  We revisit Christol's theorem on algebraic power series in positive
characteristic and propose yet another proof for it. This new proof combines
several ingredients and advantages of existing proofs, which make it very
well-suited for algorithmic purposes. We apply the construction used in the new
proof to the design of a new efficient algorithm for computing the $N$th
coefficient of a given algebraic power series over a perfect field of
characteristic~$p$. It has several nice features: it is more general, more
natural and more efficient than previous algorithms. Not only the arithmetic
complexity of the new algorithm is linear in $\log N$ and quasi-linear in~$p$,
but its dependency with respect to the degree of the input is much smaller than
in the previously best algorithm. {Moreover, when the ground field is finite,
the new approach yields an even faster algorithm, whose bit complexity is
linear in $\log N$ and quasi-linear in~$\sqrt{p}$}.
