Weight deficits of involutions and substitutions

Let $E_{2^m}$ be an ordered additive group of the finite field with $2^m$ elements. For the substitution $S: E_{2^m}\to E_{2^m}$ we introduce and investigate the index of the weight deficit of involution corresponding to the substitution $S$. By means of this index the security of $S$-boxes against the differential method of cryptoanalysis may be estimated. Some enumeration algorithms for involutions with given values of weight deficits are described.