# Generating a randomised polynomial

My question is related to this question on Cryptography SE.

In the following, all the operations and polynomials are defined over a finite field of prime order, $$\mathbb{F}_p$$, where $$p$$ is a sufficiently large prime number. All values and polynomials are non-zero.

Let the degree of each of polynomials $$R(x)$$, $$Z(x)$$ and $$P(x)$$ be $$d>1$$. Also, let $$Z$$ and $$R$$ be two polynomials picked uniformly at random from $$\mathbb{F}_p[x]$$. Note that $$P(x)$$ is a fixed polynomial. We set $$F(x)=R\cdot P+Z$$.

Let's assume we arbitrary change $$F(x)$$ to $$F'(x)$$.

Question : Is polynomial $$p'(x)=\frac{F'(x)-Z(x)}{R(x)}$$ a random polynomial uniformly distributed over the polynomial of degree d in $$\mathbb{F}_p[x]$$?

• @EthanBolker can you please explain in more details? – Ay. Oct 17 at 16:26