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 linear prover time.

翻译：本文提出了两种在单位根上进行单变量求和检验的候选方法。第一种方法采用多线性求值协议的形式，可与标准的多变量求和检验协议结合使用。另一种方法则通过将单变量求和检验直接归约为多线性求值问题来实现，该方法可与Gemini协议（Bootle等人，Eurocrypt 2022）结合使用。两种方法均支持一种极其自然的指数轮次缩减方案，可将轮数从$m$降至$\log(m)$，同时保持证明者时间为渐近线性。