The lower bounds on the second order nonlinearity of three classes of Boolean functions with high nonlinearity

Authors:
Guanghong Sun;Chuankun Wu
Affiliations:
The State Key Laboratory of Information Security, Institute of Software, Chinese Academy of Sciences, 4# South Fourth Street, Beijing 100190, China and The Graduate University of the Chinese Acade ...;The State Key Laboratory of Information Security, Institute of Software, Chinese Academy of Sciences, 4# South Fourth Street, Beijing 100190, China
Venue:
Information Sciences: an International Journal
Year:
2009

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Abstract

The rth order nonlinearity of Boolean functions is an important cryptographic criterion associated with some attacks on stream and block ciphers. It is also very useful in coding theory, since it is related to the covering radii of Reed-Muller codes. This paper tightens the lower bounds of the second order nonlinearity of three classes of Boolean functions in the form f(x)=tr(x^d) in n variables, where (1) d=2^m^+^1+3 and n=2m, or (2) d=2^m+2^m^+^1^2+1, n=2m and m is odd, or (3) d=2^2^r+2^r^+^1+1 and n=4r.