A Compact Architecture for Montgomery Elliptic Curve Scalar Multiplication Processor
We propose a compact architecture of a Montgomery elliptic curve scalar multiplier in a projective coordinate system over GF(2 m ). To minimize the gate area of the architecture, we use the common Z projective coordinate system where a common Z value is k
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University of California, Los Angeles, USA Katholieke Universiteit Leuven, Belgium {jfirst,ingrid}@ee.ucla.edu
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Abstract. We propose a compact architecture of a Montgomery elliptic curve scalar multiplier in a projective coordinate system over GF (2m ). To minimize the gate area of the architecture, we use the common Z projective coordinate system where a common Z value is kept for two elliptic curve points during the calculations, which results in one register reduction. In addition, by reusing the registers we are able to reduce two more registers. Therefore, we reduce the number of registers required for elliptic curve processor from 9 to 6 (a 33%). Moreover, a unidirectional circular shift register file reduces the complexity of the register file, resulting in a further 17% reduction of total gate area in our design. As a result, the total gate area is 13.2k gates with 314k cycles which is the smallest compared to the previous works. Keywords: Compact Elliptic Curve Processor, Montgomery Scalar Multiplication.
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Introduction
Even though the technology of ASIC advances and its implementation cost decreases steadily, compact implementations of security engines are still a challenging issue. RFID (Radio Frequency IDentification) systems, smart card systems and sensor networks are good examples which need very compact security implementations. Public key cryptography algorithms seem especially taxing for such applications. However, for some security properties such as randomized authentications and digital signatures, the use of public key cryptography algorithms is often inevitable. Among public key cryptography algorithms, elliptic curve cryptography is a good candidate due to its efficient computation and relatively small key size. In this paper, we propose an architecture for compact elliptic curve multiplication processors using the Montgomery algorithm [1]. The Montgomery algorithm is one of the most popular algorithms in elliptic curve scalar multiplication due to its resistance to side-channel attack. We use the projective coordinate system to avoid inverse operations. In order to minimize the system size, we propose new formulae for the common projective coordinate system where all the Z-coordinate values are equal. S. Kim, M. Yung, and H.-W. Lee (Eds.): WISA 2007, LNCS 4867, pp. 115–127, 2007. c Springer-Verlag Berlin Heidelberg 2007
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Y.K. Lee and I. Verbauwhede
When we use L´opez-Dahab’s Montgomery scalar multiplication algorithm [2], two elliptic curve points must be kept where X and Z-coordinate values for each point. Therefore, by the use of the common Z projective coordinate property, one register for a Z-coordinate can be reduced. Considering that the register size is quite large, e.g. 163, reducing even one register is a very effective way to minimize the gate area. Moreover, efficient register management by reuse of the registers makes it possible to reduce two additional registers. Therefore, we reduce three registers out of nine in total compared to a conventional architecture. In additio
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