b^p = 1 mod q^3
- Suppose b0^p = 1 mod q, where q is prime.
Let b = (b2 q^2 + b1 q + b0)
b^2 = (b2 b0 + b1^2) q^2 + (b1 b0 ) q + b0^2 mod q^3
b^3 = (b2 b0^2 + 2 b1^2 b0) q^2 + (2 b1 b0^2) q + b0^3 mod q^3
b^p = J2 q^2 + J1 q + b0^p mod q^3
b^p = J2 q^2 + J1 q + 1 mod q^3
b^p = 1 mod q^3 ==> J2 q^2 + J1 q = 0 mod q^3
==> J2 q + J1 = 0 mod q^2
==> J1 = 0 mod q and J2 = 0 mod q.
Use formula for (b2 q^2 + b1 q + b0)^p to find
expansion mod q^3.
If I were more familiar with that formula, I could do it here.
- --- In firstname.lastname@example.org,
Kermit Rose <kermit@...> wrote:
> Suppose b0^p = 1 mod q, where q is prime....
> Let b = (b2 q^2 + b1 q + b0)
> Use formula for (b2 q^2 + b1 q + b0)^p to findGet Pari-GP to do it for you :-)
> expansion mod q^3.
> If I were more familiar with that formula,
> I could do it here.
Assuming that p|q-1, with prime q,
we simply ask for