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Papers by Süleyman Aydınyüz

Research paper thumbnail of Error detection and correction for coding theory on k-order Gaussian Fibonacci matrices

Mathematical Biosciences and Engineering

In this study, the coding theory defined for k-order Gaussian Fibonacci polynomials is rearranged... more In this study, the coding theory defined for k-order Gaussian Fibonacci polynomials is rearranged by taking \begin{document}$ x = 1 enddocument.Wecallthiscodingtheorythek−orderGaussianFibonaccicodingtheory.Thiscodingmethodisbasedonthebegindocument\end{document}. We call this coding theory the k-order Gaussian Fibonacci coding theory. This coding method is based on the \begin{document}enddocument.Wecallthiscodingtheorythek−orderGaussianFibonaccicodingtheory.Thiscodingmethodisbasedonthebegindocument {Q_k}, {R_k} enddocumentandbegindocument\end{document} and \begin{document}enddocumentandbegindocument E_n^{(k)} enddocumentmatrices.Inthisrespect,itdiffersfromtheclassicalencryptionmethod.Unlikeclassicalalgebraiccodingmethods,thismethodtheoreticallyallowsforthecorrectionofmatrixelementsthatcanbeinfiniteintegers.Errordetectioncriterionisexaminedforthecaseofbegindocument\end{document} matrices. In this respect, it differs from the classical encryption method. Unlike classical algebraic coding methods, this method theoretically allows for the correction of matrix elements that can be infinite integers. Error detection criterion is examined for the case of \begin{document}enddocumentmatrices.Inthisrespect,itdiffersfromtheclassicalencryptionmethod.Unlikeclassicalalgebraiccodingmethods,thismethodtheoreticallyallowsforthecorrectionofmatrixelementsthatcanbeinfiniteintegers.Errordetectioncriterionisexaminedforthecaseofbegindocument k = 2 enddocumentandthismethodisgeneralizedtobegindocument\end{document} and this method is generalized to \begin{document}enddocumentandthismethodisgeneralizedtobegindocument k enddocumentanderrorcorrectionmethodisgiven.Inthesimplestcase,forbegindocument\end{document} and error correction method is given. In the simplest case, for \begin{document}enddocumentanderrorcorrectionmethodisgiven.Inthesimplestcase,forbegindocument k = 2 $\end{document}, the correct capability of the method is essentially equal to 93.33%, exceeding all well-known correction codes. It appears that for a sufficiently large va...

Research paper thumbnail of k-Order Fibonacci Polynomials on AES-Like Cryptology

Computer Modeling in Engineering & Sciences, 2022

The Advanced Encryption Standard (AES) is the most widely used symmetric cipher today. AES has an... more The Advanced Encryption Standard (AES) is the most widely used symmetric cipher today. AES has an important place in cryptology. Finite field, also known as Galois Fields, are cornerstones for understanding any cryptography. This encryption method on AES is a method that uses polynomials on Galois fields. In this paper, we generalize the AES-like cryptology on 2 × 2 matrices. We redefine the elements of k-order Fibonacci polynomials sequences using a certain irreducible polynomial in our cryptology algorithm. So, this cryptology algorithm is called AES-like cryptology on the k-order Fibonacci polynomial matrix.

Research paper thumbnail of k-ORDER FIBONACCI QUATERNIONS

Journal of Science and Arts, 2021

In this paper, we define and study another interesting generalization of the Fibonacci quaternion... more In this paper, we define and study another interesting generalization of the Fibonacci quaternions is called k-order Fibonacci quaternions. Then we obtain for Fibonacci quaternions, for Tribonacci quaternions and for Tetranacci quaternions. We give generating function, the summation formula and some properties about k-order Fibonacci quaternions. Also, we identify and prove the matrix representation for k-order Fibonacci quaternions. The matrix given for k-order Fibonacci numbers is defined for k-order Fibonacci quaternions and after the matrices with the k-order Fibonacci quaternions is obtained with help of auxiliary matrices, important relationships and identities are established.

Research paper thumbnail of k-order Gaussian Fibonacci polynomials and applications to the coding/decoding theory

Journal of Discrete Mathematical Sciences and Cryptography, 2020

In this paper we define k-order Gaussian Fibonacci polynomials with boundary conditions and give ... more In this paper we define k-order Gaussian Fibonacci polynomials with boundary conditions and give the generating function, explicit formula and some identities for k-order Gaussian Fibonacci polynomials. We introduce the matrix represent and we obtain the k-order Gaussian Fibonacci Polynomials matrix. We define a new coding theory called k-order Gaussian Fibonacci Polynomials coding theory and establish the code elements for values of k. This coding/decoding method bound to the Qk (x), Rk (x) and Ek,n (x) matrices. So, this method is different from the classical algebraic coding. Consequently, with this method, we move the coding theory onto a complex space which is a different field. Therefore, new working areas are created. © 2020 Taru Publications

Research paper thumbnail of Generalized Order Fibonacci and Lucas Polynomials and Hybrinomials

In this study, we first define generalized order Fibonacci and Lucas polynomials. We show that by... more In this study, we first define generalized order Fibonacci and Lucas polynomials. We show that by special choices one can obtain some known sequences of polynomials and numbers such as order Pell polynomials, order Jacobsthal polynomials, order Fibonacci and Lucas numbers and etc. by using the definition of order Fibonacci and Lucas polynomials. Then we consider hybrid numbers and polynomials whose importance is increasing in mathematics, physics and engineering day by day. We generalize the hybrid polynomials by moving them to the order. Hybrid polynomials that are defined with this generalization are called order Fibonacci and Lucas hybrinomials throughout this paper. We define the generalized order Fibonacci and Lucas hybrinomials using generalized order Fibonacci and Lucas polynomials. Besides this, we give the recurrence relations of the generalized order Fibonacci and Lucas hybrinomials. Also, we show that by special choices in this recurrence relations one can obtain some kno...

Research paper thumbnail of Error detection and correction for coding theory on k-order Gaussian Fibonacci matrices

Mathematical Biosciences and Engineering

In this study, the coding theory defined for k-order Gaussian Fibonacci polynomials is rearranged... more In this study, the coding theory defined for k-order Gaussian Fibonacci polynomials is rearranged by taking \begin{document}$ x = 1 enddocument.Wecallthiscodingtheorythek−orderGaussianFibonaccicodingtheory.Thiscodingmethodisbasedonthebegindocument\end{document}. We call this coding theory the k-order Gaussian Fibonacci coding theory. This coding method is based on the \begin{document}enddocument.Wecallthiscodingtheorythek−orderGaussianFibonaccicodingtheory.Thiscodingmethodisbasedonthebegindocument {Q_k}, {R_k} enddocumentandbegindocument\end{document} and \begin{document}enddocumentandbegindocument E_n^{(k)} enddocumentmatrices.Inthisrespect,itdiffersfromtheclassicalencryptionmethod.Unlikeclassicalalgebraiccodingmethods,thismethodtheoreticallyallowsforthecorrectionofmatrixelementsthatcanbeinfiniteintegers.Errordetectioncriterionisexaminedforthecaseofbegindocument\end{document} matrices. In this respect, it differs from the classical encryption method. Unlike classical algebraic coding methods, this method theoretically allows for the correction of matrix elements that can be infinite integers. Error detection criterion is examined for the case of \begin{document}enddocumentmatrices.Inthisrespect,itdiffersfromtheclassicalencryptionmethod.Unlikeclassicalalgebraiccodingmethods,thismethodtheoreticallyallowsforthecorrectionofmatrixelementsthatcanbeinfiniteintegers.Errordetectioncriterionisexaminedforthecaseofbegindocument k = 2 enddocumentandthismethodisgeneralizedtobegindocument\end{document} and this method is generalized to \begin{document}enddocumentandthismethodisgeneralizedtobegindocument k enddocumentanderrorcorrectionmethodisgiven.Inthesimplestcase,forbegindocument\end{document} and error correction method is given. In the simplest case, for \begin{document}enddocumentanderrorcorrectionmethodisgiven.Inthesimplestcase,forbegindocument k = 2 $\end{document}, the correct capability of the method is essentially equal to 93.33%, exceeding all well-known correction codes. It appears that for a sufficiently large va...

Research paper thumbnail of k-Order Fibonacci Polynomials on AES-Like Cryptology

Computer Modeling in Engineering & Sciences, 2022

The Advanced Encryption Standard (AES) is the most widely used symmetric cipher today. AES has an... more The Advanced Encryption Standard (AES) is the most widely used symmetric cipher today. AES has an important place in cryptology. Finite field, also known as Galois Fields, are cornerstones for understanding any cryptography. This encryption method on AES is a method that uses polynomials on Galois fields. In this paper, we generalize the AES-like cryptology on 2 × 2 matrices. We redefine the elements of k-order Fibonacci polynomials sequences using a certain irreducible polynomial in our cryptology algorithm. So, this cryptology algorithm is called AES-like cryptology on the k-order Fibonacci polynomial matrix.

Research paper thumbnail of k-ORDER FIBONACCI QUATERNIONS

Journal of Science and Arts, 2021

In this paper, we define and study another interesting generalization of the Fibonacci quaternion... more In this paper, we define and study another interesting generalization of the Fibonacci quaternions is called k-order Fibonacci quaternions. Then we obtain for Fibonacci quaternions, for Tribonacci quaternions and for Tetranacci quaternions. We give generating function, the summation formula and some properties about k-order Fibonacci quaternions. Also, we identify and prove the matrix representation for k-order Fibonacci quaternions. The matrix given for k-order Fibonacci numbers is defined for k-order Fibonacci quaternions and after the matrices with the k-order Fibonacci quaternions is obtained with help of auxiliary matrices, important relationships and identities are established.

Research paper thumbnail of k-order Gaussian Fibonacci polynomials and applications to the coding/decoding theory

Journal of Discrete Mathematical Sciences and Cryptography, 2020

In this paper we define k-order Gaussian Fibonacci polynomials with boundary conditions and give ... more In this paper we define k-order Gaussian Fibonacci polynomials with boundary conditions and give the generating function, explicit formula and some identities for k-order Gaussian Fibonacci polynomials. We introduce the matrix represent and we obtain the k-order Gaussian Fibonacci Polynomials matrix. We define a new coding theory called k-order Gaussian Fibonacci Polynomials coding theory and establish the code elements for values of k. This coding/decoding method bound to the Qk (x), Rk (x) and Ek,n (x) matrices. So, this method is different from the classical algebraic coding. Consequently, with this method, we move the coding theory onto a complex space which is a different field. Therefore, new working areas are created. © 2020 Taru Publications

Research paper thumbnail of Generalized Order Fibonacci and Lucas Polynomials and Hybrinomials

In this study, we first define generalized order Fibonacci and Lucas polynomials. We show that by... more In this study, we first define generalized order Fibonacci and Lucas polynomials. We show that by special choices one can obtain some known sequences of polynomials and numbers such as order Pell polynomials, order Jacobsthal polynomials, order Fibonacci and Lucas numbers and etc. by using the definition of order Fibonacci and Lucas polynomials. Then we consider hybrid numbers and polynomials whose importance is increasing in mathematics, physics and engineering day by day. We generalize the hybrid polynomials by moving them to the order. Hybrid polynomials that are defined with this generalization are called order Fibonacci and Lucas hybrinomials throughout this paper. We define the generalized order Fibonacci and Lucas hybrinomials using generalized order Fibonacci and Lucas polynomials. Besides this, we give the recurrence relations of the generalized order Fibonacci and Lucas hybrinomials. Also, we show that by special choices in this recurrence relations one can obtain some kno...