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Low-Weight Primes for Lightweight Elliptic Curve Cryptography on 8-bit AVR Processors

  • Zhe LiuEmail author
  • Johann Großschädl
  • Duncan S. Wong
Conference paper
Part of the Lecture Notes in Computer Science book series (LNCS, volume 8567)

Abstract

Small 8-bit RISC processors and micro-controllers based on the AVR instruction set architecture are widely used in the embedded domain with applications ranging from smartcards over control systems to wireless sensor nodes. Many of these applications require asymmetric encryption or authentication, which has spurred a body of research into implementation aspects of Elliptic Curve Cryptography (ECC) on the AVR platform. In this paper, we study the suitability of a special class of finite fields, the so-called Optimal Prime Fields (OPFs), for a “lightweight” implementation of ECC with a view towards high performance and security. An OPF is a finite field \(\mathbb {F}_p\) defined by a prime of the form \(p = u \cdot 2^k + v\), whereby both \(u\) and \(v\) are “small” (in relation to \(2^k\)) so that they fit into one or two registers of an AVR processor. OPFs have a low Hamming weight, which allows for a very efficient implementation of the modular reduction since only the non-zero words of \(p\) need to be processed. We describe a special variant of Montgomery multiplication for OPFs that does not execute any input-dependent conditional statements (e.g. branch instructions) and is, hence, resistant against certain side-channel attacks. When executed on an Atmel ATmega processor, a multiplication in a 160-bit OPF takes just 3237 cycles, which compares favorably with other implementations of 160-bit modular multiplication on an 8-bit processor. We also describe a performance-optimized and a security-optimized implementation of elliptic curve scalar multiplication over OPFs. The former uses a GLV curve and executes in 4.19 M cycles (over a 160-bit OPF), while the latter is based on a Montgomery curve and has an execution time of approximately 5.93 M cycles. Both results improve the state-of-the-art in lightweight ECC on 8-bit processors.

Keywords

Clock Cycle Scalar Multiplication Elliptic Curve Cryptography Modular Multiplication Wireless Sensor Node 
These keywords were added by machine and not by the authors. This process is experimental and the keywords may be updated as the learning algorithm improves.

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Copyright information

© Springer International Publishing Switzerland 2014

Authors and Affiliations

  1. 1.Laboratory of Algorithmics, Cryptology and SecurityUniversity of LuxembourgLuxembourgLuxembourg
  2. 2.Department of Computer ScienceCity University of Hong KongKowloon TongHong Kong SAR, China

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