Arithmetic properties related to the shuffle-product

Properties of the shuffle product suggest the definition of a quadratic form with domain and values in formal power series over a field of characteristic 2. This quadratic form preserves rational (respectively algebraic) power series and its restriction to the affine subspace of series with constant term 1 is bijective. Conjecturally, this bijection restricts to a bijection of rational (respectively algebraic) formal power series.