Journal of the Korea Institute of Information Security & Cryptology (정보보호학회논문지)

Korea Institute of Information Security and Cryptology (한국정보보호학회)
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On the Weight and Nonlinearity of Quadratic Rotation Symmetric Boolean Functions

회전대칭 이차 불함수의 해밍무게 및 비선형성

  • 김현진 (한국전자통신연구원 부설연구소) ;
  • 정창호 (한국전자통신연구원 부설연구소) ;
  • 박일환 (한국전자통신연구원 부설연구소)
  • Published : 2009.04.30
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Abstract

Recently, rotation symmetric Boolean functions have attracted attention since they are suitable for fast evaluation and show good cryptographic properties. For example, important problems in coding theory were settled by searching the desired functions in the rotation symmetric function space. Moreover, they are applied to designing fast hashing algorithms. On the other hand, for some homogeneous rotation symmetric quadratic functions of simple structure, the exact formulas for their Hamming weights and nonlinearity were found[2,8]. Very recently, more formulations were carried out for much broader class of the functions[6]. In this paper, we make a further improvement by deriving the formula for the Hamming weight of quadratic rotation symmetric functions containing linear terms.

회전대칭 불함수는 고속계산에 유리하고 암호학적으로 우수한 성질을 나타내어 최근 많은 주목을 받고 있다. 예를 들어, 부호이론에서 중요한 문제가 회전대칭 불함수를 이용하여 해결된 사례가 있고, 고속 해시함수 설계에 응용된 경우도 있다. 다른 한편으로, 매우 단순한 형태의 회전대칭 이차 불함수에 대한 비선형성 및 해명무게의 정확한 공식이 발견되었으며[2,8], 더 넓은 범위의 함수들에 대한 보다 일반적인 공식들도 발견되었다[6]. 본 논문에서는 이들 공식들을 조금 더 확장하여 일차항들이 포함된 회전대칭 이차 불함수에 대한 정확한 해밍무게 공식을 유도한다.

Keywords

References

  1. C, Carlet, "Boolean Functions for Cryptography and Error Correcting Codes," in: Y. Crama, P, Hammer, (Eds,) ,Boolean Methods and Models, Cambridge Univ, Press, (in press), Available at http://www-rocq.inria.fr/codes/Claude ,Carlet/pubs,html
  2. T,W, Cusick and P, Stanica, "Fast evaluation, weights and nonlinearity of rotation-symmetric functions," Disc, Math" vol. 258, no, 1-3, pp, 289-301, Dec, 2002 https://doi.org/10.1016/S0012-365X(02)00354-0
  3. S, Kavut, S, Maitra, S, Sarkar, and M,D, Yticel, "Enumeration of 9-variable rotation symmetric Boolean functions having nonlinearity ) 240," IndoCrypt 2006, LNCS 4329, pp, 266-279, 2006
  4. S, Kavut, S, Maitra, and M,D, Yticel, "Search for Boolean functions with excellent profiles in the rotation symmetric class," IEEE Trans, Inform, Theory, voL 53, no, 5, pp, 1743-1751, May 2007 https://doi.org/10.1109/TIT.2007.894696
  5. S, Kavut and M,D, Yticel, "Generalized rotation symmetric and dihedral symmetric Boolean functions - 9 variable Boolean functions with nonlinearity 242," AAECC 2007, LNCS 4851, pp, 321-329,2007
  6. H. Kim, S.M. Park, and S.G. Hahn, "On the weight and nonlinearity of homogeneous rotation symmetric Boolean functions of degree 2," Disc. Appl. Math, vol. 157, no. 2, pp. 428-432, Jan. 2009 https://doi.org/10.1016/j.dam.2008.06.022
  7. F.J. MacWilliams and N.J. Sloane, The Theory of Error-Correcting Codes, NorthHolland, Amsterdam, The Netherlands,1977
  8. J. Pieprzyk and C.X. Qu, "Fast hashing and rotation-symmetric functions," J. of Universal Computer Science, vol. 5, no.1. pp. 20-31, Jan. 1999
  9. O.S. Rothaus, "On "bent" functions," J. Combinatorial Theory (A), vol. 20, no. 3,pp. 300-305, May 1976 https://doi.org/10.1016/0097-3165(76)90024-8
  10. S. Sarkar and S. Maitra, "Idempotents in the neighbourhood of PattersonWiedemann functions having Walsh spectra zeros," Designs, Codes and Cryptography, vol. 49, no. 1-3, pp. 95-103, Dec. 2008 https://doi.org/10.1007/s10623-008-9181-y
  11. P. Stanica and S. Maitra, "Rotation symmetric functions - count and cryptographic properties," Disc. Appl. Math., vol. 156, no. 10, pp. 1567-1580, May 2008 https://doi.org/10.1016/j.dam.2007.04.029
  12. P. Stanica, S. Maitra, and J.A. Clark, "Results on rotation symmetric bent and correlation immune Boolean functions," FSE 2004, LNCS 3017, pp. 161-177,2004 https://doi.org/10.1007/978-3-540-25937-4_11