Fast algorithms of public key cryptosystem based on Chebyshev polynomials over finite field

Abstract The computation of Chebyshev polynomial over finite field is a dominating operation for a public key cryptosystem. Two generic algorithms with running time of have been presented for this computation: the matrix algorithm and the characteristic polynomial algorithm, which are feasible but not optimized. In this paper, these two algorithms are modified in procedure to get faster execution speed. The complexity of modified algorithms is still, but the number of required operations is reduced, so the execution speed is improved. Besides, a new algorithm relevant with eigenvalues of matrix in representation of Chebyshev polynomials is also presented, which can further reduce the running time of that computation if certain conditions are satisfied. Software implementations of these algorithms are realized, and the running time comparison is given. Finally an efficient scheme for the computation of Chebyshev polynomial over finite field is presented.

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

[3]  Siguna Müller,et al.  On the Computation of Square Roots in Finite Fields , 2004, Des. Codes Cryptogr..

[4]  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..

[5]  Dahu Wang,et al.  An Identity Authentication System Based on Chebyshev Polynomials , 2009, 2009 First International Conference on Information Science and Engineering.

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

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

[8]  A method for generating S-box based on iterating chaotic maps , 2010 .

[9]  F. Yu,et al.  A New Key Exchange Scheme Based on Chebyshev Polynomials , 2008, 2008 Congress on Image and Signal Processing.

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

[11]  Xing-yuan Wang,et al.  An improved key agreement protocol based on chaos , 2010 .

[12]  Vinod Patidar,et al.  A Random Bit Generator Using Chaotic Maps , 2010, Int. J. Netw. Secur..

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