AN APPLICATION OF LINEAR DIOPHANTINE EQUATIONS TO CRYPTOGRAPHY

In this chapter we propose a Key exchange protocol based on a random solution of linear Diophantine equation in n variables, where the considered linear Diophantine equation satisfies the condition for existence of infinitely many solutions. Also the crypt analysis of the protocol is analysed.

On the non-linear diophantine equation \({\boldsymbol{379}}^{\boldsymbol{x}}\boldsymbol{+}{\boldsymbol{397}}^{\boldsymbol{y}}\boldsymbol{=}{\boldsymbol{z}}^{\boldsymbol{2}}\)

In this article, authors discussed the existence of solution of non-linear diophantine equation \({379}^x+{397}^y=z^2,\) where \(x,y,z\) are non-negative integers. Results show that the considered non-linear diophantine equation has no non-negative integer solution.

Aplikasi Ring Kuadratik dalam Menyelesaikan Persamaan Pell

Artikel ini membahas tentang penerapan Metode ring (kuadratik) dalam mencari solusi pada persamaan Pell. Persamaan Pell merupakan bagian dari persamaan Diophantine non linear yang penyelesaiannya berupa bilangan bulat dengan bentuk umum persamaannya yaitu . Dalam penelitian ini persamaan Pell yang akan ditentukan solusinya yaitu . Metode yang digunakan dalam penelitian ini yaitu metode ring kuadratik. Metode ring kuadratik yang digunakan dalam menyelesaikan persamaan Pell memperhatikan konsep norm dan unit pada bilangan . Berdasarkan hasil penelitian, persamaan Pell positif memiliki paling tidak satu solusi dengan nilai yang dipilih. Sedangkan persamaan Pell negatif tidak selalu memiliki solusi, hanya pada nilai tertentu.Kata Kunci: Persamaan Pell, Ring Kuadratik, Norm This article discusses the application of the ring (quadratic) method in finding solutions of the Pell equation. The Pell equation is part of the non linear Diophantine equation whose the solution is integer with the general form of the equation is . In this research, the Pell equation which the solution will be determined is . The method used in this research is the quadratic ring method. The quadratic ring method that will be used in solving the Pell equation takes the concepts of norm and unit in number. Based on this research, positive Pell equations is has at least one solution with the value of that chosen. While the negative Pell equation is doesn’t always have a solution, just at certain values of .Keywords: Pell Equation, Quadratic Ring, Norm.

On Infinite Number of Solutions for one type of Non-Linear Diophantine Equations

In this article, we prove that the non-linear Diophantine equation 𝑦 = 2𝑥1𝑥2 …𝑥𝑘 + 1; 𝑘 ≥ 2, 𝑥𝑖 ∈ 𝑃 − {2}, 𝑥𝑖′𝑠 are distinct and P is the set of all prime numbers has an infinite number of solutions using the notion of a periodic sequence. Then we also obtained certain results concerning Euler Mullin sequence.

Solutions to Non-Linear Diophantine Equation (5κ-1)χ+(5κ)y=z2 with k is Positive Even Integer

This research seeks for a solution (if any) to non-linear Diophantine equation 2 (5 1) (5 ) k x k y    z . There are 3 possibilities of solution to the non-linear Diophantine equation, which are single solution, many solutions, or no solution. The research methodology is conducted in two stages has t, which are using simulation to seek for a solution (if any) to non-linear Diophantine equation 2 (5 1) (5 ) k x k y    z and using Catalan’s conjecture and characteristics of congruence theory. It is proven that the non-linear Diophantine equation has single solution 2 ( , , ) (1,0,5 ) k x y z  for non-negative integer x, y,z and positive even of equal to or higher than