# Constant size commitment to a bivariate polynomial

In this paper by Kate et al, a constant size polynomial commitment scheme is described.

The commitment scheme assumes a public reference string:

\begin{align*} \Big\{ \{ g^{ \tau^i }, g^{ \alpha \tau^i } \}_{i=-d}^{d} \}, \ \ \{ h^{ \tau^i }, h^{ \alpha \tau^i } \}_{i=-d}^{d}, \ \ e(g, h^{\alpha} ) \Big\} \end{align*}

where $$\alpha, \tau \in \mathbb{F}$$ are two secret values and $$d$$ is a large integer. The commitment goes as follows:

• Commit: let $$f \in \mathbb{F}[x]$$ with degree at most $$d$$. The commitment is a group element $$F$$ defined by \begin{align*} F := g^{\alpha f(\tau) }. \end{align*}

• To open the commitment to a certain value $$z$$, compute the proof $$W$$ which is another field element, defined by: \begin{align*} W := g^{ \frac{ f(\tau) -f(z) }{ \tau-z } } \end{align*}

• To verify a correct opening given $$(F, z, (v, W) )$$: \begin{align*} e(W, h^{\alpha \tau} ) \cdot e(g^v \cdot W^{-z}, h^{\alpha} ) == e(F, h^{\alpha } ) \end{align*}

We note that the the commitment to the polynomial $$f$$ is constant size, as well as the proof of correct opening for a given value $$z$$.

My question is,

Given the above reference string, or a similar reference string of roughly the same size (linear in $$d$$), is there a known constant size commitment scheme to a bivariate polynomial?

I think the main issue for the bivariate polynomials is the coefficients of the polynomial $$f(x,y)$$. Since $$\tau$$ is a secret value then to calculate $$g^{\alpha f(\tau)}$$ we can use the public parameters. As you know, the users can obtain the coefficients based on a standard basis and then solve a multiplication. $$g^{\alpha f(\tau)}=\prod_{i=0}^{n}{(g^{ \alpha \tau^i })}^{f_i}$$ For a univariate polynomial, the coefficients of $$f_i$$ are constant while if we consider bivariate polynomials then it is necessary to evaluate a second polynomial on the $$\tau$$. Precisely, the standard basis $$(1,X,X^2,...,X^d)$$ forms the univariate polynomials and the coefficients for a bivariate polynomial are changing by the second variable.