Principal Primitive Ideals in Quadratic Orders and Pell’s Equations

In this paper, we introduce a method of determining whether the primitive ideal is principal in a real quadratic order, depending on the solvability of Pell’s equation.

On Polynomial Solutions of Pell’s Equation

Polynomial Pell’s equation is x 2 − D y 2 = ± 1 , where D is a quadratic polynomial with integer coefficients and the solutions X , Y must be quadratic polynomials with integer coefficients. Let D = a 2 x 2 + a 1 x + a 0 be a polynomial in Z x . In this paper, some quadratic polynomial solutions are given for the equation x 2 − D y 2 = ± 1 which are significant from computational point of view.

On certain Diophantine equations concerning the area of right triangles

Using the theory of elliptic curve, we show that all right triangles, such that the sum of the area and the square of the sum of legs is a square, are given by an infinite set. Similarly, we get all right triangles such that the sum of the area and the square of the semi-perimeter is a square. Using the theory of Pell’s equation, we prove that there are infinitely many non-primitive right triangles such that the sum of the area and the hypotenuse (or the smaller leg) is a square, and an infinity of primitive right triangles such that the sum of the area and the smaller leg (or the perimeter, the semi-perimeter, the larger leg) is a square.

The Solvability of Polynomial Pell’s Equation

This article attempts to describe the continued fraction expansion of ÖD viewed as a Laurent series x-1. As the behavior of the continued fraction expansion of ÖD is related to the solvability of the polynomial Pell’s equation p2-Dq2=1 where D=f2+2g is monic quadratic polynomial with deg g<deg f and the solutions p, q must be integer polynomials. It gives a non-trivial solution if and only if the continued fraction expansion of ÖD is periodic.

Image encryption scheme in public key cryptography based on cubic pells quadratic case

Cryptography systems face new threats with the transformation of time and technology. Each innovation tries to contest challenges posed by the previous system by analyzing approaches that are able to provide impressive outcomes. The prime aim of this work is to urge ways in which the concept of Pell’s equation can be used in Public key Cryptography techniques.The main aim of this approach is secure and can be computed very fast. Using Cubic Pell’s equation defined in Quadratic Case, a secure public key technique for Key generation process is showcased. The paper highlights that a key generation time of proposed scheme using Pell’s Quadratic case equation is fast compared to existing methods.The strength and quality of the proposed method is proved and analyzed by obtaining the results of entropy, differential analysis, correlation analysis and avalanche effect. The superiority of the proposed method over the conventional AES and DES is confirmed by a 50% increase in the execution speed and shows that Standard diviation and Entropy analysis of proposed scheme gives immunity to guess the encryption key and also it is hard to deduce the private key from public key using Diffrential analysis.

Key Generation Using Generalized Pell’s Equation in Public Key Cryptography Based on the Prime Fake Modulus Principle to Image Encryption and Its Security Analysis

RSA is one among the most popular public key cryptographic algorithm for security systems. It is explored in the results that RSA is prone to factorization problem, since it is sharing common modulus and public key exponent. In this paper the concept of fake modulus and generalized Pell’s equation is used for enhancing the security of RSA. Using generalized Pell’s equation it is explored that public key exponent depends on several parameters, hence obtaining private key parameter itself is a big challenge. Fake modulus concept eliminates the distribution of common modulus, by replacing it with a prime integer, which will reduce the problem of factorization. It also emphasizes the algebraic cryptanalysis methods by exploring Fermat’s factorization, Wiener’s attack, and Trial and division attacks.

Formalization of the MRDP Theorem in the Mizar System

Summary This article is the final step of our attempts to formalize the negative solution of Hilbert’s tenth problem. In our approach, we work with the Pell’s Equation defined in [2]. We analyzed this equation in the general case to show its solvability as well as the cardinality and shape of all possible solutions. Then we focus on a special case of the equation, which has the form x2 − (a2 − 1)y2 = 1 [8] and its solutions considered as two sequences $\left\{ {{x_i}(a)} \right\}_{i = 0}^\infty ,\left\{ {{y_i}(a)} \right\}_{i = 0}^\infty$ . We showed in [1] that the n-th element of these sequences can be obtained from lists of several basic Diophantine relations as linear equations, finite products, congruences and inequalities, or more precisely that the equation x = yi(a) is Diophantine. Following the post-Matiyasevich results we show that the equality determined by the value of the power function y = xz is Diophantine, and analogously property in cases of the binomial coe cient, factorial and several product [9]. In this article, we combine analyzed so far Diophantine relation using conjunctions, alternatives as well as substitution to prove the bounded quantifier theorem. Based on this theorem we prove MDPR-theorem that every recursively enumerable set is Diophantine, where recursively enumerable sets have been defined by the Martin Davis normal form. The formalization by means of Mizar system [5], [7], [4] follows [10], Z. Adamowicz, P. Zbierski [3] as well as M. Davis [6].