Ihave been trying to understand why the diffie hellman key exchange algorithm works, specifically why the two exponents can be swapped in it without the result changing.

so my specific question is why:

(g^b mod m)^a mod m
= (g^a mod m)^b mod m

i know it is "because" it can be correctly rewritten as the trivially true equation:

g^(ab) mod m
= g^(ba) mod m

but that doesn't explain why it is true, i am interested in the reasoning/proof/explanation behind why the first of these equivalencies hold true.

you