Upper and Lower Bounds on Maximum Nonlinearity ofn-input m-output Boolean Function

Authors:
Tadashi Wadayama;Toru Hada;Koichiro Wakasugi;Masao Kasahara
Affiliations:
Faculty of Computer Science and System Engineering, Okayama Prefectural University, 111, Kuboki, Soja, Okayama 719-1197, Japan E-mail:wadayama@c.oka-pu.ac.jp;Kyoto Institute of Technology, Matugasaki, Kyoto, 606-8585, Japan;Kyoto Institute of Technology, Matugasaki, Kyoto, 606-8585, Japan;Kyoto Institute of Technology, Matugasaki, Kyoto, 606-8585, Japan
Venue:
Designs, Codes and Cryptography
Year:
2001

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Abstract

The paper presents lower and upper bounds on the maximumnonlinearity for an n-input m-output Booleanfunction. We show a systematic construction method for a highlynonlinear Boolean function based on binary linear codes whichcontain the first order Reed-Muller code as a subcode. We alsopresent a method to prove the nonexistence of some nonlinearBoolean functions by using nonexistence results on binary linearcodes. Such construction and nonexistence results can be regardedas lower and upper bounds on the maximum nonlinearity. For somen and m, these bounds are tighter than theconventional bounds. The techniques employed here indicate astrong connection between binary linear codes and nonlinear n-input m-output Boolean functions.