International Journal of Biology, Physics & Mathematics

Publication Date: February, 2021

**Mulugeta Andualem & Atinafu Asfaw**

Department of Mathematics, Bonga University

Bonga, Ethiopia

Journal Full Text PDF: Application of Shehu Transform to Handling Bessel Function & Cryptography.

Abstract

Cryptography is the study of art and science of preparing protected and secure data communication. The word cryptography is derived from the two Greek words; “kryptos” means “secret or hidden” and “graphos” means “to write. In this study, we will discuss the Shehu transform method to solve Bessel’s function of order of first kind and encryption and decryption method.

Keywords: Bessel function, Encryption, Cryptography, Shehu transform.

INTRODUCTION

Many problems in engineering and science can be formulated in terms of differential equations. The ordinary differential equations arise in many areas of Mathematics, as well as in Sciences and Engineering. In order to solve the certain ordinary differential equations integral transforms are widely used. In this paper, we will be discussed about the solution of Bessel’s function of order of first kind and encryption and decryption method using Shehu transform.

SHEHU TRANSFORM

Definition: A new transform called the Shehu transform of the function belonging to a class , where:

Where and is given by:

And the inverse Shehu transform is defined as

Property of the Shehu Transform

1. Property 1. Linearity property of Shehu transform. Let the functions and be in set , then , where and are nonzero arbitrary constants, and

Proof: Using the Definition (1.1) of Shehu transform, we get

Property 2. Let the function be in set , where is an arbitrary constant. Then

Using the Definition 1.1 of Shehu transform, we deduce

Substituting and in equation 1.4 yield

Derivative of Shehu transform. If the function is the nth derivative of the function with respect to , then its Shehu transform is defined by

When we obtain the following derivatives with respect to .

When we obtain the following derivatives with respect to .

Assume that equation 1.5 true for . Now we want to show that for

which implies that Eq (1.5) holds for

By induction hypothesis the proof is complete

Property 3: Let the function be in set . Then its Shehu transform is given by

Poof: Using equation 1.1

Property 4: Let the function be in set A. Then its Shehu transform is given by

Property 5: Let the function be in set A. Then its Shehu transform is given by

Property 6: Let the function be in set A. Then its Shehu transform is given by

Property 7: Let the function be in set . Then its Shehu transform is given by

Shehu transform for handling Bessel functions

Bessel function is defined for a first time by the mathematician Daniel Bernoulli and generalized by Friedrich Bessel. A differential equation of the form

Where is arbitrary real or complex number is called a Bessel equation and its solution is known as Bessel function. Bessel’s function of order of first kind is defined as

For

Relationship between and

Therefore and

Therefore,

Case 1:

Now take the Shehu transform both sides of the above result

Therefore

Case 2:

Since, . Now by applying the property of Shehu transform, we have

Since, implies

Case 3:

Since, and implies

Now by applying the property of Shehu transform of both sides of , we have

Shehu transform for handling Cryptography

In this section, we will disuse Shehu transform for encrypting the plain text and corresponding inverse Shehu transform is used for decryption.

Encryption Algorithm

A) Treat every letter in the plain text message as a number, so that , [space] = 0.

B) The plain text message is organized as finite sequence of numbers based on the above conversion. For example, our text is “BONGA”. Based on the above step; we know that, B = 2, , N = 14, G = 7, A = 1 Therefore our plaintext finite sequence is 2, 15, 14, 7, 1

C) If is the number of terms in the sequence; consider a polynomial of degree with coefficient as the term of the given finite sequence. Above finite sequence contains terms. Hence consider a polynomial of degree 4.

Take the Shehu transform of the polynomial

Next find such that ≡ mod 26 for each , . Therefore

≡ 2 mod 26, ≡ 15 mod 26, ≡ 2 mod 26, ≡ 16mod 26, ≡ 24 mod 26

D) Hence, Thus we get a key for

Therefore,

E) Now consider a new finite sequence

That is, 2, 15, 2, 16, 24

Then the cipher text is “BOBPX”

Decryption Algorithm

1. Consider the cipher text and key received from sender. In the above example cipher text is “BOBPX” and key is 0, 0, 1, 1, 0

2. Convert the given cipher text to corresponding finite sequence of numbers

2, 15, 2, 16, 24

3. Let

4.

5. Now take the Inverse Shehu transform of we obtain:

Consider the coefficient of a polynomial as a finite sequence

Now translating the number of above finite sequence to alphabets. We get the original plain text as “BONGA”

Conclusion

In this paper, we have successfully discussed the Shehu transform of Bessel’s functions and we have used Shehu transform for encrypting the plain text and corresponding inverse Shehu transform for decryption.

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