Although the conjecture regarding the existence of one-way functions remained open, there are numerous NP-based methods for constructing diverse one-way functions, including DL, lattice, and subset sum problems.
Recently, I have been endeavoring to employ multivariable polynomials in the construction of a one-way function, as follows:
$f(x_1,x_2,...,x_N)=\sum^{N}_{i=1}a_i \cdot x_i$, where $x_i\in \{0,1 \}$ and $a_i \in Z_p$, $p$ is a large prime.
The total number of potential outcomes for such a polynomial is given by the expression $\sum_{k=2}^N\binom Nk$. For the case where $N=1024$, the total number of combinations is very large.
However, I have no idea how to prove or demonstrate that such a function is a one-way function for small inputs.
Another concern arises if the constructed one-way function can be reduced to the subset-sum problem, as suggested in comments. However, this lecture note asserts that "The subset-sum problem is NP-complete but that does not necessarily imply that the function is one-way." I am perplexed by this point.
