On the number of multiplications needed to invert a monic power series over fields of characteristic two

Let P(X)=1+a1X+a2X2+⋯ be a monic power series in X with indeterminates a1,a2,… as coefficients. The coefficients b1,b2,… of the inverse of P are polynomials in the coefficients of P. We prove that if divisions are forbidden, then at least n+2⌊n/3⌋-3 essential multiplications are needed to compute b1,…,bn from a1,…,an over fields of characteristic two.