An Application of the New Integral "Aboodh Transform" in Cryptography

Abdelilah K. Hassan Sedeeg^{1, 2}, Mohand M. Abdelrahim Mahgoub^{1, 3}, Muneer A. Saif Saeed⁴

¹Mathematics Department Faculty of Sciences and Arts, Almikwah-Albaha University, Albaha, Saudi Arabia

²Mathematics Department Faculty of Education, Holy Quran and Islamic Sciences University, Khartoum, Sudan

³Mathematics Department Faculty of Sciences, Omdurman Islamic University, Khartoum, Sudan

⁴Computer Department Faculty of Sciences and Arts, Almikwah-Albaha University, Albaha, Saudi Arabia

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To cite this article:

Abdelilah K. Hassan Sedeeg, Mohand M. Abdelrahim Mahgoub, Muneer A. Saif Saeed. An Application of the New Integral "Aboodh Transform" in Cryptography. Pure and Applied Mathematics Journal. Vol. 5, No. 5, 2016, pp. 151-154. doi: 10.11648/j.pamj.20160505.12

Received: August 7, 2016; Accepted: August 23, 2016; Published: September 9, 2016

Abstract: Cryptography is the science of providing security for information, It has been used historically as a means of providing secure communication between individuals. Message encryption has become very essential to avoid the threat against possible attacks by hackers during transmission process of the message. In this paper authors have proposed a method of cryptography, in which authors have used Aboodh transform for encrypting the plain text and corresponding inverse Aboodh transform for decryption.

Keywords: Cryptography, Encryption, Decryption, Aboodh Transform

1. Introduction

Cryptography, the mathematics of encryption, plays an indispensable part in numerous fields, and a vast range of daily activities, such as electronic commerce, bank card payments and electronic building and so on. Cryptography is the only most important tool that avoids the threat against possible attacks by hackers during transmission process of the message, It is one of the cornerstones of Internet security. Cryptography is the only most important tool that avoids the threat against possible attacks by hackers during transmission process of the message.

Cryptography [5-10] referred almost exclusively to encryption, which is the process of converting ordinary information (called plaintext) into unintelligible text (called cipher text). Decryption is the reverse, in other words, moving from the unintelligible cipher text back to plaintext. A cipher (or cipher) is a pair of algorithms that create the encryption and the reversing decryption. The detailed operation of a cipher is controlled both by the algorithm and in each instance by a "key". The key is a secret (ideally known only to the communicants), usually a short string of characters, which is needed to decrypt the cipher text. (Fig. 1) [11-15].

Fig. 1. Basic encryption and Decryption.

2. Proposed Technique

In the present paper a new cryptographic scheme is proposed using Aboodh Transform [1-4]. Aboodh transform is used for encrypting the plain text and corresponding inverse Aboodh transform is used for decryption. Aboodh transform was introduced by Khalid Aboodh in 2013. Aboodh transform is a widely used integral transform in mathematics and electrical engineering that transforms a function of time into a function of complex frequency. The inverse Aboodh transform takes a complex frequency domain function and yields a function defined in the time domain. Proposed algorithm provides as many transformations as per the requirements which are the most useful factor for changing key. Therefore it is very difficult for an eyedropper to trace the key by any attack. The implementation has been done in visual basic (VB).

3. Aboodh Transform

Definition: Consider functions in the set  defined by

For a given function in the set  must be finite number, may be finite or infinite. Aboodh transform denotedby the operator  is defined by the integral equation

Properties of Aboodh transform:-

Linearity:-Aboodh transform is a linear transformation which means that the transform of a sum of waveforms is the sum of their transforms. Stated formally the linearity property is

Where  are constants.

The above result can easily be generalized to more than two functions.

Aboodh transformation &Inverse Aboodh Transform of some elementary functions:-

Elementary functions include algebraic and transcendental functions.

1.    , where  is constant.

2.     

3.    4.

4. Proposed Methodology

The following algorithm provides an insight into the proposed cryptographic scheme. The sender converts the original message or plain text into cipher text using the following steps.

4.1. Encryption Algorithm

I)       Treat every letter in the plain text message as a number, so that

II)    The plain text message is organized as finite sequence of numbers based on the above conversion. For example our text is "TEACHERS". Based on the above step; we know that,

Therefore our plain text finite sequence is

III)  If  is the number of term 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 .

Take Aboodh transform of polynomial .

Next find such that  for each . Therefore

IV)  Hence . Thus we get a key  for

V)     Now consider a new finite sequence

i.e. .

4.2. Decryption Algorithm

I) Consider the cipher text and key received from sender. In the above example cipher text is "TEBRJBLB" and key is .

II) Convert the given cipher text to corresponding finite sequence of numbers , .

III) Let ,

IV)    Let

V)      Now take the Inverse Aboodh transform of p(v).

VI)    Consider the coefficient of a polynomial  as a finite sequence.

.

VII) Now translating the number of above finite sequence to alphabets. We get the original plain text as "TEACHERS".

5. Implementation of the Efficient Algorithm

Visual basic programming language is one of the most widely use high level language today because of its advantages [16]. In this parta program has been written in visual basic language (VB), forthe implementation of the Encryption Algorithms and Implementation Decryption Algorithms in section 4.

5.1. Implementation Encryption Algorithms

Private Sub Enc_btn_Click()

Dim no_of_ltr As Integer

Dim ltrs, e(26) As String

no_of_ltr = Len(Me.text1)

Dim p(26), ris(1000), g, qis(26) As Long

Me.enc = "" Me.dcr = ""Me.mdd = ""Me.py = ""Me.dcr = ""Me.keys = ""Me.enc_txt = ""e(1) = "A"e(2) = "B"e(3) = "C"e(4) = "D"e(5) = "E"e(6) = "F"e(7) = "G"e(8) = "H"e(9) = "I"e(10) = "J"e(11) = "K"e(12) = "L"e(13) = "M"e(14) = "N"e(15) = "O"e(16) = "P"e(17) = "Q"e(18) = "R"e(19) = "S"e(20) = "T"e(21) = "U"e(22) = "V"e(23) = "W"e(24) = "X"e(25) = "Y"e(26) = "Z"

For i = 1 Tono_of_ltr

ltrs = Mid(Me.text1, (i), 1)

'MsgBox (ltrs)

For j = 1 To 26

If ltrs = e(j) Then

'MsgBox (i)

'MsgBox (j)

'MsgBoxe(j)

qis(i) = j * Factorial(i - 1)

ris(i) = qis(i) Mod 26

Me.enc = Me.enc&qis(i) & "-"

Me.mdd = Me.mdd&ris(i) & "-"

ks = ((qis(i) - ris(i)) / 26)

Me.keys = Me.keys&ks& "-"

Me.enc_txt = Me.enc_txt&e(ris(i))

End If

Next

'MsgBox (qis(i))

'LArray(i) = ltrs

'Me.enc = Me.enc& g & "-"

'Me.mdd = Me.mdd&ris(i) & "-"

'Me.dcr = Me.dcr&Chr(g)

Next

End Sub

Fig. 2. Implementation Encryption.

5.2. Implementation Decryption Algorithms

Private Sub Dec_Btn_Click()

Dim pos, intCount As Integer

Dim LArray(1000), ky(26), e(26), strTest, d, strArray(), ee(1000) As String

Dim p(26), qis(1000), no(1000), ris(26), key(26) As Long

'LArray = Split(Me.text1)

pos = Len(Me.text1)

'Me.key = ""Me.dcr = ""Me.mdd = ""Me.py = ""Me.dcr = ""Me.encrpt = ""e(1) = "A"e(2) = "B"e(3) = "C"e(4) = "D"e(5) = "E"e(6) = "F"e(7) = "G"e(8) = "H"e(9) = "I"e(10) = "J"e(11) = "K"e(12) = "L"e(13) = "M"e(14) = "N"e(15) = "O"e(16) = "P"e(17) = "Q"e(18) = "R"e(19) = "S"e(20) = "T"e(21) = "U"e(22) = "V"e(23) = "W"e(24) = "X"e(25) = "Y"e(26) = "Z"

strTest = Me.keys

strArray = Split(strTest, "-")

For intCount = LBound(strArray) To UBound(strArray) - 1

key(intCount) = (Trim(strArray(intCount)))

' MsgBox (intCount)

Next

For i = 1 Topos

d = Mid(Me.text1, (i), 1)

For j = 1 To 26

If d = e(j) Then

ris(i) = j ' * Factorial(i - 1)

'qis(i) = ris(i) Mod 26

'Me.enc = Me.enc&ris(i) & "-"

'Me.mdd = Me.mdd&qis(i) & "-"

Me.py = Me.py &ris(i) & "-"

qis(i) = ris(i) + (key(i - 1) * 26)

Me.mdd = Me.mdd&qis(i) & "-"

no(i) = (qis(i) / Factorial(i - 1))

org_txt = org_txt&e(no(i))

Me.encrpt = Me.encrpt&no(i) & "-"

'MsgBox ((ris(i))) ' - qis(i)) / 26)

End If

Next

'MsgBox (ris(i))

'LArray(i) = d

'g = Asc(d)

'p(i) = g

'qis(i) = p(i) Mod 26

'Me.py = Me.py &p(i) & "-"

'Me.enc = Me.enc& g & "-"

'Me.mdd = Me.mdd&qis(i) & "-"

'ee(i) = Chr(p(i))

'Me.encrpt = Me.encrpt&ee(i) & "-"

'Me.dcr = Me.dcr&Chr(g)

Next

End Sub

Public Function Factorial(ByVal X As Long) As Long

If X <= 1 Then

Factorial = 1

Else

Factorial = X * Factorial(X - 1)

End If

End Function

Fig. 3. Implementation Decryption.

6. Conclusion

Computers are good at doing mathematics. When we create a system to translate some piece of information into numbers (such as we do with text and ordinals or with space and coordinate systems), computer programs can process these numbers quickly and efficiently. Cryptography is one of the first lines of defense against hackers and crackers in today's world. Thus, it will stay important for a long time to come. In the proposed work a new cryptographic scheme is introduced using Aboodh Transform and the private key is the number of multiples of mod n. Visual Basic is more convenient to use in the Cryptography and is less prone to errors.

Acknowledgement

Author is thankful to Mr. Abdullah Musa, Mr. Hassan Mohammed, Mr. Khider Yousif and Ms. Nan Mohammed, for support to this work.

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