Some Notes on Odd or Even Indexed Fibonacci and Lucas Sequences

The uniqueness of the sum of the elements of finite subsets of the odd or even indexed Fibonacci and Lucas sequences are proved. Moreover, it is shown that the odd or even indexed Fibonacci and Lucas sequences are super-increasing sequences. By using the uniqueness properties established, a new cryptology method is presented and exemplified.

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Merkle–Hellman knapsack cryptosystem, https://en.wikipedia.org/wiki/Merkle%E2 %80%93Hellman_knapsack_cryptosyste m.

Koshy, T., 2001, Fibonacci and Lucas numbers with applications, John Wiley & Sons, New York-Toronto.

Vajda, S., 1989, Fibonacci and Lucas Numbers and the Golden Section: Theory and Applications, Courier Corporation.

Brannan, D.A., 2006, A First Course in Mathematical Analysis, Cambridge University Press.

Mollin, R.A., 2007, An Introduction to Cryptography, Chapman&Hall/CRC, Boca Raton.

Uçar, S., Taş, N., Özgür, N.Y., A new cryptography model via Fibonacci and Lucas numbers, arXiv: 1709.10355 [cs.CR].

N Taş, S Uçar, N.Y. Özgür, Ö.Ö. Kaymak, 2018, A new coding/decoding algorithm using Fibonacci numbers, Discrete Mathematics, Algorithms and Applications 10 (02).

M. Basu, B. Prasad, The generalized relations among the code elements for Fibonacci coding theory, Chaos Solitons Fractals 41 (2009), no. 5, 2517–2525.

S. Prajapat, A. Jain, R. S. Thakur, A Novel Approach For Information Security With Automatic Variable Key Using Fibonacci Q-Matrix, IJCCT 3 (2012), no. 3, 54–57.

A. Stakhov, V. Massingue, A. Sluchenkov, Introduction into Fibonacci Coding and Cryptography, Osnova, Kharkov (1999).

A. P. Stakhov, Fibonacci matrices, a generalization of the Cassini formula and a new coding theory, Chaos Solitons Fractals 30 (2006), no. 1, 56–66.