Leaking information when a value is blinded

Question

Assume all values are defined over a field $\mathbb{F}_p$ where $p$ is a large prime number.

Given a fixed value $a,b$ we compute $v_i=a\cdot b+r_i$, where $r_i$ is picked uniformly at random from the field. For the sake of simplicity let all the values be none zero.

Assume the adversary knows $b$.

Question 1: Given $v_i$, can the adversary learn anything about $a$ (or $r_i$)?

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Assume $p(x)$ is a monic polynomial whose coefficients are picked uniformly at random from the field. Let $x=\{x_1,x_2\}$ where $x_i$ are public values. Let $b_1$ and $b_2$ be known by the adversary. By $p(x_i)$, we mean polynomial is evaluated at $x_i$.

We compute $v_1=p(x_1)\cdot b_1+r_1$ and $v_2=p(x_2)\cdot b_2+r_2$, where $r_i$ are picked unifromly at random.

Question 2: Given $v_1$ and $v_2$ can the adversary learn anything about $p(x)$ (or $r_i$)?

Please note that the adversary knows $x_i$.

Answer 1

To make sure the values of a uniformly chosen random polynomial in $F_p[x]$ are uniformly distributed, you need to impose a maximum degree for the polynomial, since there is no uniform distribution on a countably infinite space.

Having done this and ensuring the uniformity of the polynomial values, your questions boil down to questioning the uniformity of and independence of quantities like $$a+Z$$ and pairs $$(aZ+W,aZ'+W')$$ where $Z,W,Z',W'$ are uniform and pairwise independent. And they are, since linear polynomials over $F_p$ are permutation polynomials.

To clarify, for both questions, the adversary cannot learn anything about the quantity of concern, since it is an independent random variable from the quantities they have access to. For example, $p(\cdot)$ being uniformly chosen means $p(x_1)$ is uniformly distributed and when $r_1$ which is also uniform is added to it $b_1 p(x_1)+r_1$ is distributed like $aZ+W$, which is itself uniform.