The application of modified Chebyshev polynomials in asymmetric cryptography


  • Marcin Lawnik Silesian University of Technology, Faculty of Applied Mathematics
  • Adrian Kapczyński Silesian University of Technology, Faculty of Applied Mathematics



asymmetric encryption, Chebyshev polynomials, chaos


Based on Chebyshev polynomials, you can create an asymmetric cryptosystem that allows secure communication. Such a cryptosystem uses the fact that these polynomials form a semi-group due to the composition operation. This article presents new cryptosystems that use other than semi-group property dependencies. Based on these dependencies as well as modifications of Chebyshev's polynomials, two cryptosystems have been proposed. The presented analysis shows that their security is the same as in the case of algorithms associated with the problem of discrete logarithms. The article also shows methods that allow faster calculation of Chebyshev polynomials.


Download data is not yet available.


Algehawi M., Samsudin A.: A new Identity Based Encryption (IBE) scheme usingextended Chebyshev polynomial over finite fields Zp. In:Physics Letters A, vol.374, pp. 4670–4674, 2010.

Algehawi M., Samsudin A., Jahani S.: Calculation Enhancement of ChebyshevPolynomial over Zp. In:Malaysian Journal of Mathematical Sciences, vol. 7(S),pp. 131–143, 2013.

Banasik A., Kapczy ́nski A.: Fuzzy evaluation of biometric authentication sys-tems. In:Proceedings of the 6th IEEE International Conference on IntelligentData Acquisition and Advanced Computing Systems: Technology and Applica-tions, IDAACS’2011, pp. 803–806. 2011.

Baptista M.: Cryptography with chaos. In:Physics Letters A, vol. 240(1-2), pp.50–54, 1998.

Benjamin T., Ericksen L., Jayawant P., Shattuck M.: Combinatorial trigonometrywith Chebyshev polynomials. In:Journal of Statistical Planning and Inference,vol. 140, pp. 2157–2160, 2010.

Benjamin T., Walton D.: Counting on Chebyshev Polynomials. In:MathematicsMagazine, vol. 82(2), pp. 117–126, 2009.

Bergamo P., DArco P., De Santis A., Kocarev L.: Security of Public Key Cryp-tosystems based on Chebyshev Polynomials. In:Circuits and Systems I: RegularPapers, IEEE Transactions on, vol. 52, pp. 1382–1393, 2005.

Chen Y., Xushuai J., Jiang Q., Gong L.: Key Agreement Protocol Based onChebyshev Polynomials for Wireless Sensor Network. In:Journal of Computa-tional Information Systems, vol. 10(2), p. 589594, 2014.

Cheong K.Y.: One-way Functions from Chebyshev Polynomials. CryptologyePrint Archive, Report 2012/263, 2012.

Coppersmith D.: The Data Encryption Standard (DES) and its strength againstattacks. In:IBM Journal of Research and Development, vol. 38(3), pp. 243–250,1994.

Coppersmith D., Winograd S.: Matrix multiplication via arithmetic progressions.In:Journal of Symbolic Computation, vol. 9(3), pp. 251–280, 1990.


Diffie W., Hellman M.: New Directions in Cryptography. In:IEEE TRANSAC-TIONS ON INFORMATION THEORY, vol. IT-22(6), 1976.

Elgamal T.: A Public Key Cryptosystem and a Signature Scheme Based on Dis-crete Logarithms. In:IEEE TRANSACTIONS ON INFORMATION THEORY,vol. IT-31(4), pp. 469–472, 1985.

Fee G., Monagan M.: Cryptography using Chebyshev polynomials. pp. 1–15.2004. Http://

Ghebleh M., Kanso A.: A robust chaotic algorithm for digital image steganog-raphy. In:Communications in Nonlinear Science and Numerical Simulation,vol. 19(6), pp. 1898–1907, 2014.

Hafizul Islam S.K.: Identity-based encryption and digital signature schemes usingextended chaotic maps. In:IACR Cryptology ePrint Archive, (275), p. 16, 2014.

Haifeng Q., Xiangxue L., Yu Y.: Pitfalls in Identity Based Encryption UsingExtended Chebyshev Polynomial. In:China Communications, vol. 1, pp. 58–63,2012.

Jianli Y., Dahu W.: Applying Extended Chebyshev Polynomials to Construct aTrap-door One-way Function in Real Field. pp. 1680–1682. 2009.

Kapczy ́nski A., Banasik A.: Model of intelligent detection mechanism againstfalse biometric data injection in fingerprint-based authentication systems. In:Proceedings of the 5th IEEE International Workshop on Intelligent Data Ac-quisition and Advanced Computing Systems:Technology and Applications,IDAACS’2009, pp. 496–498. 2009.

Kocarev L., Makraduli J., Amato P.: Public-Key Encryption Based on ChebyshevPolynomials. In:Circuits, Systems and Signal Processing, vol. 24(5), pp. 497–517,2005.

Kocarev L., Tasev Z.: Public-Key Encryption Based on Chebyshev Maps. vol. 3,pp. 28–31. 2003.

Lai H., Xiao J., Li L., Yang Y.: Applying Semigroup Property of EnhancedChebyshev Polynomials to Anonymous Authentication Protocol. In:Mathemat-ical Problems in Engineering, (454823), pp. 1–17, 2012.

Lawnik M.: Applications of Viete-Lucas polynomials in public-key cryptography.In:Stud. Informat., vol. 38(4), pp. 69–77, 2017. In polish.

Lawnik M.: Generalized logistic map and its application in chaos based cryptog-raphy. In:J. Phys.: Conf. Ser., vol. 936(1742-6588), pp. 1–4, 2017.

Lawnik M.: Combined logistic and tent map. In:J. Phys.: Conf. Ser., vol.1141(012132), pp. 1–6, 2018.

Lawnik M.: The problem of the inverse Lyapunov exponent and its application.2019/05/28; 20:52 str. 12/13

In:Nonlinear Anal., Model. Control, vol. 23(6), pp. 951–960, 2018.

Li Z., Cui Y., Xu H.: Fast algorithms of public key cryptosystem based onChebyshev polynomials over finite field. In:The Journal of China Universitiesof Posts and Telecommunications, vol. 18(2), pp. 86–93, 2011.

Liao X., Chen F., Wong K.W.: On the Security of Public-Key Algorithms Basedon Chebyshev Polynomials over the Finite Field ZN. In:IEEE Transactions OnComputers, vol. 59(10), pp. 1392–1401, 2010.

Lima J., Panario D., Campello de Souza R.: Public-key encryption based onChebyshev polynomials over GF(q). In:Information Processing Letters, vol.111(2), pp. 51–56, 2010.

Mishkovski I., Kocarev L.:Chaos-Based Public-Key Cryptography, pp. 27–65.Springer Berlin Heidelberg, 2011.

Rivest R., Shamir A., Adleman L.: A Method for Obtaining Digital Signaturesand Public-key Cryptosystems. In:Commun. ACM, vol. 21(2), pp. 120–126,1978.

Rivlin T.:Chebyshev polynomials: from approximation theory to algebra andnumber theory. Wiley, 1990.

Roy R., Sarkar A., Changder S.: Chaos based Edge Adaptive Image Steganogra-phy. In:Procedia Technology, vol. 10, pp. 138–146, 2013.

Sun J., Zhao G., Li X.: An Improved Public Key Encryption Algorithm Basedon Chebyshev Polynomials. In:TELKOMNIKA, vol. 11(2), pp. 864–870, 2013.

Toan-Thinh T., Minh-Triet T., Anh-Duc D.: Improved Chebyshev Polynomials-Based Authentication Scheme in Client-Server Environment. In:Security andCommunication Networks, vol. 2019(4250743), pp. 1–11.

Vairachilai S., KavithaDevi M., Gnanajeyaraman R.: Public Key Cryptosystemsusing Chebyshev Polynomials Based on Edge Information. pp. 243–245. 2014.

Witu la R., S lota D.: On modified Chebyshev polynomials. In:Journal of Math-ematical Analysis and Applications, vol. 324(1), pp. 321–343, 2006.

Xiang T., Wong K.W., Liao X.: On the security of a novel key agreement protocolbased on chaotic maps. In:Chaos, Solitons & Fractals, vol. 40(2), pp. 672–675,2009.

Xiao D., Liao X., Deng S.: A novel key agreement protocol based on chaoticmaps. In:Information Sciences, vol. 177(4), pp. 1136–1142, 2007.




How to Cite

Lawnik, M., & Kapczyński, A. (2019). The application of modified Chebyshev polynomials in asymmetric cryptography. Computer Science, 20(3).