Open Access Open Access  Restricted Access Subscription Access

A Cryptosystem based on Topological Space

Sunil Kumar Kashyap

Abstract


The study of surface is topology. This paper uses some characteristics of knot theory and develops a public key cryptosystem. A topologically, algebraically one way trapdoor function-based knot cryptography. A link is a set of one or more closed loops in three-dimensional space. The individual loops are called the components of the link. The loops can be twisted or knotted. If there is only one loop, the link is called knot. The knot theory becomes fruitful after the discovery of Jones Polynomial. These knots are represented by three-dimensional geometry and polynomials both. But here we took an example on those knots, which can be represented by two variables; this is known by HOMFLY polynomial. HOMFLY is the compact version of the first letter of its discoverer, Hoste, Ocneanu, Millette, Freyd, Lickorish, and Yetter. There are several types of knot polynomials. Our proposed cryptography is based on all those knot polynomials, which are fit on the machinery of topology.


Keywords


Topology, knot theory, cryptography, HOMFLY, polynomial.\

Full Text:

PDF

References


Adams CC. The knot book: an elementary introduction to the mathematical theory of knots. New York: W.H. Freeman; 1994. pp. 171–2.

Doll H, Hoste J. A tabulation of oriented links. Math Comput. 1991;57(196):747–61. doi: 10.1090/S0025–5718–1991–1094946–4.

Freyd P, Yetter D, Hoste J, Lickorish WBR, Millett K, Ocneanu A. A new polynomial invariant of knots and links. Bull Am Math Soc. 1985;12(2):239–46. doi: 10.1090/S0273–0979–1985–15361–3.

Jones VFR. Hecke algebra representations of braid groups and link polynomials. Ann Math. 1987;126(2):335–88. doi: 10.2307/1971403.

Kanenobu T. Infinitely many knots with the same polynomial invariant. Proc Am Math Soc. 1986;97(1):158–62. doi: 10.1090/S0002–9939–1986–0831406–7.

Kanenobu T, Sumi T. Polynomial invariants of 2-bridge knots through 22 crossings. Math Comput. 1993;60:771–8 and S17–28.

Kauffman LH. Knots and physics. Singapore: World Scientific Publishing; 1991. p. 52.

Livingston C. Knot theory. Washington, DC: Math. Association Am; 1993. p. 213–7.

Morton HR, Short HB. Calculating the 2-variable polynomial for knots presented as closed braids. J Algor. 1990;11(1):117–31. doi: 10.1016/0196–6774(90)90033-B.

Przytycki JH, Traczyk P. Conway algebras and skein equivalence of links. Proc Am Math Soc. 1987;100(4):744–8. doi: 10.1090/S0002–9939–1987–0894448–2.

Stoimenow A. Jones polynomials. Available from: http://www.ms.u-tokyo.ac.jp/~stoimeno/ptab/j10.html.

Kashyap Sunil Kumar, Sharma Birendra Kumar, Banerjee Amitabh. A Cryptosystem based on DLP. Int J Network Sec. 2006;3(1):95–100.

Kashyap Sunil Kumar, Sharma Birendra Kumar, Banerjee Amitabh. On Mega Series of the two distinct types of discrete logarithms in free groups and public Key cryptosystem. Asian. J Inf Technol. 2007;6(10):1041–5.

Kashyap Sunil Kumar, Sharma Birendra Kumar, Banerjee Amitabh. Mathematical directions for curing the cancer disease: theory and practice. J Exp Ther Oncol. 2008;7(3):98–109.

Kashyap Sunil Kumar, Sharma Birendra Kumar, Banerjee Amitabh. The theory of relativity in algebraic structures and public Key cryptosystems. International e-journal of Engineering, management, Technology and Applications. 2008;4(1):90–94.

Kashyap Sunil Kumar, Sharma Birendra Kumar, Banerjee Amitabh. A public key cryptosystem based on discrete logarithms in metacyclic Groups. J Mod Math Stat. 2008;2(1):28–9.

Kashyap Sunil Kumar, Sharma Birendra Kumar, Banerjee Amitabh. On discrete logarithms in Ihara zeta function and public Key cryptosystem. International e-journal of Engineering, Management, Technology and Applications. 2008;5(3):176–197.

Kashyap Sunil Kumar, Sharma Birendra Kumar, Banerjee Amitabh. New cryptography on discrete logarithm problem under Definite Integral Calculus, The Institute of Chartered Financial Analysts of India University Press. J Comput Sci. 2009;3(2):54–63.

Kashyap Sunil Kumar, Sharma Birendra Kumar, Banerjee Amitabh. Automatic cryptosystem, The Institute of Chartered Financial Analysts of India University Press. J Comput Sci. 2009;3(2):98–109.

Kashyap Sunil Kumar, Sharma Birendra Kumar, Banerjee Amitabh. On Einstein’s Summation Convention based Discrete Logarithm Problem under Exterior Algebra and Cryptography. Math Comput Sci. 2010;1(2):189–92.

Kashyap Sunil Kumar, Sharma Birendra Kumar, Banerjee Amitabh. A new discrete logarithm problem in m-dimensional manifolds and cryptography. Math Comput Sci. 2010;1(2):201–3.

Kashyap Sunil Kumar, Sharma Birendra Kumar. Amitabh Banerjee, GF(p12) based PKC, International e-journal of Engineering, Management, Technology and Applications. 2010;8(1):61–86.

Kashyap Sunil Kumar, Sharma Birendra Kumar, Banerjee Amitabh. Discrete logarithm problem over the levi-Civita tensor in the Poincare algebra and cryptography. Int J Comput Sci Softw Eng. 2010;4(1):52–70.

Kashyap Sunil Kumar, Sharma Birendra Kumar, Banerjee Amitabh. New discrete logarithm problems over the electromagnetic Field tensor F under the Poincare converse lemma and cryptography. Int J Comput Sci Softw Eng. 2010;4(1):86–100.

Kashyap Sunil Kumar, Sharma Birendra Kumar, Banerjee Amitabh. Murthy’s DLP based PKC, The Institute of Chartered Financial Analysts of India University Press. J Comput Sci. 2011;5(4):7–17.

Kashyap Sunil Kumar, Sharma Birendra Kumar, Banerjee Amitabh, Shrivastava Subhas Chandra. On Krasnoselskii Fixed point theorem and fractal. Chaos, Solutions and Fractals. 2014;61(1):44–45.

Kashyap Sunil Kumar, Sharma Birendra Kumar, Banerjee Amitabh. On Discrete Logarithm Problem based on Algebraic Varieties over Finite Field and Public Key Cryptosystem. Int J Eng Dev Res. 2016;4(4):72–9.

Patle BK, Parhi DR, Jagdesh A, Kashyap Sunil Kumar. Probabilistic Fuzzy Controller based Robotics Path Decision Theory. World. J Eng. 2017;13(2):181–92.

Patle BK, Parhi DR, Jagdesh A, Kashyap Sunil Kumar. On Firfly Algorithm: optimization and application in mobile robot navigation. World J Eng. 2017;14(1):105–21.

Jain Swati, Jain Vikas Kumar, Kashyap Sunil Kumar, Kumar Sanjay. Academic performance evaluator over the Cluster L2-Metric. Res J Eng Technol. 2017;8(1): 45–60. doi: 10.5958/2321–581X.2017.00001.0.

Jain Swati, Jain Vikas Kumar, Kashyap Sunil Kumar, Kumar Sanjay. Academic Data Modeling based on Fuzzy Genetic Algorithm. Research Journal of Engineering and Technology. 2017;8(2):61–62.




DOI: https://doi.org/10.37591/joces.v11i2.849

Refbacks

  • There are currently no refbacks.


Copyright (c) 2021 Journal of Communication Engineering & Systems