Browse > Article

NAP and Optimal Normal Basis of Type II and Efficient Exponentiation in $GF(2^n)$  

Kwon, Soon-Hak (성균관대학교 수학과)
Go, Byeong-Hwan (성균관대학교 수학과)
Koo, Nam-Hun (성균관대학교 수학과)
Kim, Chang-Hoon (대구대학교 컴퓨터.IT공학부)
Publication Information
The Journal of Korean Institute of Communications and Information Sciences / v.34, no.1C, 2009 , pp. 21-27 More about this Journal
Abstract
We present an efficient exponentiation algorithm for a finite field $GF(2^n)$ determined by an optimal normal basis of type II using signed digit representation of the exponents. Our signed digit representation uses a non-adjacent form (NAF) for $GF(2^n)$. It is generally believed that a signed digit representation is hard to use when a normal basis is given because the inversion of a normal element requires quite a computational delay. However our result shows that a special normal basis, called an optimal normal basis (ONB) of type II, has a nice property which admits an effective exponentiation using signed digit representations of the exponents.
Keywords
Gaussian normal basis; Optimal normal basis; Exponentiation; Signed digit representation; NAF (non-adjacent form);
Citations & Related Records
연도 인용수 순위
  • Reference
1 S. Gao, J. von zur Gathen, and D.Panario, "Orders and cryptographicalapplications," Math. Comp., Vol.67,pp.343-352, 1998   DOI   ScienceOn
2 S.Feisel, J. von zur Gathen, M.Shokrollahi, "Normal bases via eneralGauss periods," Math. Comp., Vol.68,pp.271-290, 1999   DOI   ScienceOn
3 J.H. van Lint, Introduction to CodingTheory, 3rd, Springer-Verlag, 1999
4 S. Gao and S. Vanstone, "On orders ofoptimal normal basis generators,," MathComp., Vol.64, pp.1227-1233, 1995   DOI   ScienceOn
5 D.M. Gordon, "A survey of fastexponentiation methods," J. Algorithm,Vol.27, pp.129-146, 1998   DOI   ScienceOn
6 J. von zur Gathen and I. Shparlinski,"Orders of Gauss periods in finite fields,"ISAAC 95, Leture Notes in ComputerScience, Vol.1004, pp.208-215, 1995   DOI
7 H. Wu, "On complexity of polynomialbasis squaring in GF($2^{m}$)," SAC 00,Lecture Notes in Computer Science,Vol.2012, pp.118-129, 2001   DOI
8 C.H. Lim and P.J. Lee, "More flexibleexponentiation with precomputation," Crypto 94, Lecture Notes in ComputerScience, Vol.839, pp.95-107, 1994   DOI
9 E.F. Brickel, D.M. Gordon, K.S.McCurley, and D.B. Wilson, 'Fastexponentiation with precomputation,' Eurocrypt 92, Lecture Notes in ComputerScience, Vol.658, pp.200-207, 1992   DOI
10 P. de Rooij, "Efficient exponentiation usingprecomputation and vector addition chains,"Eurocrypt 94, Lecture Notes in ComputerScience, Vol.950, pp.389-399, 1994   DOI
11 S. Arno and F.S. Wheeler, "Signed digitrepresentation of minimal hammingweight," IEEE Trans. Computers, Vol.42,pp.1007-1010, 1993   DOI   ScienceOn
12 S. Gao, J. von zur Gathen, and D.Panario, "Gauss periods and fastexponentiation in finite fields," Latin 95,Lecture Notes in Computer Science, vol911, pp.311-322, 1995   DOI   ScienceOn
13 D.E. Knuth, The Art of ComputerProgramming, 3rd : Seminumericalalgorithms, Vol.II, Addison-Wesley, 2001
14 A.J. Menezes, P.C. van Oorschot, and S.A.Vanstone, Handbook of AppliedCryptography, CRC Press, 1996
15 A.J. Menezes, I.F. Blake, S. Gao, R.C.Mullin, S.A. Vanstone, and T. Yaghoobian,Applications of Finite Fields, KluwerAcademic Publisher, 1993
16 H. Wu and M.A. Hasan, "Efficientexponentiation of a primitive root inGF($2^{m}$)," IEEE Trans. Computers, Vol.46,pp.162-172, 1997   DOI   ScienceOn