Lower Bounds On The Second-Order Nonlinearity Of Some Boolean Functions

Boolean functions used in stream ciphers and block ciphers should have high r-th order nonlinearity to resist several known attacks and some potential ones.The r-th order nonlinearity of Boolean functions also play an important role in coding theory,since its maximal value equals the covering radius of the r-th order Reed-Muller code.It is very difficult to give the lower bounds of r-th order nonlinearity for Boolean functions with large number of variables and higher algebraic degrees in general.The known results on the lower bounds of r-th order nonlinearity are mainly based on the relationship between the(r-1)-th nonlinearity of derivative and r-th order nonlinearity of Boolean functions.In this thesis,based on the permutation nonlinearity and differential uniformity of the class of Maiorana-McFarland(M-M)bent functions,a lower bound of the second-order nonlinearity for M-M bent functions is derived.The new lower bound is applicable to the arbitrary class of M-M bent functions.The lower bound of second-order nonlinearity of some special M-M Bent are just special cases of the new proposed new bound.As a byproduct,a lower bound of the second-order nonlinearity for Canteaut-Leander Bent functions is also given.In addition,a lower bound on the second-order nonlinearity for the generalized M-M Boolean functions is derived.