Generating a MAC on a Polynomial

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 polynomials and values below are non-zero.

Let $$F(x)= R(x)\cdot \phi(x)+Z(x)$$, where the degree of each of the polynomials $$R,Z$$ and $$\phi$$ is $$d>1$$. Also, let $$Z$$ and $$R$$ be two polynomials picked uniformly at random from the field. Note $$\phi$$ is a fixed polynomial.

Question 1: Is $$F(x)$$ a MAC of $$\phi$$(x)?

Question 2: Is polynomial $$\frac{F'(x)-Z(x)}{R(x)}$$ a random polynomial distributed uniformly among polynomials of degree $$d$$ over $$\mathbb{F}_p$$? where $$F'(x)$$ is a fixed polynomial known by the adversary.

• What have you done to try to figure out these questions so far? What's your definition of a MAC for Q1? Obviously if $Z$ and $R$ are random variables then some function of them is also a random variable; in Q2 did you mean to ask about a particular distribution, like whether $(F' - Z)/R$ is uniformly distributed among the degree-$d$ polynomials in $\mathbb F_p[x]$? Can $R$ be zero? – Squeamish Ossifrage Oct 17 '19 at 16:10
• @SqueamishOssifrage Thanks for useful comments. I have edited the question. – Ay. Oct 17 '19 at 16:19