International Journal of Computer Applications Technology and Research Volume 3– Issue 5, 312 - 317, 2014, ISSN: 2319–8656 www.ijcat.com 312 Implementation of Matrix based Mapping Method Using Elliptic Curve Cryptography Geetha G Dept. of Electronics and Communication BNM Institute of Technology Bangalore, India Padmaja Jain Dept. of Electronics and Communication BNM Institute of Technology Bangalore, India Abstract: Elliptic Curve Cryptography (ECC) gained a lot of attention in industry. The key attraction of ECC over RSA is that it offers equal security even for smaller bit size, thus reducing the processing complexity. ECC Encryption and Decryption methods can only perform encrypt and decrypt operations on the curve but not on the message. This paper presents a fast mapping method based on matrix approach for ECC, which offers high security for the encrypted message. First, the alphabetic message is mapped on to the points on an elliptic curve. Later encode those points using Elgamal encryption method with the use of a non-singular matrix. And the encoded message can be decrypted by Elgamal decryption technique and to get back the original message, the matrix obtained from decoding is multiplied with the inverse of non-singular matrix. The coding is done using Verilog. The design is simulated and synthesized using FPGA. Keywords: Cryptography; Elliptic Curve; Finite Field; Mapping; Non-singular matrix; Elgamal Encryption; Elgamal Decryption 1. INTRODUCTION With the rapid development of technology, people find various methods to hack information. For secured data communication, Cryptography is one of the techniques. It basically deals with encryption and decryption of a given data. The two types of cryptography being Public and Private key cryptography, where in two types of keys are used in former and a single key is used in later case. The advantage of public key cryptography is that it is more secure than private key cryptography. ECC is one such method of public key cryptography along with RSA. The key attraction of ECC over RSA is that it offers equal security even for smaller bit size, thus reducing the band width, processing complexity [1]. In ECC, the operations such as point inverse, point addition, point subtraction, scalar multiplication are performed on the points obtained from an elliptic curve. These point operations are useful in performing encryption and decryption operations. In paper [2], Static (One to One) and dynamic (One to N) mapping methods are explained. In static, though it is a simple technique, the same alphanumeric characters from the different words are always mapped onto the same x-y coordinates of the elliptic curve points. When encrypted, points obtained will also be same. So, an intruder can easily interpret data with trial and error method. Hence the secrecy of data transmission by using this methodology is very low. In dynamic mapping, the alphanumeric characters are mapped dynamically on to the points of EC. Thus it is difficult for an intruder to guess which particular character is mapped to which point on EC. But mapping method using matrix method as in paper [3], guarantees the security for the data. And no intruder can hack it. Since this method avoids the regularity in the resultant encrypted text. Thus strengthens the crypto systems and provides better performance. This paper is organized as follows. The brief introduction to cryptography is given in section 1, cryptography using elliptic curves followed by the point operations, encryption and decryption operations is given in section 2, section 3 describes the proposed method, and the mapping technique followed by illustration and results in section 4, section 5 is about the future enhancements, section 6 gives conclusion and section 7 is about the acknowledgement followed by references. 2. CRYTOGRAPHY USING ELLIPTIC CURVES 2.1. Elliptic Curve In elliptic curve cryptography, a restricted form of elliptic curve defined over a finite field F p is considered. One particular interest for cryptography is referred to elliptic group mod p, where p is prime number. Eq.1 defines the condition for choosing the elliptic curve. 4a 3 +27b 2 (mod p) ≠ 0 (1) Where ‘a’ and ‘b’ are two nonnegative integers less than p. Then E p (a, b) indicates the elliptic group mod p whose elements (x, y) are pairs of nonnegative integers less than p. Eq. 2 refers to the general form of elliptic curve. y 2 =x 3 +ax+b (2) 2.2. Modular Arithmetic Modular arithmetic is the principal mathematical concept in Cryptography. Here for every operation, modulus is taken w.r.t the prime number. Eg: Prime number considered in this work is 31. 2.3. ECC Point Operations 2.3.1. Point Inverse If J = (x, y)  E (F p ), then (x, y) + (x, – y) = ∞. The point (x, – y)  E (F p ) and is called the inverse of J. Given a point J(x 1 , y 1 ) on an elliptic curve, -J(x 1 , y 1 ) represents its inverse. The inverse of a given point can be computed using Eq. 3. -J(x 1 , y 1 ) = J(x 1 , p- y 1 ) (3) Fig.1 shows the graphical representation of point inverse.