Protocols for Univariate Sumcheck

Three 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 two both consist of a direct reduction from univariate sumcheck to multilinear evaluation, which can be combined with Gemini (Bootle et al., Eurocrypt 2022). Two of our approaches optionally support a natural exponential round reduction from m to log(m) while retaining asymptotically linear prover time.