The global avalanche characteristics of two Boolean functions and algebraic immunity.

The global avalanche characteristics criterion of two Boolean functions was introduced by Zhou et al. [On the global avalanche characteristics criterion of two Boolean functions and the higher order nonlinearity, Inform. Sci. 180(2) (2010), pp. 256–265] to measure the cryptographic behaviour in a global characteristic. The two indicators σ f, g and Δ f, g of Boolean functions f and g were presented. In this paper, a new upper bound on σ f, g is derived, and a technique on constructing Boolean functions to attain the lower bound on the sum-of-squares indicator is described by using the disjoint spectra method. Some new upper bounds on Δ f, g and σ f, g are deduced for two special Boolean functions. Two relationships between σ f, g and algebraic immunity of the two Boolean functions are obtained. Finally, some links among different cryptographic indicators are shown. [ABSTRACT FROM AUTHOR]