Padma Bh et. al. / (IJCSE) International Journal on Computer Science and Engineering Vol. 02, No. 05, 2010, 1904-1907 Encoding And Decoding of a Message in the Implementation of Elliptic Curve Cryptography using Koblitz’s Method Padma Bh 1 , D.Chandravathi 2 , P.Prapoorna Roja 3 1.Asst.Professor,Dept.of MCA,GVP College for Degree &PG Courses,Visakhapatnam. 2.Asst.Professor,Dept of MCA, GVP College for Degree & PG Courses, Visakhapatnam. 3.Professor, SSN College Of Engineering, Department Of IT, Chennai. Abstract: Elliptic Curve Cryptography recently gained a lot of attention in industry. The principal attraction of ECC compared to RSA is that it offers equal security for a smaller bit size, thereby reducing processing overhead. ECC is ideal for constrained environment such as pager, PDAs, cellular phones and smart cards. For the implementation of elliptic curve cryptography (ECC) the plaintext encoding should be done before encryption and decoding should be done after decryption. ECC Encryption and Decryption methods can only encrypt and decrypt a point on the curve and not messages. The Encoding(converting message to a point) and Decoding (converting a point to a message) are important functions in Encryption and Decryption in ECC. The paper discusses Koblitz’s method to represent a message to a point and vice- versa. The paper also describes implementation results of Koblitz’s Encoding and Decoding methods. Keywords: Encryption, Decryption, Elliptic Curve cryptography, Encoding, Decoding. Introduction: Elliptic Curve Cryptography[2] is a public key Cryptography. ECC is ideal for environments such as pager, PDAs, cellular phones and smart cards. Moreover, because of the apparent hardness of the underlying elliptic curve discrete logarithm problem (ECDLP), ECC systems are also well suited for applications that need long-term security requirements. Elliptic Curve Cryptography (ECC) is a public key technology that offers performance advantages at higher security levels. Every user taking part in public key cryptography will take a pair of keys, a public key and a private key. Only the particular user knows the private key whereas the public keys are distributed to all users taking part in the communication. Some public key algorithm may require a set of predefined constants to be known by all the devices taking part in the communication. In ECC we call these predefined constants as ‘Domain Parameters. Understanding ECC needs full mathematical background on elliptic curves. Elliptic curves are not ellipses. The general cubic equation of elliptic curves is y 2 +axy+by=x 3 +cx 2 +dx+e. But for our purpose it is sufficient to limit the equation to the form y 2 = x 3 + ax + b. Say E P (a,b) consisting of all the points (x,y) that satisfy the above equation together with element at infinity O. A group can be defined based on the set E P (a,b) for specific values of a and b[8]. If P,Q R are points on E P (a,b) the relations commutativity, associativity, existence of an identity element and existence of inverse hold good[4]. The heart of ECC is discrete logarithm problem that can be stated as “it should be very hard to find a value k such that Q=kP where P and Q are known’. But ‘it should be relatively easy to find Q where k and P are known’ P, Q are points on the elliptic curve [5]. Elliptic Curve Example: Let the equation of the curve is y 2 mod p = x 3 + ax + b mod p Inputs : a, b, p (p is key of the ECC algorithm) Choose two non-negative integers a, b and a large prime number such that 4a 3 + 27b 2 mod p !=0. For Example, the following figure (fig 1) shows the elliptic curve, y 2 mod 23 = x 3 + x + 1 mod 23 . Here points P, Q lie on the curve and P+Q gives another point that lie on the line that connects P and Q as shown in the fig 1 below. Fig 1 ISSN : 0975-3397 1904