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• 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 +
Oct 2, 2011 1 of 2
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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.

Kermit
• ... Get Pari-GP to do it for you :-) Assuming that p|q-1, with prime q, we simply ask for bsol(p,q)=lift(znprimroot(q^3)^(q^2*(q-1)/p)); example:
Oct 2, 2011 1 of 2
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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 find
> expansion mod q^3.
> If I were more familiar with that formula,
> I could do it here.

Get Pari-GP to do it for you :-)

Assuming that p|q-1, with prime q,