Journal of KIISE:Computer Systems and Theory (한국정보과학회논문지:시스템및이론)

Korean Institute of Information Scientists and Engineers (한국정보과학회)
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Improvement on Bailey-Paar's Optimal Extension Field Arithmetic

Bailey-Paar 최적확장체 연산의 개선

  • 이문규 (인하대학교 컴퓨터정보공학부)
  • Published : 2008.08.15
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Abstract

Optimal Extension Fields (OEFs) are finite fields of a special form which are very useful for software implementation of elliptic curve cryptosystems. Bailey and Paar introduced efficient OEF arithmetic algorithms including the $p^ith$ powering operation, and an efficient algorithm to construct OEFs for cryptographic use. In this paper, we give a counterexample where their $p^ith$ powering algorithm does not work, and show that their OEF construction algorithm is faulty, i.e., it may produce some non-OEFs as output. We present improved algorithms which correct these problems, and give improved statistics for the number of OEFs.

최적확장체(Optimal Extension Field: OEF)는 유한체의 일종으로서, 타원곡선 암호시스템의 소프트웨어 구현에 있어 매우 유용하다. Bailey 및 Paar는 $P^i$ 거듭제곱 연산을 비롯하여 다수의 효율적인 OEF 연산 알고리즘을 제안하였으며, 또한 암호 응용에 적합한 OEF를 생성하기 위한 효과적인 알고리즘을 제안하였다. 본 논문에서는 Bailey-Paar의 $P^i$ 거듭제곱 알고리즘이 적용되지 않는 반례를 제시하며, 또한 그들의 OEF 생성 알고리즘은 실제로 OEF가 아닌 유한체를 OEF로 출력하는 오류가 있음을 보인다. 본 논문에서는 이러한 문제들을 해결한 개선된 알고리즘들을 제시하고, OEF의 개수에 관한 수정된 통계치를 제시한다.

Keywords

References

  1. N. Koblitz, "Elliptic curve cryptosystems," Mathematics of Computation, Vol.48, 1987, pp. 203-209 https://doi.org/10.1090/S0025-5718-1987-0866109-5
  2. V. Miller, "Use of elliptic curves in cryptography," Advances in Cryptology-CRYPTO '85, LNCS, Vol. 218, 1986, pp. 417-428, Springer
  3. D.V. Bailey and C. Paar, "Efficient arithmetic in finite field extensions with application in elliptic curve cryptography," Journal of Cryptology, Vol. 14, 2001, pp.153-176 https://doi.org/10.1007/s001450010012
  4. N.P. Smart, "A comparison of different finite fields for elliptic curve cryptosystems," Computers and Mathematics with Applications, Vol.42, 2001, pp. 91-100 https://doi.org/10.1016/S0898-1221(01)00133-X
  5. S. Baktir and B. Sunar, "Optimal Tower Fields," IEEE Transactions on Computers, Vol.53, No.10, 2004, pp. 1231-1243 https://doi.org/10.1109/TC.2004.83
  6. T. Kobayashi, "Base-$\phi$ method for elliptic curves over OEF," IEICE Trans. Fundamentals, Vol. E83-A, No.4, 2000, pp. 679-686