# MatrixPower with Modulus

I have a matrix that will be multiplied with itself by an extremely large amount of times under $Z/pZ$. The matrix itself contains small numbers, so MatrixPower[Mod[mat, p], pow, vec] won't work; and taking $\bmod$ after calculation is infeasible since the result is extremely large. Can I have similar functionality like some other function's Modulus -> p with MatrixPower?

Using an undocumented function:

AlgebraMatrixPowerMod[{{2, 3, 1}, {5, 2, 4}, {0, 3, 2}}, 4, 6]

{{1, 0, 5}, {1, 1, 5}, {3, 3, 4}}

Mod[MatrixPower[{{2, 3, 1}, {5, 2, 4}, {0, 3, 2}}, 4], 6]

{{1, 0, 5}, {1, 1, 5}, {3, 3, 4}}


Unfortunately, the undocumented function does not support the action form of MatrixPower[].

• This is exactly what I want, PE502 part 2 solved:) – vapor Aug 5 '16 at 13:10
• Another one for (805). @happyfish There are over 500 PE problems now? That's a lot to work through! I stopped when there were only 330 or so, and I was getting repeatedly stumped by ones in the 130's. How many have you solved? – Mr.Wizard Aug 5 '16 at 13:26
• @Mr.Wizard I have done nearly 200, most of them are simple ones. But 502 is on the top of their difficulty list and I am nearly done :) – vapor Aug 5 '16 at 16:29
• This function also support negative exponent (modular inverse). – user202729 Jan 2 at 1:53

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}}} *)

• This is, of course, the "Russian peasant" algorithm for exponentiation. Here is a compact implementation: Fold[Mod[If[#2 == 1, a.#, #].#, m] &, a, Rest[IntegerDigits[n, 2]]]`. – J. M.'s discontentment Aug 6 '16 at 15:21