Determine dlog in quotient rings of polynomial rings Question:

Determine $\operatorname{dlog}_x (x^2 + 1)$ in $\Bbb Z_5[x]/\langle\,x^3 + x + 1\,\rangle$

So I know the elements of $F = \Bbb Z_5[x]/\langle\,x^3 + x + 1\,\rangle $ are of the form $ax^2 + bx + c \bmod x^3 + x + 1$ ($a, b, c \in \Bbb Z_5$). I know how to calculate inverse of elements in F and all, but I dont know how to solve the discrete log problem. Any solution, partial or complete, would be great. Thanks...

*

*$x^1 = x$.


*$x^2= x^2$.


*$x^3 \equiv -x-1 \equiv 4x+4$. Here we use that $x^3 + x +1 \equiv 0$ in this field (assuming it is a field, haven't checked), and coefficients are mod 5.


*$x^4 \equiv 4x^2 + 4x$.


*$x^5 \equiv 4x^3 + 4x^2 \equiv 4(-x-1)+ 4x^2 \equiv 4x^2 + x +1$.


*$x^6 \equiv 4x^3 + x^2 + x \equiv 4(-x-1) + x^2 +x \equiv x^2 + 2x + 1$.


*$x^7 \equiv x^3 + 2x^2 + x \equiv (-x-1) + 2x^2 + x \equiv 2x^2 + 4$.


*$x^8 \equiv 2x^3 + 4x \equiv 2(-x-1) + 4x = 2x+3$.


*$x^9 \equiv 2x^2 + 3x$.


*$x^{10} \equiv 2x^3 + 3x \equiv 2(-x-1) + 3x \equiv x + 3$.


*$x^{11} \equiv x^2 + 3x$.


*$x^{12} \equiv x^3 + 3x^2 \equiv -x-1 + 3x^2 = 3x^2 + 4x + 4$.


*$x^{13} \equiv 3x^3 + 4x^2 + 4x \equiv 3(-x-1) + 4x^2 + 4x \equiv 4x^2 + x + 2$.


*$x^{14} \equiv 4x^3 + x^2 + 2x \equiv 4(-x-1) + x^2 + 2x \equiv x^2 + 3x +1$.
etc. Continue until we get $x^n \equiv x^2+1$, and the answer is $n$.
Or use a computer algebra package, or write your own program...
