New results on nonexistence of generalized bent functions

详细信息    查看全文

• 作者：Yupeng Jiang ; Yingpu Deng
• 关键词：Field descent method ; Generalized bent functions ; Cyclotomic fields ; Prime ideal factorization ; Principal ideal ; Ideal norm ; 11R04 ; 94A15
• 刊名：Designs, Codes and Cryptography
• 出版年：2015
• 出版时间：June 2015
• 年：2015
• 卷：75
• 期：3
• 页码：375-385
• 全文大小：190 KB
• 参考文献：1.Agoh T., Dilcher K., Skula L.: Fermat quotients for composite muduli. J. Number Theory 66, 29-0 (1997).
  2.Akyildiz E., Gülo?lu I.S., ?keda M.: A note of generalized bent functions. J. Pure Appl. Algebra 106, 1- (1996).
  3.Dillon J.F.: Elementary Hadamard difference sets. PhD Dissertation, University of Maryland (1974).
  4.Feng K.: Generalized bent functions and class group of imaginary quadratic fields. Sci. China A 44, 562-70 (2001).
  5.Feng K., Liu F.: New results on the nonexistence of generalized bent functions. IEEE Trans. Inform. Theory 49, 3066-071 (2003a).
  6.Feng K., Liu F.: Non-existence of some generalized bent functions. Acta Math. Sin. (Engl. Ser.). 19, 39-0 (2003b).
  7.Fr?hlich A., Taylor M.J.: Algebraic Number Theory. Cambridge University Press, Cambridge (1991).
  8.Helleseth T., Kholosha A.: On generalized bent functions. In: Information Theory and Applications Workshop (ITA), 2010.
  9.?keda M.: A remark on the non-existence of generalized bent functions. Lect. Notes Pure Appl. Math. 204, 109-19 (1999).
  10.Kumar P.V., Scholtz R.A., Welch L.R.: Generalized bent functions and their properties. J. Comb. Theory A 40, 90-07 (1985).
  11.Liu F., Ma Z., Feng K.: New results on non-existence of generalized bent functions (II). Sci. China A 45, 721-30 (2002).
  12.Marcus D.A.: Number Fields. Springer, Berlin (1977).
  13.Olsen J.D., Scholtz R.A., Welch L.R.: Bent-function sequences. IEEE Trans. Inform. Theory 28, 858-64 (1982).
  14.Pei D.: On nonexistence of generalized bent functions. Lect. Notes Pure Appl. Math. 141, 165-72 (1993).
  15.Rothaus O.S.: On “bent-functions. J. Comb. Theory. A 20, 300-05 (1976).
  16.Schmidt B.: Cyclotomic integers and finite geometry. J. Am. Math. Soc. 12(4), 929-52 (1999).
  
• 作者单位：Yupeng Jiang (1)
  Yingpu Deng (1)
  
  1. Key Laboratory of Mathematics Mechanization, NCMIS, Academy of Mathematics and Systems Science, Chinese Academy of Sciences, Beijing, 100190, People’s Republic of China
  
• 刊物类别：Mathematics and Statistics
• 刊物主题：Mathematics
  Combinatorics
  Coding and Information Theory
  Data Structures, Cryptology and Information Theory
  Data Encryption
  Discrete Mathematics in Computer Science
  Information, Communication and Circuits
  
• 出版者：Springer Netherlands
• ISSN：1573-7586

文摘

We obtain two kinds of new results on nonexistence of generalized bent functions (GBFs). Based on the results of Feng, Liu and Ma, we use Schmidt’s field descent method to get the first kind. For the second kind, we use both decomposition law in cyclotomic fields and bent requirements to prove that no GBFs with type \([3,\,2\cdot 23^{e}]\) exist.