In each round of a degree-$d$ sumcheck, the prover sends $d$ independent values of a degree-$d$ univariate $g_i(X)$ (the $(d{+}1)$-th is fixed by $g_i(0) + g_i(1) = s_{i-1}$). The standard evaluation set is $\lbrace 0, 1, \ldots, d \rbrace$. The optimization from Speeding Up Sum-Check Proving (Extended Version) replaces this with $\lbrace \infty, 0, 1, \ldots, d{-}1 \rbrace$, where $g_i(\infty)$ is the leading coefficient.
In the bookkeeping-table approach, evaluations at $X = 0$ and $X = 1$ reuse existing table entries directly. Swapping the highest integer point $d$ for $\infty$ restructures the per-round computation so that the leading coefficient can be obtained more cheaply than an evaluation at a large integer point. The paper reports a 3x speedup for the first sumcheck of Spartan-in-Jolt (see Section 6.2).
In each round of a degree-$d$ sumcheck, the prover sends$d$ independent values of a degree-$d$ univariate $g_i(X)$ (the $(d{+}1)$ -th is fixed by $g_i(0) + g_i(1) = s_{i-1}$ ). The standard evaluation set is $\lbrace 0, 1, \ldots, d \rbrace$ . The optimization from Speeding Up Sum-Check Proving (Extended Version) replaces this with $\lbrace \infty, 0, 1, \ldots, d{-}1 \rbrace$ , where $g_i(\infty)$ is the leading coefficient.
In the bookkeeping-table approach, evaluations at$X = 0$ and $X = 1$ reuse existing table entries directly. Swapping the highest integer point $d$ for $\infty$ restructures the per-round computation so that the leading coefficient can be obtained more cheaply than an evaluation at a large integer point. The paper reports a 3x speedup for the first sumcheck of Spartan-in-Jolt (see Section 6.2).