Sumcheck optimization 1: linear space and linear time old implementation

protocol:

Optimization in special case:

linear time, linear space prover:

Analysis:

To what I can think of:

\(d\cdot (d-1) \cdot 2^{l-1}\) small multiplication come from that \(s_1(u)\) is a \(d\) degree polynomial, that means prover needs to send \(d\) points to the verifier(shouldn’t be d+1?), for each points, there will be \(d – 1\) multiplication to compute this product:

But isn’t the following part also need multiplication?

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