The Optimal Normal Elements for Massey-Omura Multiplier

Massey-Omura 승산기를 위한 최적 정규원소

  • Published : 2004.06.01


Finite field multiplication and division are important arithmetic operation in error-correcting codes and cryptosystems. The elements of the finite field GF($2^m$) are represented by bases with a primitive polynomial of degree m over GF(2). We can be easily realized for multiplication or computing multiplicative inverse in GF($2^m$) based on a normal basis representation. The number of product terms of logic function determines a complexity of the Messay-Omura multiplier. A normal basis exists for every finite field. It is not easy to find the optimal normal element for a given primitive polynomial. In this paper, the generating method of normal basis is investigated. The normal bases whose product terms are less than other bases for multiplication in GF($2^m$) are found. For each primitive polynomial, a list of normal elements and number of product terms are presented.


  1. IEEE Trans. Information Theory v.28 Bit-Serial Reed-Solomon Encoder E.R.Berlekamp
  2. IEEE Trans. Computers v.C-34 VLSI Architectures for Computing Multiplications and Inverses in GF($2^m$) C.C.Wang;T.K.Truong;H.M.Shao;J.K.Omura;I.S.Reed
  3. J. Soc. Electro. Comm. A Fast Algorithm for Computing Multiplicative Inverses in GF($2^m$)Using Normal Bases T.Itoh;O.Teechai;S.Tsujii
  4. 한국통신정보보호학회 종합학술발표회 논문집 v.6 no.1 유한체 GF($2^m$)상에서의 빠른 역원계산기법 박정식;안금혁;김영길;장청룔
  5. 정보보호논문지 v.13 no.2 GF($2^m$)에서 정규기저를 이용한 고속 곱셈 역원 역산 방법 장용희;권용진
  6. Electronics Letters v.33 no.3 Improved Normal Basis Inversion in GF($2^m$) S.M.Ten
  7. IEEE Trans. on Inform. Theory v.43 no.2 Normal Basis of the Finite Field $F_{2(p-1)/p^m}$ over F₂ M.Wang;F.Blake
  8. ISSPA'99 Normal Basis Inversion in Some Finite Fields J.H.Jeng
  9. IEEE Trans. Computers v.41 no.8 Modular Construction of Low Complexity Parallel Multipliers for a Class of Finite Fields GF($2^m$) M.A.Hansan;;V.K.Bhargava
  10. thesis for Ph. D. in Combinatorics and Optimization Normal Bases over Finite Fields S.Gao
  11. IEEE Trans. Computers v.51 no.1 A New Hardware Architecture for Operation in GF($2^m$) C.H.Kim;S.Oh;J.Lim
  12. Error Control Systems for Digital Communication and Storage S.B.Wicker