Cryptography is a concept of protecting information and conversations which are transmitted through a public source, so that the send and receiver only read and process it. There are several encryption and decryption algorithm which involves mathematical concepts to provide more security to the text which has to be shared through a medium. In this paper, an algorithm is provided for both coding and decoding using cyclic symmetric matrices. Also Euler totient function, prime numbers are employed here. Furthermore, algorithm using prime number in integers is extended to prime numbers in Gaussian integers. This concept increases the security of the text.
For centuries, people have sent secret messages by various means. But some messages were not maintained secretly as there was no proper security. In order to maintain secrecy, cryptography was developed. It is the process of converting ordinary plain text (message to be sent) into some unintelligible text and vice versa. It helps to transmit data in a particular form so that the intended persons can read and process it. It is also useful for user authentication.
In olden days, an algorithm in cryptography was based on concepts which are well- known by all. But now- a- days, it is mainly based on mathematical theory and computer applications. Especially, number theory is playing a vital role in it, which employs the concepts such as congruence, Euler’s theorem.
In modern days, one can make use of any mathematical concepts to make their algorithm. As much as mathematics imposed, as much as security increases. Motivated by , this work aims to propose an algorithm to improve the security based on things such as prime numbers, Euler phi function, cyclic symmetric matrices. By modifying the assignments of alphabets in , this work is developed.
This paper involves two algorithms. First one uses integer prime assignment whereas second one uses Gaussian prime assignment. For the second case, Euler phi function on Gaussian integers φZ[i] is employed.
Common notations and definitions:
II. ALGORITHM BASED ON INTEGER PRIMES
B. Algorithm for Encryption
Assign the first 26 prime numbers to the alphabets. i.e.,
III. ALGORITHM BASED ON GAUSSIAN PRIMES
In this paper, there are two algorithms. One involving integer primes and the other uses Gaussian primes. One can maintain comparatively more secrecy in second one than that of first one. To improve much more security, one can modify the assignment by taking large primes as well as Gaussian primes.
Apostol, T. M., Introduction to Analytic Number Theory (1st ed. 1976. Corr. 5th printing 1998 ed.). Springer, 1976.
May, C. A., Application of the Euler Phi Function in the Set of Gaussian Integers, 2015.
Thiagarajan, K., Balasubramanian, P., Nagaraj, J., &Padmashree, J., Encryption and decryption algorithm using algebraic matrix approach. In Journal of Physics: Conference Series (Vol. 1000, No. 1, p. 012148). IOP Publishing, 2018.
Trappe, W., & Washington, L. C., Introduction to Cryptography. Prentice Hall, 2006.
Andre weil, Number Theory : An Approach through History, From Hammurapito to Legendre, Bikahsuser, Boston, 1987.
Bibhotibhusan Batta and Avadhesh Narayanan Singh, History of Hindu Mathematics, Asia Publishing House, 1983.
Boyer. C. B., History of mathematics, John Wiley & sons lnc., New York, 1968.
L.E. Dickson, History of Theory of Numbers, Vol.2, Chelsea Publishing Company, New York, 1952.
Davenport, Harold, The higher Arithmetic: An introduction to the Theory of Numbers (7th ed.) Cambridge University Press, 1999.
James Matteson, M.D. “A Collection of Diophantine problems with solutions” Washington, ArtemasMartin, 1888.
Tituandreescu, DorinAndrica, “An introduction to Diophantine equations” Springer Publishing House, 2002.
Manju Somanath, J. Kannan, K.Raja, “Gaussian integer solutions to space Pythagorean Equation x^2+y^2+z^2=w^2”, International Journal of Modern Trends in Engineering and Research, Volume 3, Issue 4, April 2016, pp . 287 - 289.
Manju Somanath, J. Kannan, K.Raja , “Gaussian Pythagorean Triples”, International Journal of Engineering Research and Management (IJERM), Volume 03, Issue 04, April 2016, pp . 131 - 132.
Manju Somanath, J. Kannan, “Congruum Problem”, International Journal of Pure and Applied Mathematical Sciences (IJPAMS), Volume 9, Number 2 (2016), pp. 123-131.
M. A. Gopalan, J. Kannan, Manju Somanath, K.Raja, “Integral Solutions of an Infinite Elliptic Cone? x?^2=4y^2+5z^2”, IJIRSET, Volume 5, Issue10, October 2016, pp .17551 - 17557.
Manju Somanath, J. Kannan, K.Raja, “Lattice Points of an Infinite Cone x^2+y^2=85z^2”, International Journal of Recent Innovation in Engineering and Research, Vol. 1 Issue. 5, September 2016, pp. 14 -17.
Manju Somanath, J. Kannan, K.Raja, “Lattice Points of an Infinite Cone x^2+y^2=(?^2n+?^2n ) z^2”, International Journal of Mathematical Trends and Technology, Vol. 38 No. 2, October 2016, pp(95 - 98).
Manju Somanath, K. Raja, J. Kannan and V. Sangeetha, “On the Gaussian Integer Solutions for an Elliptic Diophantine Equation”, Advances and Applications in Mathematical Sciences, Vol. 20, No. 05, March, 2021, pp. 815 – 822.
Manju Somanath, K. Raja, J. Kannan and K. Kaleeswari, “Solutions Of Negative Pell Equation Involving Chen Prime”, Advances and Applications in Mathematical Sciences, Vol. 19, No. 11, Sep. 2020, pp. 1089 – 1095
Manju Somanath, K. Raja, J. Kannan and M. Mahalakshmi, “On A Class of Solutions for A Quadratic Diophantine Equation”, Advances and Applications in Mathematical Sciences, Vol. 19, No. 11, Sep. 2020, pp. 1097 – 1103.
Manju Somanath, K. Raja, J. Kannan and B. Jeyashree, “Non Trivial Integral Solutions of Ternary Quadratic Diophantine Equation”, Advances and Applications in Mathematical Sciences, Vol. 19, No. 11, Sep. 2020, pp. 1105 – 1112.
Manju Somanath, K. Raja, J. Kannan and S. Nivetha, “Exponential Diophantine Equation in Three Unknowns”, Advances and Applications in Mathematical Sciences, Vol. 19, No. 11, Sep. 2020, pp. 1113 – 1118.
Manju Somanath, K. Raja, J. Kannan and A. Akila, “Integral Solutions of An Infinite Elliptic Cone x^2=9y^2+11z^2 ”, Advances and Applications in Mathematical Sciences, Vol. 19, No. 11, Sep. 2020, pp. 1119 – 1124.
J. Kannan, “Ternary Exponential Diophantine Equation”, Adalya Journal, Vol. 08, No. 09, September 2019, pp. 1283-1286.
J. Kannan, “On the Diophantine Equation ?47?^x+?48?^y=z^2”, Infokara Research, Vol.:08, No. 09, September 2019, pp. 778-781.
J. Kannan, “On Sequences of special Diophantine Triples generated through Linear Polynomials”, A Journal of Composition Theory, Vol.:12, No. 09, September 2019, pp. 703-706.
J. Kannan, Manju Somanath and K. Raja, “On a Class of Solutions for the Hyperbolic Diophantine Equation”, International Journal of Applied Mathematics, Vol.: 32, No. 03, July 2019, pp. 443- 449. [Scopus Indexed ].
J. Kannan, Manju Somanath and K. Raja, “Solutions of Negative Pell Equation Involving Twin Prime”, JP Journal of Algebra, Number Theory and Applications, Vol.: 40, No.: 5, October 2018, pp. 869-874.