A FPGA Implementation of an Elliptic Curve Cryptosystem

Abstract

This paper describes a FPGA implementation of The EC can take many forms. In the context of this paper, an elliptic curve cryptosystem. Such systems are becoming the EC will be constrained to the form given by (1) as provided increasingly popular as they provide the highest strength per by the NIST' [4]. bit of any cryptosystem commonly used today. The cryptosystem was built exploiting the wNAF representation of the private key y2 +y = 3 +ax2+b( as well as parallelism through the arithmetic units. Equations (2) to (6) describe the typical operations which I. INTRODUCTION can be performed on an EC defined by (1) on a Galois field Elliptic curve (EC) cryptosystems (ECC) were first pro-GF(2N) [5]. First, P3 can be computed from P3 (see Figure posed in 1985 independently by Neil Koblitz [1] and Victor 1) as follows: Miller [2]. Several solutions to improve the computational P3 (X3, Y3) (X3, Y3 + X3)(2) efficiency have since been proposed. As a result, ECC gained favor as an efficient and attractive alternative to more con-The standard addition operation, as described by Figure 1, ventional public key cryptosystems as it provides the highest can be performed in a sequence of three steps, i.e. strength per bit of any cryptosystem commonly used today. Y2 + Y EC Diffie-Hellman [3] key exchange protocol relies on the A x2 + Yl(3) relative facility of performing EC multiplication (kP where kis a scalar and P, a point on an EC) compared to the difficulty A2X+ A +1a + Yl + 1,( of retrieving k knowing kP and P. Y3The multiplication kP is based on the addition of a point Finally, the addition of a point with itself, referenced as P by itself k times. The point addition can be interpreted point doubling, is performed following the same mechanism geometrically as illustrated by Figure 1. The addition of 2 as addition. However, the line between P1 and P2 is replaced points, P1 = (x1, Yl) and P2 = (X2, Y2), consist of finding by the tangent to P1. It follows that (3) and (4) are replaced P3 = (X3, y3), the third intersection point between the EC by the following: and the line linking P1 and P2. A fourth point, P3, is defined Yi as the second intersection between the EC and the vertical line A = + 1, (5) passing through P3. The addition is then defined as: P, +P2 +l (6) P3.