Nonlinearity, local and global avalanche characteristics of balanced Boolean functions Original Research Article

For a Boolean function f, define Δf(α)=∑xf̂(x)f̂(x⊕α), f̂(x)=(−1)f(x), the absolute indicator Δf=maxα≠0 |Δf(α)|, and the sum-of-squares indicator σf=∑αΔf2(α). We construct a class of functions with good local avalanche characteristics, but bad global avalanche characteristics, namely we show that 22n(1+p)⩽σf⩽23n−2,Δf=2n, where p is the number of linear structures (with even Hamming weight) of the first half of a strict avalanche criterion balanced Boolean function f. We also derive some bounds for the nonlinearity of such functions. It improves upon the results of Son et al. (Inform. Process. Lett. 65 (3) (1998) 139) and Sung et al. (Inform. Process. Lett. 69 (1) (1999) 21). In our second result we construct a class of highly nonlinear balanced functions with good local and global avalanche characteristics. We show that for these functions, 22n+2⩽σf⩽22n+2+ε (ε=0 for n even and ε=1 for n odd).