Moments of the weight deficit of a random equiprobable involution acting on a vector space over a field of two elements

For the additive group of a vector space of increasing dimension $m$ over a field of two elements we study the moments of a random variable equal to the weight deficit of a random equiprobable involution formed by the product of $2^{m-1}$ independent binary cycles. Exact and asymptotic formulas for the binomial moments and for the variance are obtained.