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 Saeed^{4}
^{1}Mathematics Department Faculty of Sciences and Arts, Almikwah-Albaha University, Albaha, Saudi Arabia
^{2}Mathematics Department Faculty of Education, Holy Quran and Islamic Sciences University, Khartoum, Sudan
^{3}Mathematics Department Faculty of Sciences, Omdurman Islamic University, Khartoum, Sudan
^{4}Computer 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].
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
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
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.
References