On the Amortized Complexity of Zero Knowledge Protocols for Multiplicative Relations

We present a protocol that allows to prove in zero-knowledge that committed values $x_i, y_i, z_i$, $i=1,\dots,l$ satisfy $x_iy_i=z_i$, where the values are taken from a finite field $K$, or are integers. The amortized communication complexity per instance proven is $O(\kappa + l)$ for an error probability of $2^{-l}$, where $\kappa$ is the size of a commitment. When the committed values are from a field of small constant size, this improves complexity of previous solutions by a factor of $l$. When the values are integers, we improve on security: whereas previous solutions with similar efficiency require the strong RSA assumption, we only need the assumption required by the commitment scheme itself, namely factoring. We generalize this to a protocol that verifies $l$ instances of an algebraic circuit $D$ over $K$ with $v$ inputs, in the following sense: given committed values $x_{i,j}$ and $z_i$, with $i=1,\dots,l$ and $j=1,\dots,v$, the prover shows that $D(x_{i,1},\dots,x_{i,v})= z_i$ for $i=1,\dots,l$. For circuits with small multiplicative depth, this approach is better than using our first protocol: in fact, the amortized cost may be asymptotically smaller than the number of multiplications in $D$