Construction of Key Echange Protocol over Max-Plus Algebra to En-crypt and Decrypt Arabic Documents

In this paper, we consider the operation of matrix in Max-plus Algebra to construct a key public to encrypt and decrypt arabic document. Matrix operation in max-plus algebra is slight diﬀerent with matrix operation in classical. In this paper, we also use a modular 37 to translate it into arabic text (arabic letter with number and special character)


INTRODUCTION
Cryptography is one of mathematics subject which provides some techniques to secure data in communication.Nowadays, the development of internet allow people to communicate more efficient.Unfortunately, this invention encourage people to steal data for their private needs.The encryption of data is important way to secure online transmission throughout the internet.Therefore, many researcher developed cryptography as a technique for secure information which has two systems to build key public are symmetric and asymmetric key.
During this time, cryptography is built on classical algebra and number theory.In 70s, a Brazil Mathematician, Imre Simon introduced Tropical Algebra.Tropical algebra known as max-plus algebra or minplus algebra.In max-plus scheme, tropical algebra is defined by a semiring T = R max ∪{− ∝} endowed with ⊕ (maximum) and ⊗ (addition), for min-plus algebra ⊕ is replaced by minimum.
In this paper, we will define Tropical algebra as max-plus algebra.There are many application of maxplus algebra such as job scheduling, transportation, communication and networking system.Max-plus algebra has some properties look like classical algebra but ⊕ is non invertible since there does not an element x in R such that a ⊕ x = x ⊕ a = − ∝.Therefore, tropical algebra has idempotency property, a ⊕ a = max(a, a) = a.[8] A few research discuss appplication of cryptography to secure arabic documentation.The new modification of arabic text was introduced by adapting the classical cipher of modular 26 and change it with modular 37. [1] In Order to encrypt and decrypt arabic documents, we will construct key public with matrices over maxplus algebra dan use classical modular 37.
Cryptography has two methods key exchange protocols that are called symmetric key and asymmetric key.In this research, we consider symmetric key as a method to encrypt and decrypt plain text.Symmet-Copyright c 2015, JNSMR, ISSN: 2460-4453 ric key applies the same key in the process to encrypt and decrypt plain text as the following table Identity element of addition and multiplication are − ∝ and 0 ∈ R, respectively.The properties of maxplus algebra analogue with classical algebra as follow: 1. Associative For all a, b, c are elements in R max then

MATRICES OVER MAX-PLUS ALGE-BRA
Let A, B are matrices over max-plus algebra then operation of addition and multiplication are defined as below:

STICKEL'S PROTOCOL OVER MAX-PLUS ALGEBRA
The implementation of matrix over min-plus algebra in Stickel's Protocol was introduced by Dima Grigerov and Vladimir Shpilran [3][7] [6].We will adopt the protocol into max-plus algebra.We define the protocol of Stickel in max-plus algebra as follows: 1. Alice and Bob pick at random two pairs of natural numbers m,n and r,s, respectively Therefore, we can conclude that Alice and Bob shared the same secret key K = K 1 = K 2

APPLICATION OF KEY EXCHANGE PROTOCOL TO ENCRYPT AND DE-CRYPT ARABIC DOCUMENT
To encrypt Arabic documents, we implement block cipher modular 37 [5] into a synthetics table as follow: Copyright c 2015, JNSMR, ISSN: 2460-4453 We process using stickel's algorithm as below: 1. Alice and Bob pick at random two pairs of natural numbers 2,1 and 1,1, respectively 2 Alice computes and send it to Bob and send it to Bob 4 Alice and Bob compute key public and Therefore, we get key public

Encription Process
Alice want to send a message to Bob : and the encryption process of this message is • Divide the message into 4 characters : • convert the block cipher using this table • Alice transform the message and convert into matrices • We use the key public to get cipher text : after we get the cipher text, Bob will decrypt the message with the same key and obtain the original message from Alice

COCLUSION
In this paper, we applying the cryptography over maxplus algebra to secure arabic documents based on classical stickel's protocol to generate key public.for future, we could discuss the complexity of computation and other aspect such as the possibility of attacker to break this can.Finally, max-plus algebra could be implemented into cryptography

2
Alice computes u = A n B m and send it to Bob 3 Bob computes v = A r B s and send it to Bob 4 Alice and Bob compute key public K 1 = A n vB m and K 2 = A r uB s