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

1. Since for any relationship a=x^y, the relationship is invariant with respect to the numeric base you are using (base 2, base 6, base 16, etc).

2. Since the mod N operation is equivalent to extracting the least significant digit (LSD) in base N

3. Since the LSD of the result A in base N can only be affected by the LSD of X in base N, and not digits in higher places. (e.g. 34*56 = 30*50+30*6+50*4+4*5 = 10*(3+50+3*6+5*4)+4*6)

Therefore, from LSD(A)=LSD(X^Y) we can deduce

LSD(A)=LSD(LSD(X)^Y)

Therefore

A mod N = ((X mod N) ^ Y) mod N

and

(X ^ Y) mod N = ((X mod N) ^ Y) mod N)

Therefore you can do the mod before each power step, which keeps your result in the range of integers.

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This assumes a is not negative, and for any x^y, a^y < MAXINT

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This answer answers the wrong question. (alex)