We know that we can reduce the base of an exponent modulo m : a^ k (a mod m)^ k (mod m). But the same is not true of the exponent itself ! That is,...

We know that we can reduce the base of an exponent modulo m : a^ k ≡ (a mod m)^ k (mod m). But the same is not true of the exponent itself ! That is, we cannot write a ^k ≡ [a ^(k mod m)] (mod m). This is easily seen to be false in general. Consider, for instance, that 2 ^10 mod 3 = 1 but 2^ (10 mod 3) mod 3 = 2^1 mod 3 = 2.