We study the problem of low-bandwidth non-linear computation on Reed-Solomon encoded data. Given an [n,k] Reed-Solomon encoding of a message vector 𝐟 ∈ 𝔽_q^k, and a polynomial g ∈ 𝔽_q[X₁, X₂, …, X_k], a user wishing to evaluate g(𝐟) is given local query access to each codeword symbol. The query response is allowed to be the output of an arbitrary function evaluated locally on the codeword symbol, and the user’s aim is to minimize the total information downloaded in order to compute g(𝐟). This problem has been studied before for linear functions g; in this work we initiate the study of non-linear functions by starting with quadratic monomials. For q = p^e and distinct i,j ∈ [k], we show that any scheme evaluating the quadratic monomial g_{i,j} := X_i X_j must download at least 2 log₂(q-1) - 3 bits of information when p is an odd prime, and at least 2log₂(q-2) -4 bits when p = 2. When k = 2, our result shows that one cannot do significantly better than the naive bound of k log₂(q) bits, which is enough to recover all of 𝐟. This contrasts sharply with prior work for low-bandwidth evaluation of linear functions g(𝐟) over Reed-Solomon encoded data, for which it is possible to substantially improve upon this bound [Venkatesan Guruswami and Mary Wootters, 2016; Tamo et al., 2018; Shutty and Wootters, 2021; Kiah et al., 2024; Con and Tamo, 2022]. Some proofs have been omitted from this extended abstract; the full version can be found at [Keller Blackwell and Mary Wootters, 2025].

A Homomorphic Secret Sharing (HSS) scheme is a secret-sharing scheme that shares a secret $x$ among $s$ servers, and additionally allows an output client to reconstruct some function $f(x)$, using information that can be locally computed by each server. A key parameter in HSS schemes is download rate, which quantifies how much information the output client needs to download from each server. Recent work (Fosli, Ishai, Kolobov, and Wootters, ITCS 2022) established a fundamental limitation on the download rate of linear HSS schemes for computing low-degree polynomials, and gave an example of HSS schemes that meet this limit. In this paper, we further explore optimal-rate linear HSS schemes for polynomials. Our main result is a complete characterization of such schemes, in terms of a coding-theoretic notion that we introduce, termed optimal labelweight codes. We use this characterization to answer open questions about the amortization required by HSS schemes that achieve optimal download rate. In more detail, the construction of Fosli et al. required amortization over $\ell$ instances of the problem, and only worked for particular values of $\ell$. We show that -- perhaps surprisingly -- the set of $\ell$'s for which their construction works is in fact nearly optimal, possibly leaving out only one additional value of $\ell$. We show this by using our coding-theoretic characterization to prove a necessary condition on the $\ell$'s admitting optimal-rate linear HSS schemes. We then provide a slightly improved construction of optimal-rate linear HSS schemes, where the set of allowable $\ell$'s is optimal in even more parameter settings. Moreover, based on a connection to the MDS conjecture, we conjecture that our construction is optimal for all parameter regimes.