On Encryption with Continued Fraction

Year 2022, Volume: 13 Issue: 2, 149 - 152, 28.06.2022

Merve Güney Duman

https://doi.org/10.24012/dumf.1038230

Abstract

Many mathematicians have investigated the properties of continued fractions. They made continued fraction expansions of the Pi number, the golden ratio and many more special numbers. With the help of continued fractions, solutions of some Diophantine equations are obtained. In this study, encryption was made using continued fractional expansions of the square root of non-perfect-square integers. Each of the 29 letters in the alphabet is represented by the root of nonperfect square integers starting from 2. Then, continued fraction expansions of the square root of each letter’s number equivalent were calculated. Afterwards, all numbers in the continued fraction expansion were considered as an integer by removing the comma. This information was tabulated for later usage. Each word is considered as individual letters, and a space is left between the encrypted versions of each letter. After the encryption process, the process of deciphering the encrypted text was dealt with. In the deciphering process, since there is a blank between the numbers, the numbers are written as a continued fraction and the integer expansion is calculated. Later, the letter corresponding to this number was found.

Keywords

Continued fractions, Cryptology, Information security, Cryptography

References

• [1] D. C. Collins, “Continued Fractions,” The MIT Undergraduate J. of Mathematics, vol. 1, pp. 11-20, 1999.
• [2] M. Kline, Mathematical Thought from Ancient to Modern Times, New York, USA: Oxford University Press, 1972. [3] Koshy, T., “Fibonacci and Lucas Numbers with Application”, New York, USA: Wiley, 2001.
• [4] Brezinski, C., “History of Continued Fractions and Pade Approximants”, Berlin, Germany: Springer-Verlag, 1990.
• [5] Ozyılmaz, C., Nallı, A., “Restructuring of Discrete Logarithm Problem and Elgamal Cryptosystem by Using the Power Fibonacci Sequence Module M”, Journal of Science and Arts, ss. 61-70, 2019.
• [6] Koblitz, N., “Elliptic Curve Cryptosystems”, Mathematics of Computation, 48, 203-209, 1987.
• [7] Basu, M., Prasad, B., “The Generalized Relations Among the Code Elements for Fibonacci Coding Theory”, Chaos Solitons Fractals, 41, no.5, 2517-2525, 2019.
• [8] Prajapat, S., Jain, A., Thakur, R. S., “A Novel Approach For Information Security With Automatic Variable Key Using Fibonacci Q-Matrix”, IJCCT 3, no. 3, 54–57, 2012.
• [9] Prasad, B., “Coding Theory on Lucas p Numbers”, Discrete Mathematics, Algorithms and Applications, 8, no.4, 2016.
• [10] Stakhov, A., Massingue, V., Sluchenkov, A., “Introduction into Fibonacci Coding and Cryptography”, Osnova, Kharkov, 1999.
• [11] Stakhov, P., “Fibonacci matrices, a Generalization of the Cassini Formula and a New Coding Theory”, Chaos Solitons Fractals, 30, no. 1, 56–66, 2006.
• [12] Kodaz, H., Botsalı, F. M., “Simetrik ve Asimetrik Şifreleme Algoritmalarının Karşılaştırılması”, Selçuk Üniversitesi Teknik Bilimler Meslek Yüksekokulu Teknik-Online Dergi, 9, 10-23, 2010.
• [13] Kraft J. S., Washington L. C., “An Introduction to Number Theory with Cryptography”, Boca Raton, New York, London, CRC Press Taylor & Francis Group, 2014.
• [14] Stinson, D. R., “Cryptography Theory and Practise. Third edition”, London, England: Chapman & Hall/CRC Press Taylor & Francis Group, 2006.
• [15] Kahn, D., “The Codebreakers”, New York, USA: The Macmilian Company, 1996.
• [16] Stinson, D. R., “Cryptography Theory and Practice”, New York, USA: Chapman & Hall / CRC, 2002.
• [17] National Bureau of Standard., Data Encryption Standard, Federal ˙Information Processing Standards, NBS., 1977.
• [18] Mollin, R. A., “An Introduction to Cryptography”, Boca Raton, New York, London, Chapman and Hall/CRC, 2006.
• [19] Redmond, D., “Number Theory: An Introduction”, New York, USA: Markel Dekker, Inc, 1996.
• [20] Adler, A., Cloury, J.E., “The Theory of Numbers, A Text and Source Book of Problems”, Boston, London, Singapore, Jones and Bartlett Publishers, 1995.
• [21] Mollin, R. A., “Fundamental Number Theory with Applications”, Boca Raton, New York, London, Tokyo, CRC Press, 1998.
• [22] Kalman, D., Mena, R., “The Fibonacci Numbers Exposed”, Mathematics Magazine, 76, 2003.