Ultimately I'm trying to define all the steps necessary to go from this toy quartic polynomial modulus:
$$x^4 + 21x^3 + 5x^2 + 7x + 1 \equiv 0 \mod 23$$ to: $$x = 18, 19$$
One of the recommended first steps is to use the polynomial version of Euclid's Algorithm like this:
$$\gcd(x^4 + 21 x^3 + 5 x^2 + 7 x + 1,x^{23}-x)$$
From what I've gathered, the solution is: $$x^2+9x+20$$
For this question, can someone guide me through the steps involved in reaching $x^2+9x+20$? Bonus points if steps to $x = -5, -4$ were provided. :)
I've read many wiki pages, watched several youtube videos, and countless stackexchange posts on the matter and I'm more confused now than when I started. In each example, it seems the initial setup introduces forms of polynomials I have no idea how they're are obtained.
Part of my confusion stems from the fact that polynomial long division is often mentioned as one of the steps and illustrations indicate that when using my $x^{23}-x$, that one of the steps is a polynomial with an x of all powers from 19 to 0.
For example:
$$x^{19} + 2 x^{18} + 22 x^{17} + 4 x^{16} + 21 x^{15} + 4 x^{14} + 14 x^{13} + 18 x^{12} + 9 x^{11} + 10 x^{10} + 19 x^{9} + 22 x^{8} + 8 x^{7} + 16 x^{6} + 3 x^{5} + 9 x^{4} + 21 x^{3} + 6 x^{2} + 2 x + 2$$
I need to use a solution that does not do this because my real problem involves a modulus n of $115792089237316195423570985008687907852837564279074904382605163141518161494337$ and I couldn't even write this out in this universe's lifetime.
This post suggests that Square-and-multiply can be used to quickly find this answer. I am familiar with the Square-and-multiply technique using numbers, but am all thumbs where polynomials are involved.
Any clarity would be helpful here. Steps would be great. Thanks!
By the way, the modulus here is prime so the Chinese remainder theorem and Hensel lifting aren't necessary.
Update
Here is what I'm actually trying to solve. Hope it sheds some light into the right approach with my toy example.
$$\gcd(x^4 + 104904789243076764769272258331008861968773673797810778195683332912883890030960x^3 + 0x^2 + 7476413576101162797801345879216150799700321200725070869821865316357935848858x + 115422971207940030049746909572711733051107198341303901253516904650890299203952, x^{115792089237316195423570985008687907852837564279074904382605163141518161494337} - x)$$
two distinct zeros modulo 23
were determined from this solution. I think that partially answers that :) – Levitikon Aug 02 '18 at 14:03