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Hash-based Polynomial Commitment Scheme for Small Polynomials

I am building a SNARK project which needs to use PCS (polynomial commitment scheme). Because of some constraints, I want the field of PCS to have no additional structures and thus I only want to use ...
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Succinct proof of evaluation of known polynomial

Consider the zeroes polynomial $$ zeroes_n(X) = \prod_{0\leq i< n} (X-i) . $$ Fix a large prime $p$, and fix some $n$ that is less than $p$ but which may still be very large (e.g. $p\approx 2^{256}...