Nonlinearity, local and global avalanche characteristics of balanced Boolean functions

For a Boolean function f , define ∆f (α) = ∑ x f̂(x)f̂(x⊕ α), f̂(x) = (−1)f(x), the absolute indicator ∆f = maxα6=0 |∆f (α)|, and the sum-of-squares indicator σf = ∑ α ∆ 2 f (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 ≤ 2,∆f = 2n, where p is the number of linear structures (with even Hamming weight) of the first half of an SAC balanced Boolean function f . We also derive some bounds for the nonlinearity of such functions. It improves upon the results of Son et al. [5] and Sung et al. [7]. 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+ǫ (ǫ = 0 for n even and ǫ = 1 for n odd).