Super Power Mod by Ilan Vardi, from "Computational Recreations in Mathematica," 
		Addison-Wesley Publishing Co., Redwood City, CA, 1991, pages 226-229.

SuperPowerMod[a_, k_, n_] := 0 /; Mod[a, n] == 0;
SuperPowerMod[a_, k_, n_] := Mod[a, n] /; k == 1;
SuperPowerMod[a_, k_, n_] := PowerMod[a, a, n] /; k == 2;
SuperPowerMod[a_, k_, n_] := Mod[2, a] /; n == 2;
SuperPowerMod[a_, k_, n_] := SuperPowerMod[a, k, n] = 
  Block[{fi = FactorInteger[n], m = Mod[a, n], i, rprimem, gcdn}, 
   rprimem = m; 
   gcdlist = 
    Block[{i = 0}, 
       While[ Mod[rprimem, #[[1]]] == 0, rprimem /= #[[1]]; 
        i++]; {#[[1]], #[[2]], i}] & /@ 
     Select[fi, Mod[m, #[[1]]] == 0 &]; 
   If[gcdlist == {}, 
    PowerMod[m, SuperPowerMod[a, k - 1, EulerPhi[n]], n], 
    gcdn = Times @@ (#[[1]]^#[[2]] & /@ gcdlist); 
    If[ LogStar[a, Max[#[[2]] - #[[3]] & /@ gcdlist]] > k - 1, 
     PowerMod[m, SuperPower[a, k - 1], n], 
     If[ gcdn == n, 0, 
      Mod[ gcdn PowerMod[gcdn, -1, n/gcdn] PowerMod[ m/rprimem, 
         SuperPowerMod[a, k - 1, EulerPhi[n/gcdn]], n/gcdn] PowerMod[
         rprimem, SuperPowerMod[a, k - 1, EulerPhi[n]], n], n]]]]]

LogStar[a_, n_] := 0 /; n < a;
LogStar[a_, n_] := LogStar[a, Log[a, N[n]]] + 1;
LogStarStar[a_, n_] := 0 /; n < a;
LogStarStar[a_, n_] := LogStarStar[a, LogStar[a, n]] + 1

SuperSuperPowerMod[a_, k_, n_] := Mod[a, n] /; k == 1
SuperSuperPowerMod[a_, k_, n_] := SuperPowerMod[a, a, n] /; k == 2
SuperSuperPowerMod[a_, k_, n_] := 
 SuperPowerMod[a, SuperSuperPower[a, k - 1], n] /; 
  LogStarStar[Log[2, n]] > k
SuperSuperPowerMod[a_, k_, n_] := SuperPowerMod[a, n, n]
SuperPower[a_, 1] := a
SuperPower[a_, k_] := a^SuperPower[a, k - 1]
SuperSuperPower[a_, 1] := 1
SuperSuperPower[a_, k_] := SuperPower[a, SuperSuperPower[a, k - 1]]