The Journal of Korean Institute of Communications and Information Sciences (한국통신학회논문지)

The Korean Institute of Commucations and Information Sciences (한국통신학회)
권호 표지

On p-ary Bent Functions Defined on Finite Fields

유한체 상에서 정의된 p진 Bent 함수

  • 김영식 (서울대학교 전기컴퓨터공학부 부호 및 암호 연구실) ;
  • 장지웅 (서울대학교 전기컴퓨터공학부 부호 및 암호 연구) ;
  • 노종선 (서울대학교 전기컴퓨터공학부 부호 및 암호 연구실)
  • Published : 2004.06.01
Download PDF
⟨ Previous Next ⟩

Abstract

It is known that a bent function corresponds to a perfect nonlinear function, which makes it difficult to do the differential cryptanalysis in DES and in many other block ciphers. In this paper, for an odd prime p, quadratic p-ary bent functions defined on finite fields are given from the families of p-ary sequences with optimal correlation properly. And quadratic p-ary bent functions, that is, perfect nonlinear functions from the finite field F $_{p^{m}}$ to its prime field $F_{p}$ are constructed by using the trace functions. trace functions.

Bent 함수가 DES나 많은 블록 암호 시스템에서 차분 암호분석법이 어렵도록 만들어 주는 완전 비선형 함수와 상응관계가 있다는 것이 알려져 있다. 본 논문에서는 홀수인 소수 p에 대해서 유한체에서 정의된 2차 p진 bent 함수가 최적의 상관 특성을 갖는 p진 시퀀스의 군으로부터 주어졌다. 그리고 이차 p진 bent 함수, 즉 유한체 F $_{p^{m}}$에서 소수체 $F_{p}$ 로의 완전 비선형 함수가 race 함수를 사용해서 생성되었다.e 함수를 사용해서 생성되었다.

Keywords

References

  1. Preprint Highly nonlinear mappings C.Carlet;C.Ding
  2. Designs, Codes and Cryptography v.10 Planar functions and planes of Lenz-Barlotti class Ⅱ R.Coulter;R.W.Matthews https://doi.org/10.1023/A:1008292303803
  3. Math. Z. v.103 Planes ofn order n with collineation groups of order $n^2$ P.Dembowski;t.G.Ostrom https://doi.org/10.1007/BF01111042
  4. Linear groups with exposition of the Galois field theory L.E.Dickson
  5. IEEE Trans. Inform. Theory v.45 no.2 New families ofn almost perfect nonlinear power mappings T.Helleseth;C.Rong;d.Sandberg https://doi.org/10.1109/18.748997
  6. Applicable Algebra in Engineering, communication and Computing v.8 Some power mappings with low differential uniformity T.Helleseth;d.Sandberg https://doi.org/10.1007/s002000050073
  7. IEEE Trans. Inform. Theory v.37 no.2 Prime-phase sequence with periodic correlation properties better than binary sequences P.V.Kumar;O.Moreno https://doi.org/10.1109/18.79916
  8. J. Combinatorial Theory, Series A. v.40 Generalized bent functions and their properties P.V.Kumar;R.A.Scholtz;L.R.Welch https://doi.org/10.1016/0097-3165(85)90049-4
  9. The Theory of Matrices with Applications, 2nd ed. P.Lancaster;M.Tismenetsky
  10. Encyclopedia of Mathematics and its Applications v.20 Finite Fields R.Lidl;H.Niederreiter
  11. Lecture Notes in Computer Science v.473 Constructions ofn bent functions and difference sets K.Nyberg https://doi.org/10.1007/3-540-46877-3_13
  12. Lecture Notes in Computer Science v.765 Differentially uniform mappings for cryptography K.Nyberg https://doi.org/10.1007/3-540-48285-7_6
  13. IEEE Trans. Inform. Theory v.28 no.2 Bent-function sequences J.D.Olsen;r.A.Scholtz;L.R.Welch https://doi.org/10.1109/TIT.1982.1056589
  14. J. Combinatorial Theory, Series A. v.20 On bent functions O.S.rothaus https://doi.org/10.1016/0097-3165(76)90024-8
  15. SIAM J. Comput. v.9 no.4 Factorization of symmetric matrices and trace-orthogonal bases in finite fields G.Seroussi;A.Lempel https://doi.org/10.1137/0209059
  16. IEEE Trans. Inform. Theory v.20 Lower bounds on the maximal cross correlation of signals L.R.Welch
  17. Mathematical Properties of Sequences and Other Combinatorial Structures, The Kluwer International Series in Engineering and Computer Seicene, On P-ary bent function defined on finite field Young-Sik Kim;Ji Woong Jang;Jong Seon No;tor Helleseth