A New Application to Coding Theory via Fibonacci and Lucas Numbers

Abstract

Coding/decoding algorithms are of great importance to help in improving information security since information security is a more significant problem in recent years. In this paper we introduce two new coding/decoding algorithms using Fibonacci Q-matrices and R-matrices. Our models are based on the blocked message matrices and the encryption of each message matrix with different keys. These new algorithms will not only increase the security of information but also has high correct ability.

Genelleştirilmiş Fibonacci ve Lucas polinomlarıyla birlikte sağ devirsel matrisler ve kodlama teorisi

“…By differently taking p and q , we can obtain different algorithms. Furthermore it can be mixed the above new blocking methods with the previous methods given in [5][6][7]. It is possible to produce new blocking methods similar to minesweeper algorithm given in [7].…”

“…Proof. For = 1 n , det 1 ( ) = 1 G satisfies the equation (7). Let us consider the case 2 n  and we focus on the following matrices:…”

“…There are many studies including Fibonacci, Fibonacci quaternion, Lucas, Pell, Pell (p,i)-numbers and their applications such as coding theory in the literature [1][2][3][4][5][6][7]. Here we consider two classes of right circulant matrices whose entries are generalized Fibonacci and Lucas polynomials to obtain new coding/decoding algorithms.…”

The Deep Learning Solutions on Lossless Compression Methods for Alleviating Data Load on IoT Nodes in Smart Cities

“…One of the main entropy coding types creates and assigns every single symbol of the entry into a unique prefix-free code. There are more than 16 algorithms support entropy algorithms such as Arithmetic Coding [ 71 , 72 , 73 , 74 ], Asymmetric Numeral Systems (ANS) [ 75 , 76 , 77 ], Golomb Coding [ 78 , 79 ], Adaptive Huffman [ 80 , 81 , 82 ], Canonical Huffman [ 83 ], Modified Huffman [ 84 ], Range encoding [ 85 , 86 ], Shannon [ 87 ], Shannon–Fano [ 88 , 89 , 90 ], Shannon–Fano–Elias [ 91 ], Tunstall coding [ 92 , 93 ], Unary coding [ 94 , 95 , 96 ], Universal Exp-Golomb [ 97 , 98 ], Universal Fibonacci Coding [ 99 , 100 , 101 ], Universal Gamma Coding [ 102 , 103 ], Universal Levenshtein Coding [ 104 ].…”

On the zeros of R-Bonacci polynomials and their derivatives

“…Fibonacci polynomials, a broad class of polynomials, were first described by Belgian mathematician Eugene Charles Catalan (1814-1894), German mathematician E. Jacobsthal and Lucas polynomials in 1970 by M. Bicknell. The starting point of this polynomial class is based on well-known Golden Ratio and Fibonacci numbers, which are still of great interest in the world of modern applied sciences and whose new applications are still found (see, for instance, [1]- [16] and [18]- [20]). For any positive real number x, the Fibonacci polynomials are defined by…”