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I am reading the Pinocchio paper. The paper says, in paragraph "polynomial asymptotics" of section 4.2.1, a worker, in order to include $h(s)$ into the proof, has to interpolate $p(x)$, and then divide it with $t(x)$ to get $h(x)$, then calculate $h(s)$.

As a worker, I would calculate $p(s)$, then $t(s)$, and let $h(s)=p(s)*(t(s))^{-1}$. Suppose this $h(s)$ is good, this approach is much less complicated, compared to the approach described in the paper, which dictates at least $O(n (log(n))^2)$ complexity.

I am wondering why this simpler approach to get $h(s)$ for the worker does not work. Any help is appreciated.

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The worker does not know $s, t(s), v_k(s), w_k(s), y_k(s)$, the workers knows only the content of the public evaluation key and the public verification key. That's to say, the worker knows only $g^s, g^{t(s)}, g^{v_k(s)}$ etc.

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