I'm trying to calculate 19v^2 + 49v + 8 to the 67th power over the finite field Z/67Z using Sage where v = sqrt(-2).
Here's what I have so far (using t instead of v):
R.<t> = PolynomialRing(GF(67))
poly = (19 * (t^2) + 49*t + 8)
poly^67
This works fine.
But now, I want to evaluate the resulting polynomial with the value sqrt(-2).
I'm not sure how to proceed. Any ideas?
Computing with a square root of -2 amounts to working modulo the polynomial t^2 + 2.
The function power_mod can be used for that.
Instead of first powering and then reducing modulo t^2 + 2, which would be wasteful, it performs the whole powering process modulo t^2 + 2, which is a lot more efficient.
Here are two ways to write the (same) computation.
sage: t = polygen(GF(67))
sage: p = 19 * t^2 + 49 * t + 8
sage: power_mod(p, 67, t^2 + 2)
sage: R.<t> = GF(67)[]
sage: p = R([8, 49, 19])
sage: power_mod(p, 67, R([2, 0, 1]))