On the symmetrical hyperbolic functions and a cryptographic application

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Thesis Type: Postgraduate

Institution Of The Thesis: Gazi University, Fen Bilimleri Enstitüsü, Turkey

Approval Date: 2013

Thesis Language: Turkish

Student: Özge Bostancı

Supervisor: DURSUN TAŞCI

Abstract:

A. Stakhov, B. Rozin defined symmetrical hyperbolic functions which are the being extensions of Binet formulas for the Fibonacci and Lucas numbers in continuous domain. By concidering the characteristical polynomial of the Fibonacci sequences, the sequences n U and n V for the characteristical polynomial x2 = ax +1, a∈Z+ can be obtained with following reccurence relations n 2 n 1 n U aU U + + = + , n ≥ 0 ; 0 U = 0 , 1 U =1 and n 2 n 1 n V aV V + + = + , n ≥ 0 ; 0 V = 2 , 1 V = a . In this thesis, the proporties of n U and n V , the relations between n U and n V , also the relations between hyperbolic functions and them were given. Furthermore, the symmetrical hyperbolic functions U and V were defined, which are continuous cases of n U and n V . By the help of these functions, Generalized Golden Matrices were defined. Finally, a cryptographic application were given, in which the Generalized Golden Matrices are used.