Fix a finite set of integers $S$ and a prime number $p$. Let $(a\_1, a\_2, \dotsc, a\_n)$, $(b\_1, b\_2, b\_3, \dotsc, b\_n)$ be two sequences of integers where the numbers $a\_i$ and $b\_i$ are chosen uniformly from the set $S$ (In the case I care about S = {-2,-1,1,2 }). I would like to understand what is the probability that $$\sum\_{i\_1 < i\_2 < \dotsb < i\_k } a\_{i\_1}a\_{i\_2}\dots a\_{i\_k} = 0 \mod p$$ an the probability that $$\sum\_{i\_1 \leq j\_1 < i\_2 \leq j\_2 < \dotsb < i\_k \leq j\_k} a\_{i\_1}b\_{j\_1}\dots a\_{j\_k}b\_{j\_k} = 0 \mod p.$$

I am interested in the asymptotic probabilities as $n$ increases and $k$ is fixed (or $k$ grows much slower than $n$) and expect some type of equidistribution as $n$. One particular case is [Question about estimating random symmetric sums modulo p](https://mathoverflow.net/questions/402034/question-about-estimating-random-symmetric-sums-modulo-p), where one particular sum is reduced to the linear case. Can this trick be for the general case?

Any suggestion, trick or vaguely related information related to this is highly appreciated.

**P.S.** Can the second type of sum (the one involving $a\_i$, $b\_i$) be somehow related to a sum of the first type?, the ordering makes things much more tricky and I am wondering whether there is a procedure to reduce this sum to the first type. (Also the second sum even though not so natural is the one that I really need to understand).

Thanks!