Tips and tricks

This section contains various ideas and snippets that you might find useful while writing halo2 circuits.

Small range constraints

A common constraint used in R1CS circuits is the boolean constraint: $b\ast (1-b)=0$. This constraint can only be satisfied by $b=0$ or $b=1$.

In halo2 circuits, you can similarly constrain a cell to have one of a small set of values. For example, to constrain $a$ to the range $[0..5]$, you would create a gate of the form:

$$a\cdot (1-a)\cdot (2-a)\cdot (3-a)\cdot (4-a)=0$$

while to constraint $c$ to be either 7 or 13, you would use:

$$(7-c)\cdot (13-c)=0$$

The underlying principle here is that we create a polynomial constraint with roots at each value in the set of possible values we want to allow. In R1CS circuits, the maximum supported polynomial degree is 2 (due to all constraints being of the form $a\ast b=c$). In halo2 circuits, you can use arbitrary-degree polynomials - with the proviso that higher-degree constraints are more expensive to use.

Note that the roots don't have to be constants; for example $(a-x)\cdot (a-y)\cdot (a-z)=0$ will constrain $a$ to be equal to one of $\{x,y,z\}$ where the latter can be arbitrary polynomials, as long as the whole expression stays within the maximum degree bound.