptosystem based on Chebyshev Polynomials over Finite Field

A recently proposed public key cryptosystem based on Chebyshev polynomials suggests a new approach to data encryption. But the security of the cryptosystem has not been investigated in depth, for lack of an appropriate analysis method. In this paper, a new representation of Chebyshev polynomial is introduced to study security issues of the  cryptosystem. The properties of Chebyshev polynomial sequence are presented, and their impact on the cryptosystem are discussed. Finally some principles for parameter selection for the cryptosystem are proposed. The methodology used in this paper is supposed to offer a useful means for future researches on this topic.

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

[2]  Kwok-Wo Wong,et al.  On the Security of Public-Key Algorithms Based on Chebyshev Polynomials over the Finite Field $Z_N$ , 2010, IEEE Transactions on Computers.

[3]  Adi Shamir,et al.  A method for obtaining digital signatures and public-key cryptosystems , 1978, CACM.

[4]  Leonard M. Adleman,et al.  A subexponential algorithm for the discrete logarithm problem with applications to cryptography , 1979, 20th Annual Symposium on Foundations of Computer Science (sfcs 1979).

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

[6]  Martin E. Hellman,et al.  An improved algorithm for computing logarithms over GF(p) and its cryptographic significance (Corresp.) , 1978, IEEE Trans. Inf. Theory.

[7]  Andrew M. Odlyzko,et al.  Discrete Logarithms: The Past and the Future , 2000, Des. Codes Cryptogr..

[8]  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..

[9]  Daniel Panario,et al.  Security of public-key cryptosystems based on Chebyshev polynomials over prime finite fields , 2008, 2008 IEEE International Symposium on Information Theory.

[10]  Xiaofeng Liao,et al.  A novel key agreement protocol based on chaotic maps , 2007, Inf. Sci..

[11]  K. Ramar,et al.  Public key cryptosystems based on chaotic-Chebyshev polynomials , 2009, 2009 International Conference on Intelligent Agent & Multi-Agent Systems.

[12]  Takeshi Koshiba,et al.  More on Security of Public-Key Cryptosystems Based on Chebyshev Polynomials , 2007, IEEE Transactions on Circuits and Systems II: Express Briefs.

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

[14]  G. J. Fee,et al.  Cryptography Using Chebyshev Polynomials , 2004 .

[15]  M. Rabin DIGITALIZED SIGNATURES AND PUBLIC-KEY FUNCTIONS AS INTRACTABLE AS FACTORIZATION , 1979 .

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

[17]  Liu Yun,et al.  Public Key Encryption Algorithm Based on Chebyshev Polynomials over Finite Fields , 2006, 2006 8th international Conference on Signal Processing.

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