One of the most popular and appealing theories in mathematics is number theory, which is a basket of characteristics of numbers, particularly integers. It is more advantageous subject to master because it encompasses a variety of elements. Diophantine equations are polynomial equations with integer coefficients and two or more unknowns such that the only solutions of interest are the integer ones. In number theory, geometric shapes were important, and most researchers used these Diophantine equations to develop new kinds of geometric shapes [5, 4, 6]. An isosceles trapezoid is a special type of trapezoid where the two non-parallel sides (legs) are equal in length, and the base angles are also equal. In other words, it’s a trapezoid with a line of symmetry bisecting the two equal sides. Let the two non-parallel sides as c and two parallel sides as a, b for the isosceles trapezoid, then the area can be calculated by A = (a − b + 2c)(b − a + 2c). Here we find the isosceles trapezoid with the parallel sides as n, n + r and non parallel sides as 2n − 1 where n, r ∈ N having integer area.

Diophantine analysis, the mathematical study of Diophantine problems, comprises of a special type of equation, commonly known as Pell equation [1]. It is one among the various Diophantine equations and it takes the form x − dy = 1 (0.1) where d is a fixed positive integer, not a perfect square. It also has been proved that equation (0.1) has infinitely many solutions in positive integers. This Pell equation is generalized as x − dy = n (0.2) for some integer n.

Let p be a prime and (xk, yk) be positive integer solutions of the equation x − py = 1. Using these solutions p∗ p∗ x1 py1 th we can define the matrix Q as Q = [3]. The specialty of this matrix is its k power can be y1 x1 th 2 2 p∗ k xk pyk obtained directly from the k solution (xk, yk) of equation x − py = 1. That is, (Q ) = . yk xk

p∗ x1 py1 2 2 2 In this paper, we construct a similar matrix Qa = for the equation x1 − py1 = a , where a ∈ Z. The y1 x1 th p∗ r r−1 xr pyr 2 2 2 r power of this matrix is (Qa ) = a where (xr, yr) is a positive integer solution of x −py = a . yr xr

In [3], an encryption and decryption algorithms are developed using the Pell equation x − py = 1 and its corresponding matrix Q . Likewise, in this work, we deal with the same algorithm with little modifications but

2 2 2 p∗

we employ the generalized Pell equation x − py = a and its corresponding matrix Qa . In particular, here we restrict a as a Sphenic number, a number which is a product of three distinct primes, but here a is chosen as product of three consecutive primes which has been chosen with the help of p in the equation x − py = a .

Here we collect all isosceles Trapezoid with integer area. Let ABCD be a isosceles Trapezoid with the sides AB = n, BC = AD = 2n − 1, DC = n + r, where n, r ∈ N(as shown in figure 1). The area of Isosceles Trapezoid

is calculated as A = (n + n + r) (n + r − n + 2(2n − 1))(n − n − r + 2(2n − 1))

(2n+r) 2 2 2 2 2

and after solving we get A = (4n − 2) − r . For area A ∈ Z, (4n − 2) − r = l for some integer l. This is of the form pythagorean equation [10] and it can be written as (4n − 2) = (u + v )k, r = 2uvk, l = (u − v )k, for some integers u, v, k. From this n can be written as (u + v )k 1 n = + (0.3)

. We consider four cases for n.

The notations used in this article are as follows.

(1) P - Set of all primes

(2) B - 2n × 2n matrix which is constructed using the given message.

th

(3) Bk - k block of B with the size 2 (ie, 2 × 2 matrix).

(4) b - number of blocks of the matrix B.

(5) s = min{q ∈ P : q|b}

2 if b is 1 or 2n (n ∈ N) (6) p = s if b is 2n + 1(n ∈ N)

b if b ≤ p (7) r = p if b > p

(8) dk - determinant of the matrix Bk.

bk1 bk2 (9) - elements of Bk bk3 bk4

(10) E=[dk, bki]i=1,2,4 - encrypted matrix.

q q ∗

1 2 p r

(11) - elements of (Qa ) q3 q4

(12) δ -notation for space.

(13) (x, y) - Positive integer solutions of x − py = a

(14) a - Sphenic number

Here, the integer area for the isosceles trapezoid that was gathered from the previous section is used to allocate the alphabets’ positions.

In this section, we present the encryption and decryption algorithms.

p r 2 2 2

In this paper, (Qa ) constructed using the solutions of x − py = a using the choices of a and p. We encrypt and decrypt the message using the generalized Pell equation of the form x − dy = n where d and n are prime (p) and square of a sphenic number (a ) respectively. This content is the generalization of [3]. We may also extend this with any other Diophantine equation or we may use any other choices of a. We employ a few additional geometric forms for enhanced safety.