Bressoud & Wagon define MatrixPowerMod on page 34 of their book, A Course in Computational Number Theory.
MatrixPowerMod[a_, n_, m_] :=
Block[{b = a, d = IntegerDigits[n, 2]},
Do[
b = Mod[b.b, m];
If[d[[i]] == 1, b = Mod[b.a, m]],
{i, 2, Length[d]}];
b]
An example involving Fibonacci numbers is the following:
AbsoluteTiming[MatrixPowerMod[{{0, 1}, {1, 1}}, 10^8, 10^10]]
(* {0.005576, {{1300390626, 7760546875}, {7760546875, 9060937501}}} *)
