PCS ⇒ Vector commitments

Fully-black-box reduction · standard model · folklore · security loss: tight: a position-binding adversary is forwarded unchanged as an evaluation-binding adversary

Statement

A PCS over $\mathbb{F}$ for degree bound $n-1$, with $n\le |\mathbb{F}|$, yields a vector commitment for ${\mathbb{F}}^n$: fix distinct ${\omega }_1,\dots ,{\omega }_n\in \mathbb{F}$, commit to $(v_1,\dots ,v_n)$ as a commitment to the polynomial $f$ of degree $<n$ with $f({\omega }_i)=v_i$, and open position $i$ with an evaluation proof for $f({\omega }_i)=v_i$. Position binding follows from evaluation binding at ${\omega }_i$ — folklore.

Sketch

Two accepted openings of one commitment at position $i$ to $v\ne v^{\prime }$ are two accepted evaluation proofs at ${\omega }_i$ with values $v$ and $v^{\prime }$ against that commitment, an evaluation-binding break.