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

We improve and simplify the result of the part 4 of "Counting curves and their projections" (Joachim von zur Gathen, Marek Karpinski, Igor Shparlinski) by showing that counting roots of a sparse polynomial over $\mathbb{F}_{2^n}$ is #P- and $\oplus$P-complete under deterministic reductions.