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<p>A common constraint used in R1CS circuits is the boolean constraint: <spanclass="katex"><spanclass="katex-html"aria-hidden="true"><spanclass="base"><spanclass="strut"style="height:0.69444em;vertical-align:0em;"></span><spanclass="mord mathnormal">b</span><spanclass="mspace"style="margin-right:0.2222222222222222em;"></span><spanclass="mbin">∗</span><spanclass="mspace"style="margin-right:0.2222222222222222em;"></span></span><spanclass="base"><spanclass="strut"style="height:1em;vertical-align:-0.25em;"></span><spanclass="mopen">(</span><spanclass="mord">1</span><spanclass="mspace"style="margin-right:0.2222222222222222em;"></span><spanclass="mbin">−</span><spanclass="mspace"style="margin-right:0.2222222222222222em;"></span></span><spanclass="base"><spanclass="strut"style="height:1em;vertical-align:-0.25em;"></span><spanclass="mord mathnormal">b</span><spanclass="mclose">)</span><spanclass="mspace"style="margin-right:0.2777777777777778em;"></span><spanclass="mrel">=</span><spanclass="mspace"style="margin-right:0.2777777777777778em;"></span></span><spanclass="base"><spanclass="strut"style="height:0.64444em;vertical-align:0em;"></span><spanclass="mord">0</span></span></span></span>.
This constraint can only be satisfied by <spanclass="katex"><spanclass="katex-html"aria-hidden="true"><spanclass="base"><spanclass="strut"style="height:0.69444em;vertical-align:0em;"></span><spanclass="mord mathnormal">b</span><spanclass="mspace"style="margin-right:0.2777777777777778em;"></span><spanclass="mrel">=</span><spanclass="mspace"style="margin-right:0.2777777777777778em;"></span></span><spanclass="base"><spanclass="strut"style="height:0.64444em;vertical-align:0em;"></span><spanclass="mord">0</span></span></span></span> or <spanclass="katex"><spanclass="katex-html"aria-hidden="true"><spanclass="base"><spanclass="strut"style="height:0.69444em;vertical-align:0em;"></span><spanclass="mord mathnormal">b</span><spanclass="mspace"style="margin-right:0.2777777777777778em;"></span><spanclass="mrel">=</span><spanclass="mspace"style="margin-right:0.2777777777777778em;"></span></span><spanclass="base"><spanclass="strut"style="height:0.64444em;vertical-align:0em;"></span><spanclass="mord">1</span></span></span></span>.</p>
<p>In halo2 circuits, you can similarly constrain a cell to have one of a small set of
values. For example, to constrain <spanclass="katex"><spanclass="katex-html"aria-hidden="true"><spanclass="base"><spanclass="strut"style="height:0.43056em;vertical-align:0em;"></span><spanclass="mord mathnormal">a</span></span></span></span> to the range <spanclass="katex"><spanclass="katex-html"aria-hidden="true"><spanclass="base"><spanclass="strut"style="height:1em;vertical-align:-0.25em;"></span><spanclass="mopen">[</span><spanclass="mord">0..5</span><spanclass="mclose">]</span></span></span></span>, you would create a gate of
<p>while to constraint <spanclass="katex"><spanclass="katex-html"aria-hidden="true"><spanclass="base"><spanclass="strut"style="height:0.43056em;vertical-align:0em;"></span><spanclass="mord mathnormal">c</span></span></span></span> to be either 7 or 13, you would use:</p>
<p>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 <spanclass="katex"><spanclass="katex-html"aria-hidden="true"><spanclass="base"><spanclass="strut"style="height:0.46528em;vertical-align:0em;"></span><spanclass="mord mathnormal">a</span><spanclass="mspace"style="margin-right:0.2222222222222222em;"></span><spanclass="mbin">∗</span><spanclass="mspace"style="margin-right:0.2222222222222222em;"></span></span><spanclass="base"><spanclass="strut"style="height:0.69444em;vertical-align:0em;"></span><spanclass="mord mathnormal">b</span><spanclass="mspace"style="margin-right:0.2777777777777778em;"></span><spanclass="mrel">=</span><spanclass="mspace"style="margin-right:0.2777777777777778em;"></span></span><spanclass="base"><spanclass="strut"style="height:0.43056em;vertical-align:0em;"></span><spanclass="mord mathnormal">c</span></span></span></span>).
In halo2 circuits, you can use arbitrary-degree polynomials - with the proviso that
higher-degree constraints are more expensive to use.</p>
</blockquote>
<p>Note that the roots don't have to be constants; for example <spanclass="katex"><spanclass="katex-html"aria-hidden="true"><spanclass="base"><spanclass="strut"style="height:1em;vertical-align:-0.25em;"></span><spanclass="mopen">(</span><spanclass="mord mathnormal">a</span><spanclass="mspace"style="margin-right:0.2222222222222222em;"></span><spanclass="mbin">−</span><spanclass="mspace"style="margin-right:0.2222222222222222em;"></span></span><spanclass="base"><spanclass="strut"style="height:1em;vertical-align:-0.25em;"></span><spanclass="mord mathnormal">x</span><spanclass="mclose">)</span><spanclass="mspace"style="margin-right:0.2222222222222222em;"></span><spanclass="mbin">⋅</span><spanclass="mspace"style="margin-right:0.2222222222222222em;"></span></span><spanclass="base"><spanclass="strut"style="height:1em;vertical-align:-0.25em;"></span><spanclass="mopen">(</span><spanclass="mord mathnormal">a</span><spanclass="mspace"style="margin-right:0.2222222222222222em;"></span><spanclass="mbin">−</span><spanclass="mspace"style="margin-right:0.2222222222222222em;"></span></span><spanclass="base"><spanclass="strut"style="height:1em;vertical-align:-0.25em;"></span><spanclass="mord mathnormal"style="margin-right:0.03588em;">y</span><spanclass="mclose">)</span><spanclass="mspace"style="margin-right:0.2222222222222222em;"></span><spanclass="mbin">⋅</span><spanclass="mspace"style="margin-right:0.2222222222222222em;"></span></span><spanclass="base"><spanclass="strut"style="height:1em;vertical-align:-0.25em;"></span><spanclass="mopen">(</span><spanclass="mord mathnormal">a</span><spanclass="mspace"style="margin-right:0.2222222222222222em;"></span><spanclass="mbin">−</span><spanclass="mspace"style="margin-right:0.2222222222222222em;"></span></span><spanclass="base"><spanclass="strut"style="height:1em;vertical-align:-0.25em;"></span><spanclass="mord mathnormal"style="margin-right:0.04398em;">z</span><spanclass="mclose">)</span><spanclass="mspace"style="margin-right:0.2777777777777778em;"></span><spanclass="mrel">=</span><spanclass="mspace"style="margin-right:0.2777777777777778em;"></span></span><spanclass="base"><spanclass="strut"style="height:0.64444em;vertical-align:0em;"></span><spanclass="mord">0</span></span></span></span> will constrain <spanclass="katex"><spanclass="katex-html"aria-hidden="true"><spanclass="base"><spanclass="strut"style="height:0.43056em;vertical-align:0em;"></span><spanclass="mord mathnormal">a</span></span></span></span> to be equal to one of <spanclass="katex"><spanclass="katex-html"aria-hidden="true"><spanclass="base"><spanclass="strut"style="height:1em;vertical-align:-0.25em;"></span><spanclass="mopen">{</span><spanclass="mord mathnormal">x</span><spanclass="mpunct">,</span><spanclass="mspace"style="margin-right:0.16666666666666666em;"></span><spanclass="mord mathnormal"style="margin-right:0.03588em;">y</span><spanclass="mpunct">,</span><spanclass="mspace"style="margin-right:0.16666666666666666em;"></span><spanclass="mord mathnormal"style="margin-right:0.04398em;">z</span><spanclass="mclose">}</span></span></span></span> where the latter can be arbitrary polynomials, as long as the whole expression stays within the maximum degree bound.</p>
where <spanclass="katex"><spanclass="katex-html"aria-hidden="true"><spanclass="base"><spanclass="strut"style="height:0.69444em;vertical-align:0em;"></span><spanclass="mord mathnormal"style="margin-right:0.03148em;">k</span></span></span></span> is the number of data points. (<spanclass="katex"><spanclass="katex-html"aria-hidden="true"><spanclass="base"><spanclass="strut"style="height:0.69444em;vertical-align:0em;"></span><spanclass="mord mathnormal"style="margin-right:0.03148em;">k</span><spanclass="mspace"style="margin-right:0.2777777777777778em;"></span><spanclass="mrel">=</span><spanclass="mspace"style="margin-right:0.2777777777777778em;"></span></span><spanclass="base"><spanclass="strut"style="height:0.64444em;vertical-align:0em;"></span><spanclass="mord">4</span></span></span></span> in our example above.)</p>
<p>Recall that the Lagrange basis polynomial <spanclass="katex"><spanclass="katex-html"aria-hidden="true"><spanclass="base"><spanclass="strut"style="height:1.036108em;vertical-align:-0.286108em;"></span><spanclass="mord"><spanclass="mord mathnormal"style="margin-right:0.01968em;">l</span><spanclass="msupsub"><spanclass="vlist-t vlist-t2"><spanclass="vlist-r"><spanclass="vlist"style="height:0.311664em;"><spanstyle="top:-2.5500000000000003em;margin-left:-0.01968em;margin-right:0.05em;"><spanclass="pstrut"style="height:2.7em;"></span><spanclass="sizing reset-size6 size3 mtight"><spanclass="mord mathnormal mtight"style="margin-right:0.05724em;">j</span></span></span></span><spanclass="vlist-s"></span></span><spanclass="vlist-r"><spanclass="vlist"style="height:0.286108em;"><span></span></span></span></span></span></span><spanclass="mopen">(</span><spanclass="mord mathnormal"style="margin-right:0.07847em;">X</span><spanclass="mclose">)</span></span></span></span> evaluates to <spanclass="katex"><spanclass="katex-html"aria-hidden="true"><spanclass="base"><spanclass="strut"style="height:0.64444em;vertical-align:0em;"></span><spanclass="mord">1</span></span></span></span> at
<spanclass="katex"><spanclass="katex-html"aria-hidden="true"><spanclass="base"><spanclass="strut"style="height:0.68333em;vertical-align:0em;"></span><spanclass="mord mathnormal"style="margin-right:0.07847em;">X</span><spanclass="mspace"style="margin-right:0.2777777777777778em;"></span><spanclass="mrel">=</span><spanclass="mspace"style="margin-right:0.2777777777777778em;"></span></span><spanclass="base"><spanclass="strut"style="height:0.716668em;vertical-align:-0.286108em;"></span><spanclass="mord"><spanclass="mord mathnormal">x</span><spanclass="msupsub"><spanclass="vlist-t vlist-t2"><spanclass="vlist-r"><spanclass="vlist"style="height:0.311664em;"><spanstyle="top:-2.5500000000000003em;margin-left:0em;margin-right:0.05em;"><spanclass="pstrut"style="height:2.7em;"></span><spanclass="sizing reset-size6 size3 mtight"><spanclass="mord mathnormal mtight"style="margin-right:0.05724em;">j</span></span></span></span><spanclass="vlist-s"></span></span><spanclass="vlist-r"><spanclass="vlist"style="height:0.286108em;"><span></span></span></span></span></span></span></span></span></span> and <spanclass="katex"><spanclass="katex-html"aria-hidden="true"><spanclass="base"><spanclass="strut"style="height:0.64444em;vertical-align:0em;"></span><spanclass="mord">0</span></span></span></span> at all other <spanclass="katex"><spanclass="katex-html"aria-hidden="true"><spanclass="base"><spanclass="strut"style="height:0.8888799999999999em;vertical-align:-0.19444em;"></span><spanclass="mord"><spanclass="mord mathnormal">x</span><spanclass="msupsub"><spanclass="vlist-t vlist-t2"><spanclass="vlist-r"><spanclass="vlist"style="height:0.31166399999999994em;"><spanstyle="top:-2.5500000000000003em;margin-left:0em;margin-right:0.05em;"><spanclass="pstrut"style="height:2.7em;"></span><spanclass="sizing reset-size6 size3 mtight"><spanclass="mord mathnormal mtight">i</span></span></span></span><spanclass="vlist-s"></span></span><spanclass="vlist-r"><spanclass="vlist"style="height:0.15em;"><span></span></span></span></span></span></span><spanclass="mpunct">,</span><spanclass="mspace"style="margin-right:0.16666666666666666em;"></span><spanclass="mord mathnormal"style="margin-right:0.05724em;">j</span><spanclass="mspace"style="margin-right:0.2777777777777778em;"></span><spanclass="mrel"><spanclass="mrel"><spanclass="mord vbox"><spanclass="thinbox"><spanclass="rlap"><spanclass="strut"style="height:0.8888799999999999em;vertical-align:-0.19444em;"></span><spanclass="inner"><spanclass="mord"><spanclass="mrel"></span></span></span><spanclass="fix"></span></span></span></span></span><spanclass="mrel">=</span></span><spanclass="mspace"style="margin-right:0.2777777777777778em;"></span></span><spanclass="base"><spanclass="strut"style="height:0.65952em;vertical-align:0em;"></span><spanclass="mord mathnormal">i</span><spanclass="mord">.</span></span></span></span></p>