On the weight and nonlinearity of homogeneous rotation symmetric Boolean functions of degree 2

• Published in: Discrete Applied Mathematics Vol. 157; no. 2; pp. 428 - 432
• Main Authors: Kim, Hyeonjin, Park, Sung-Mo, Hahn, Sang Geun
• Format: Journal Article
• Language: English
• Published: Kidlington Elsevier B.V 01.01.2009; Elsevier
• Subjects: Algebraic combinatorics; Algorithmics. Computability. Computer arithmetics; Applied sciences; Boolean function; Combinatorics; Combinatorics. Ordered structures; Computer science; control theory; systems; Exact sciences and technology; Hamming weight; Mathematics; Nonlinearity; Order, lattices, ordered algebraic structures; Rotation symmetric; Sciences and techniques of general use; Theoretical computing; Rotation symmetric; Boolean function; Nonlinearity; Hamming weight; Evaluation; Computer theory; Subgroup; Symmetric function; Rotation; Optimization; N order; Discrete function; Cyclic; N variable function; Hashing; Combinatorics
• ISSN: 0166-218X; 1872-6771
• DOI: 10.1016/j.dam.2008.06.022

We improve parts of the results of [T. W. Cusick, P. Stanica, Fast evaluation, weights and nonlinearity of rotation-symmetric functions, Discrete Mathematics 258 (2002) 289–301; J. Pieprzyk, C. X. Qu, Fast hashing and rotation-symmetric functions, Journal of Universal Computer Science 5 (1) (1999) 20–31]. It is observed that the n -variable quadratic Boolean functions, f n , s ( x ) ≔ ∑ i = 1 n x i x i + s − 1 for 2 ≤ s ≤ ⌈ n 2 ⌉ , which are homogeneous rotation symmetric, may not be affinely equivalent for fixed n and different choices of s . We show that their weights and nonlinearity are exactly characterized by the cyclic subgroup 〈 s − 1 〉 of Z n . If n gcd ( n , s − 1 ) , the order of s − 1 , is even, the weight and nonlinearity are the same and given by 2 n − 1 − 2 n 2 + gcd ( n , s − 1 ) − 1 . If the order is odd, it is balanced and nonlinearity is given by 2 n − 1 − 2 n + gcd ( n , s − 1 ) 2 − 1 .