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  • https://doi.org/10.1007/11751632_64Copy DOI Icon

Efficient Exponentiation in GF(p m ) Using the Frobenius Map

  • Jan 1, 2006
  • Mun-Kyu Lee +3 more
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Abstract

The problem of exponentiation over a finite field is to compute A e for a field element A and a positive integer e. This problem has many useful applications in cryptography and information security. In this paper, we present an efficient exponentiation algorithm in optimal extension field (OEF) GF(p m ), which uses the fact that the Frobenius map, i.e., the p-th powering operation is very efficient in OEFs. Our analysis shows that the new algorithm is twice as fast as the conventional square-and-multiply exponentiation. One of the important applications of our new algorithm is random generation of a base point for elliptic curve cryptography, which is an attractive public-key mechanism for resource-constrained devices. We present a further optimized exponentiation algorithm for this application. Our experimental results show that the new technique accelerates the generation process by factors of 1.62–6.55 over various practical elliptic curves.

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