#P- and $\oplus$P- completeness of counting roots of a sparse polynomial

Consider the field F2n . Its elements are presented as polynomials from F2[x] modulo some irreducible polynomial of degree n. This polynomial can be found in time polynomial in n, as well as the matrix that related two representation corresponding to different irreducible polynomials [2]. Therefore, we do not need to specify a choice of the irreducible polynomial speaking about polynomial reductions. Consider the following counting problem (SparcePolynomialRoots): given n and a polynomial from F2n[x], find the number of its roots in F2n . The polynomial is given in a sparse representation, i.e., as a list of coefficients and degrees. The size of input is the total bit size of all this information (each coefficient takes n bits).