Highly Nonlinear 0-1 Balanced Boolean Functions Satisfying Strict Avalanche Criterion

Nonlinearity, 0-1 balancedness and strict avalanche criterion (SAC) are important criteria for cryptographic functions. Bent functions have maximum nonlinearity and satisfy SAC however they are not 01 balanced and hence cannot be directly used in many cryptosystems where 0-1 balancedness is needed. In this paper we construct (i) 0-1 balanced boolean functions on V2k+1 (k ≥ 1) having nonlinearity 22k 2k and satisfying SAC, (ii) 0-1 balanced boolean functions on V2k (k ≥ 2) having nonlinearity 22k-1 2k and satisfying SAC. We demonstrate that the above nonlinearities are very high not only for the 0-1 balanced functions satisfying SAC but also for all 0-1 balanced functions. Disciplines Physical Sciences and Mathematics Publication Details Jennifer Seberry and Xian-Mo Zhang, Highly nonlinear 0-1 balanced boolean functions satisfying strict avalanche criterion, ( Jennifer Seberry and Yuliang Zheng, (Eds.)), Advances in Cryptography Auscrypt'92, Conference held at the Gold Coast, Australia, December 1992, 718, Lecture Notes in Computer Science, Springer-Verlag, Berlin--Heidelberg--New York, (1993), 145-155. This conference paper is available at Research Online: http://ro.uow.edu.au/infopapers/1082 Highly Nonlinear 0-1 Balanced Boolean Functions Satisfying Strict Avalanche Criterion (Extended Abstract) Jennifer Seberry * and Xian-Mo Zhang ** Department of Computer Science The University of Wollongong Wollongong, NSW 2522, AUSTRALIA Abstract. Nonlinearity, 0-1 balanced ness and strict avalanche criterion (SAC) are important criteria for cryptographic functions. Bent functions have maximum nonlinearity and satisfy SAC however they are not 01 balanced and hence cannot be directly used in many cryptosystems where 0-1 balanced ness is needed. In this paper we construct (i) 0-1 balanced boolean functions on Y2k+l (k 2: 1) having nonlinearity 22k _ 2k and satisfying SAC, (ii) 0-1 balanced boolean functions on Y2k (k 2: 2) having nonlinearity 2 1 _ 2k and satisfying SAC. We demonstrate that the above nonlinearities are very high not only for the 0-1 balanced functions satisfying SAC but also for all 0-1 balanced functions. Nonlinearity, 0-1 balanced ness and strict avalanche criterion (SAC) are important criteria for cryptographic functions. Bent functions have maximum nonlinearity and satisfy SAC however they are not 01 balanced and hence cannot be directly used in many cryptosystems where 0-1 balanced ness is needed. In this paper we construct (i) 0-1 balanced boolean functions on Y2k+l (k 2: 1) having nonlinearity 22k _ 2k and satisfying SAC, (ii) 0-1 balanced boolean functions on Y2k (k 2: 2) having nonlinearity 2 1 _ 2k and satisfying SAC. We demonstrate that the above nonlinearities are very high not only for the 0-1 balanced functions satisfying SAC but also for all 0-1 balanced functions.