On Univariate Sumcheck

Two candidate approaches for univariate sumcheck over roots of unity are presented. The first takes the form of a multilinear evaluation protocol, which can be combined with the standard multivariate sumcheck protocol. The other consists of a direct reduction from univariate sumcheck to multilinear evaluation, which can be combined with Gemini (Bootle et al., Eurocrypt 2022). Both approaches optionally support a very natural exponential round reduction from $m$ to $\log(m)$ while retaining asymptotically optimal linear prover time.