Cryptology ePrint Archive: Report 2004/284

ON THE DEGREE OF HOMOGENEOUS BENT FUNCTIONS

Qingshu Meng and Huanguo Zhang and Min Yang and Jingsong Cui

Abstract: It is well known that the degree of a $2m$-variable bent function is at most $m.$ However, the case in homogeneous bent functions is not clear. In this paper, it is proved that there is no homogeneous bent functions of degree $m$ in $2m$ variables when $m>3;$ there is no homogenous bent function of degree $m-1$ in 2m variables when $m>4;$ Generally, for any nonnegative integer $k$, there exists a positive integer $N$ such that when $m>N$, there is no homogeneous bent functions of degree $m-k$ in $2m$ variables. In other words, we get a tighter upper bound on the degree of homogeneous bent functions. A conjecture is proposed that for any positive integer $k>1$, there exists a positive integer $N$ such that when $m>N$, there exists homogeneous bent function of degree $k$ in $2m$ variables.

Category / Keywords: secret-key cryptography / bent functions, Walsh transform, algebraic degree

Date: received 1 Nov 2004, last revised 1 Dec 2005

Contact author: mqseagle at sohu com

Available format(s): Postscript (PS) | Compressed Postscript (PS.GZ) | PDF | BibTeX Citation

Note: a scholar told me one spelling error due to my unfamilarity to latex when i submitted this paper to eprint in 2004.

Version: 20051201:071440 (All versions of this report)

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