Inner Product Functional Commitments with Constant-Size Public Parameters and Openings

Abstract

Functional commitments (Libert et al.~[ICALP\u2716]) allow a party to commit to a vector $\vec v$ of length $n$ and later open the commitment at functions of the committed vector succinctly, namely with communication logarithmic or constant in $n$. Existing constructions of functional commitments rely on trusted setups and have either $O(1)$ openings and $O(n)$ parameters, or they have short parameters generatable using public randomness but have $O(\log n)$-size openings. In this work, we ask whether it is possible to construct functional commitments in which both parameters and openings can be of constant size. Our main result is the construction of FC schemes matching this complexity. Our constructions support the evaluation of inner products over small integers; they are built using groups of unknown order and rely on succinct protocols over these groups that are secure in the generic group and random oracle model