This:
$$x^4 + 21x^3 + 5x^2 + 7x + 1 \equiv 0 \mod 23$$
Leads to:
$$x = 18 || x =19$$
I know this because of this WolframAlpha example and because a fellow member posted it in a since deleted & related question.
What I don't understand are the steps involved in arriving at x = 18 || x = 19 from this equation.
My question starts with the reduced terms mod 23 example in the linked question. I'm now trying understand how to reduce this equation to x = 18 || x = 19.
I have come across a few posts and theorems that hint a solutions, but I lack the math skills to connect any of it together. I am a software developer, not a mathematician. So if anyone can walk me through some steps on how to get from the equation to 18 || 19, that would be great!
This is a toy example representing a new Elliptic Curve Crypto operation where the actual modulus is $2^{256}$ large. So, trying all possible values x is not practical. WolframAlpha is capable of producing solutions to my large modulo equations in a fraction of a second so I know they aren't trying all possible values x.
Fermat’s Little Theorem seems the most promising so far, but I don't understand how to apply it to this equation. This post describes a solution but unfortunately their example is very basic and not very relatable to my equation.
Anything would be helpful here. Steps would be great. Thanks!
Best Answer
The OP requested that I link my other answer as an answer to this one as well.