finding a^b^c^… mod m

Question

I would like to calculate:

a^bcd... mod m

Do you know any efficient way since this number

Answer 1

Look at the behavior of A^X mod M as X increases. It must eventually go into a cycle. Suppose the cycle has length P and starts after N steps. Then X >= N implies A^X = A^(X+P) = A^(X%P + (-N)%P + N) (mod M). Therefore we can compute A^B^C by computing y=B^C, z = y < N ? y : y%P + (-N)%P + N, return A^z (mod m).

Notice that we can recursively apply this strategy up the power tree, because the derived equation either has an exponent < M or an exponent involving a smaller exponent tower with a smaller dividend.

The only question is if you can efficiently compute N and P given A and M. Notice that overestimating N is fine. We can just set N to M and things will work out. P is a bit harder. If A and M are different primes, then P=M-1. If A has all of M's prime factors, then we get stuck at 0 and P=1. I'll leave it as an exercise to figure that out, because I don't know how.

///Returns equivalent to list.reverse().aggregate(1, acc,item => item^acc) % M
func PowerTowerMod(Link list, int M, int upperB = M)
    requires M > 0, upperB >= M
    var X = list.Item
    if list.Next == null: return X
    var P = GetPeriodSomehow(base: X, mod: M)
    var e = PowerTowerMod(list.Next, P, M)
    if e^X < upperB then return e^X //todo: rewrite e^X < upperB so it doesn't blowup for large x
    return ModPow(X, M + (e-M) % P, M)