A proposed method for encrypting and sending confidential data using polynomials

With the improvements of cyberspace and communications, an essential problem was raised and that is how to secure the transmitted data and keep it confidential. Many techniques have been used for this purpose, some of them were broken but others were stayed immune against different attacks. Complexity of the used technique is one of the major reasons that kept it secure. To increase complexity, mathematics was used. In this paper, a method for encrypting and sending confidential data was proposed. The method depends on mathematical equations for encrypting data, sending and decrypting it. The method is complex, secure and workable.


Introduction
In the last decades, transmitting data was one of the most important fields which concerns data security researchers.The extreme development and usage expansion of Internet caused a huge problem, which is, protecting transmitted data from unauthorized manipulation like modifying, damaging or even just reading transmitted data [1].Cryptography was invented to secure data from any outsiders.Many methods and techniques were invented and improved through time.Complexity is one of the essential concepts in creating any new cryptography method [2].El-Gamal cryptosystem was invented to encrypt and decrypt confidential messages with specific equations and numbers to be chosen carefully and mathematically [3].Sending specific data by mathematical equations to construct polynomials was also one of the powerful mathematical methods used in cryptosystems [4].Shamir and Lagrange are two related equations that could be used for sending information in cryptosystems [5].
In this paper a combination of the previous methods will be used to encrypt a secrete message then use the result in Shamir's polynomial, sending few numbers to the receiver who is going to reconstruct a polynomial with Lagrange to find out the encrypted data which is going to be decrypted to reproduce the plain text.
The next section will explain the methodology of Shamir, Lagrange and El-Gamal.Section three will explain the proposed method.Section four will explain the implementation of the proposed method and finally, the conclusions and the references.

El-Gamal, Shamir, and Lagrange Techniques
Each technique used in the proposed method will be explained below [6]:

El-Gamal cryptosystem
El-Gamal encryption is a public key system.Its complexity depends on the difficulty of finding logarithm of modules numbers.The keys of El-Gamal found by a person (receiver) by:  Choosing a prime number P.


Choosing random number g.  a is a primitive number g.  Compute Y ≡ ga mod P.
The public key is (P, a, g), and the private key is K.
For encrypting the message M: Compute C1 ≡ gK mod P. Compute C2 ≡ YK * M mod P Send (C1, C2) to the receiver.
After encrypting a message, K could be left and use another number for the next transmission.This is an advantage for El-Gamal system since the same message may have different cipher text in each different transmission.

Shamir's polynomial
Shamir's algorithm was used in sharing secrete data by first choosing k different values known as xi, where i between 1 and k, then, the distribution of shared data [7]:  If the secrete data is N, then N-1 value will be chosen randomly (a1, a2, ., an-1).
 Compute yi = a (xi) for all i <= k, and a (x) = ∑ ajx j mod p n−1 j=0 …… [1] Where p is a prime number.
 Each person Pi will be given yi. The secrete data for Shamir's system is the constant value in the polynomial  The polynomial couldn't be constructed unless all distributed values are joined in the polynomial.

Lagrange's Method
Lagrange interpolation formula gives a unique polynomial of degree k-1 for k known points y1, y2, …, yk where yi = a (xi) and [8]: For a given set of n + 1 nodes xi; the Lagrange polynomials are the n + 1 polynomials defined by Then, the interpolating polynomial is defined as …….. [4] If each Lagrange Polynomial is of degree of at most n; then Pn also has this property.The Lagrange polynomials can be characterized as follows:

The suggested method
In this paper, the method suggested is combining the previous techniques in one process, where the data to be transmitted has to be encrypted first to ensure the secrecy of the confidential data.The encryption is done using El-Gamal cryptosystem.Then, for each cipher text, Shamir's algorithm is applied and three points are sent.
The suggested method has one common step between the sender and the receiver that step is the keys of El-Gamal.Then, there are five steps for the sender and four steps for the receiver.
 Use C1 and C2 to decrypt the message using El-Gamal algorithm.

The Implementation of the suggested method
To implement the proposed method, the consecutive steps of the proposed method will be followed as below:

The keys of El-Gamal
Each part (sender and receiver) should have his public and private key; let's suppose that the sender has the following keys: Let a = 6, g = 11, a is primitive of P Let K = 7, K is a random number Let P = 23, P is a prime number Y = ak mod P = 6 Public key = (g, a, Y); private key = K

The decryption process
The receiver will use these Y's in Lagrange interpolation to reconstruct the pair (C1, C2) which are going to be decrypted by El_Gamal decryption algorithm to find out the original message.
For decryption use Lagrange equations to extract C1, C2 Then: So the decrypted message is the letter 'M' The benefit of El_Gamal algorithm is that if the sender use different random number in his process then C1 and C2 will be different even for the same letter.Let the message be the word "DOOR", then, by applying the algorithm, the result will be as in table 1 below: The decryption will not affected by the choice of the different random numbers of the sender and the original letters will surly reconstructed successfully.

Conclusion
Transmitting secrete message between different parities required an attention to be sure that the data will not be manipulated or read by unauthorized outers.Cryptography system used to encrypt the data before transmission so that even if it is hijacked it won't be readable.El-Gamal algorithm is one of the well known cryptosystems that had been used for decades.In the other hand, Shamir sharing system and Lagrange interpolation were proved to be complex, secure, and workable for data transmission.Combining the previous methods in one process (the suggested method) ensures having the benefits of all methods together.It provides encryption, secrete distribution, and complex reconstruction of the received data before decryption.

Table 1
Different results for the same letter