Nonlinearities of S-boxes

We introduce an indicator of the non-balancedness of functions defined over Abelian groups, and deduce a new indicator, denoted by NB, of the nonlinearity of such functions. We prove an inequality relating NB and the classical indicator NL, introduced by Nyberg and studied by Chabaud and Vaudenay, of the nonlinearity of S-boxes. This inequality results in an upper bound on NL which unifies Sidelnikov–Chabaud–Vaudenay's bound and the covering radius bound. We also deduce from bounds on linear codes three new bounds on NL that improve upon Sidelnikov–Chabaud–Vaudenay's bound and the covering radius bound in many cases.