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< ol class = "chapter" > < li class = "chapter-item expanded affix " > < a href = "../../index.html" > halo2< / a > < / li > < li class = "chapter-item expanded " > < a href = "../../concepts.html" > < strong aria-hidden = "true" > 1.< / strong > Concepts< / a > < / li > < li > < ol class = "section" > < li class = "chapter-item expanded " > < a href = "../../concepts/proofs.html" > < strong aria-hidden = "true" > 1.1.< / strong > Proof systems< / a > < / li > < li class = "chapter-item expanded " > < a href = "../../concepts/arithmetization.html" > < strong aria-hidden = "true" > 1.2.< / strong > PLONKish Arithmetization< / a > < / li > < li class = "chapter-item expanded " > < a href = "../../concepts/chips.html" > < strong aria-hidden = "true" > 1.3.< / strong > Chips< / a > < / li > < li class = "chapter-item expanded " > < a href = "../../concepts/gadgets.html" > < strong aria-hidden = "true" > 1.4.< / strong > Gadgets< / a > < / li > < / ol > < / li > < li class = "chapter-item expanded " > < a href = "../../user.html" > < strong aria-hidden = "true" > 2.< / strong > User Documentation< / a > < / li > < li > < ol class = "section" > < li class = "chapter-item expanded " > < a href = "../../user/dev-tools.html" > < strong aria-hidden = "true" > 2.1.< / strong > Developer tools< / a > < / li > < li class = "chapter-item expanded " > < a href = "../../user/simple-example.html" > < strong aria-hidden = "true" > 2.2.< / strong > A simple example< / a > < / li > < li class = "chapter-item expanded " > < a href = "../../user/lookup-tables.html" > < strong aria-hidden = "true" > 2.3.< / strong > Lookup tables< / a > < / li > < li class = "chapter-item expanded " > < a href = "../../user/gadgets.html" > < strong aria-hidden = "true" > 2.4.< / strong > Gadgets< / a > < / li > < li class = "chapter-item expanded " > < a href = "../../user/tips-and-tricks.html" > < strong aria-hidden = "true" > 2.5.< / strong > Tips and tricks< / a > < / li > < li class = "chapter-item expanded " > < a href = "../../user/wasm-port.html" > < strong aria-hidden = "true" > 2.6.< / strong > WASM Guide< / a > < / li > < / ol > < / li > < li class = "chapter-item expanded " > < a href = "../../dev.html" > < strong aria-hidden = "true" > 3.< / strong > Developer Documentation< / a > < / li > < li > < ol class = "section" > < li class = "chapter-item expanded " > < a href = "../../dev/features.html" > < strong aria-hidden = "true" > 3.1.< / strong > Feature development< / a > < / li > < / ol > < / li > < li class = "chapter-item expanded " > < a href = "../../design.html" > < strong aria-hidden = "true" > 4.< / strong > Design< / a > < / li > < li > < ol class = "section" > < li class = "chapter-item expanded " > < a href = "../../design/proving-system.html" > < strong aria-hidden = "true" > 4.1.< / strong > Proving system< / a > < / li > < li > < ol class = "section" > < li class = "chapter-item expanded " > < a href = "../../design/proving-system/lookup.html" > < strong aria-hidden = "true" > 4.1.1.< / strong > Lookup argument< / a > < / li > < li class = "chapter-item expanded " > < a href = "../../design/proving-system/permutation.html" > < strong aria-hidden = "true" > 4.1.2.< / strong > Permutation argument< / a > < / li > < li class = "chapter-item expanded " > < a href = "../../design/proving-system/circuit-commitments.html" > < strong aria-hidden = "true" > 4.1.3.< / strong > Circuit commitments< / a > < / li > < li class = "chapter-item expanded " > < a href = "../../design/proving-system/vanishing.html" class = "active" > < strong aria-hidden = "true" > 4.1.4.< / strong > Vanishing argument< / a > < / li > < li class = "chapter-item expanded " > < a href = "../../design/proving-system/multipoint-opening.html" > < strong aria-hidden = "true" > 4.1.5.< / strong > Multipoint opening argument< / a > < / li > < li class = "chapter-item expanded " > < a href = "../../design/proving-system/inner-product.html" > < strong aria-hidden = "true" > 4.1.6.< / strong > Inner product argument< / a > < / li > < li class = "chapter-item expanded " > < a href = "../../design/proving-system/comparison.html" > < strong aria-hidden = "true" > 4.1.7.< / strong > Comparison to other work< / a > < / li > < / ol > < / li > < li class = "chapter-item expanded " > < a href = "../../design/protocol.html" > < strong aria-hidden = "true" > 4.2.< / strong > Protocol Description< / a > < / li > < li class = "chapter-item expanded " > < a href = "../../design/implementation.html" > < strong aria-hidden = "true" > 4.3.< / strong > Implementation< / a > < / li > < li > < ol class = "section" > < li class = "chapter-item expanded " > < a href = "../../design/implementation/proofs.html" > < strong aria-hidden = "true" > 4.3.1.< / strong > Proofs< / a > < / li > < li class = "chapter-item expanded " > < a href = "../../design/implement
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< h1 id = "vanishing-argument" > < a class = "header" href = "#vanishing-argument" > Vanishing argument< / a > < / h1 >
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< p > Having committed to the circuit assignments, the prover now needs to demonstrate that the
various circuit relations are satisfied:< / p >
< ul >
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< li > The custom gates, represented by polynomials < span class = "katex" > < span class = "katex-html" aria-hidden = "true" > < span class = "base" > < span class = "strut" style = "height:1em;vertical-align:-0.25em;" > < / span > < span class = "mord" > < span class = "mord text" > < span class = "mord" > gate< / span > < / span > < span class = "msupsub" > < span class = "vlist-t vlist-t2" > < span class = "vlist-r" > < span class = "vlist" style = "height:0.2175em;" > < span style = "top:-2.4559em;margin-right:0.05em;" > < span class = "pstrut" style = "height:2.7em;" > < / span > < span class = "sizing reset-size6 size3 mtight" > < span class = "mord mathnormal mtight" > i< / span > < / span > < / span > < / span > < span class = "vlist-s" > < / span > < / span > < span class = "vlist-r" > < span class = "vlist" style = "height:0.2441em;" > < span > < / span > < / span > < / span > < / span > < / span > < / span > < span class = "mopen" > (< / span > < span class = "mord mathnormal" style = "margin-right:0.07847em;" > X< / span > < span class = "mclose" > )< / span > < / span > < / span > < / span > .< / li >
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< li > The rules of the lookup arguments.< / li >
< li > The rules of the equality constraint permutations.< / li >
< / ul >
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< p > Each of these relations is represented as a polynomial of degree < span class = "katex" > < span class = "katex-html" aria-hidden = "true" > < span class = "base" > < span class = "strut" style = "height:0.6944em;" > < / span > < span class = "mord mathnormal" > d< / span > < / span > < / span > < / span > (the maximum degree
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of any of the relations) with respect to the circuit columns. Given that the degree of the
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assignment polynomials for each column is < span class = "katex" > < span class = "katex-html" aria-hidden = "true" > < span class = "base" > < span class = "strut" style = "height:0.6667em;vertical-align:-0.0833em;" > < / span > < span class = "mord mathnormal" > n< / span > < span class = "mspace" style = "margin-right:0.2222em;" > < / span > < span class = "mbin" > − < / span > < span class = "mspace" style = "margin-right:0.2222em;" > < / span > < / span > < span class = "base" > < span class = "strut" style = "height:0.6444em;" > < / span > < span class = "mord" > 1< / span > < / span > < / span > < / span > , the relation polynomials have degree
< span class = "katex" > < span class = "katex-html" aria-hidden = "true" > < span class = "base" > < span class = "strut" style = "height:1em;vertical-align:-0.25em;" > < / span > < span class = "mord mathnormal" > d< / span > < span class = "mopen" > (< / span > < span class = "mord mathnormal" > n< / span > < span class = "mspace" style = "margin-right:0.2222em;" > < / span > < span class = "mbin" > − < / span > < span class = "mspace" style = "margin-right:0.2222em;" > < / span > < / span > < span class = "base" > < span class = "strut" style = "height:1em;vertical-align:-0.25em;" > < / span > < span class = "mord" > 1< / span > < span class = "mclose" > )< / span > < / span > < / span > < / span > with respect to < span class = "katex" > < span class = "katex-html" aria-hidden = "true" > < span class = "base" > < span class = "strut" style = "height:0.6833em;" > < / span > < span class = "mord mathnormal" style = "margin-right:0.07847em;" > X< / span > < / span > < / span > < / span > .< / p >
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< blockquote >
< p > In our < a href = "../proving-system.html#example" > example< / a > , these would be the gate polynomials, of
2023-01-20 13:40:27 -08:00
degree < span class = "katex" > < span class = "katex-html" aria-hidden = "true" > < span class = "base" > < span class = "strut" style = "height:0.7278em;vertical-align:-0.0833em;" > < / span > < span class = "mord" > 3< / span > < span class = "mord mathnormal" > n< / span > < span class = "mspace" style = "margin-right:0.2222em;" > < / span > < span class = "mbin" > − < / span > < span class = "mspace" style = "margin-right:0.2222em;" > < / span > < / span > < span class = "base" > < span class = "strut" style = "height:0.6444em;" > < / span > < span class = "mord" > 3< / span > < / span > < / span > < / span > :< / p >
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< ul >
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< li > < span class = "katex" > < span class = "katex-html" aria-hidden = "true" > < span class = "base" > < span class = "strut" style = "height:1em;vertical-align:-0.25em;" > < / span > < span class = "mord" > < span class = "mord text" > < span class = "mord" > gate< / span > < / span > < span class = "msupsub" > < span class = "vlist-t vlist-t2" > < span class = "vlist-r" > < span class = "vlist" style = "height:0.207em;" > < span style = "top:-2.4559em;margin-right:0.05em;" > < span class = "pstrut" style = "height:2.7em;" > < / span > < span class = "sizing reset-size6 size3 mtight" > < span class = "mord mtight" > 0< / span > < / span > < / span > < / span > < span class = "vlist-s" > < / span > < / span > < span class = "vlist-r" > < span class = "vlist" style = "height:0.2441em;" > < span > < / span > < / span > < / span > < / span > < / span > < / span > < span class = "mopen" > (< / span > < span class = "mord mathnormal" style = "margin-right:0.07847em;" > X< / span > < span class = "mclose" > )< / span > < span class = "mspace" style = "margin-right:0.2778em;" > < / span > < span class = "mrel" > =< / span > < span class = "mspace" style = "margin-right:0.2778em;" > < / span > < / span > < span class = "base" > < span class = "strut" style = "height:1em;vertical-align:-0.25em;" > < / span > < span class = "mord" > < span class = "mord mathnormal" > a< / span > < span class = "msupsub" > < span class = "vlist-t vlist-t2" > < span class = "vlist-r" > < span class = "vlist" style = "height:0.3011em;" > < span style = "top:-2.55em;margin-left:0em;margin-right:0.05em;" > < span class = "pstrut" style = "height:2.7em;" > < / span > < span class = "sizing reset-size6 size3 mtight" > < span class = "mord mtight" > 0< / span > < / span > < / span > < / span > < span class = "vlist-s" > < / span > < / span > < span class = "vlist-r" > < span class = "vlist" style = "height:0.15em;" > < span > < / span > < / span > < / span > < / span > < / span > < / span > < span class = "mopen" > (< / span > < span class = "mord mathnormal" style = "margin-right:0.07847em;" > X< / span > < span class = "mclose" > )< / span > < span class = "mspace" style = "margin-right:0.2222em;" > < / span > < span class = "mbin" > ⋅< / span > < span class = "mspace" style = "margin-right:0.2222em;" > < / span > < / span > < span class = "base" > < span class = "strut" style = "height:1em;vertical-align:-0.25em;" > < / span > < span class = "mord" > < span class = "mord mathnormal" > a< / span > < span class = "msupsub" > < span class = "vlist-t vlist-t2" > < span class = "vlist-r" > < span class = "vlist" style = "height:0.3011em;" > < span style = "top:-2.55em;margin-left:0em;margin-right:0.05em;" > < span class = "pstrut" style = "height:2.7em;" > < / span > < span class = "sizing reset-size6 size3 mtight" > < span class = "mord mtight" > 1< / span > < / span > < / span > < / span > < span class = "vlist-s" > < / span > < / span > < span class = "vlist-r" > < span class = "vlist" style = "height:0.15em;" > < span > < / span > < / span > < / span > < / span > < / span > < / span > < span class = "mopen" > (< / span > < span class = "mord mathnormal" style = "margin-right:0.07847em;" > X< / span > < span class = "mclose" > )< / span > < span class = "mspace" style = "margin-right:0.2222em;" > < / span > < span class = "mbin" > ⋅< / span > < span class = "mspace" style = "margin-right:0.2222em;" > < / span > < / span > < span class = "base" > < span class = "strut" style = "height:1.0641em;vertical-align:-0.25em;" > < / span > < span class = "mord" > < span class = "mord mathnormal" > a< / span > < span class = "msupsub" > < span class = "vlist-t vlist-t2" > < span class = "vlist-r" > < span class = "vlist" style = "height:0.3011em;" > < span style = "top:-2.55em;margin-left:0em;margin-right:0.05em;" > < span class = "pstrut" style = "height:2.7em;" > < / span > < span class = "sizing reset-size6 size3 mtight" > < span class = "mord mtight" > 2< / span > < / span > < / span > < / span > < span class = "vlist-s" > < / span > < / span > < span class = "vlist-r" > < span class = "vlist" style = "height:0.15em;" > < span > < / span > < / span > < / span > < / span > < / span > < / span > < span class = "mopen" > (< / span > < span class = "mord mathnormal" style = "margin-right:0.07847em;" > X< / span > < span class = "mord" > < span class = "mord mathnormal" style = "margin-right:0.03588em;" > ω< / span > < span class = "msupsub" > < span class = "vlist-t" > < span class = "vlist-r" > < span class = "vlist" style = "height:0.8141em;" > < span style = "top:-3.063em;margin-right:0.05em;" > < span class = "pstrut" style = "height:2.7em;" > < / span > < span class = "sizing reset-size6 size3 mtight" > < span class = "mord mtight" > < span class = "mord mtight" > − < / span > < span class = "mord mtight" > 1< / span > < / span > < / span > < / span > < / span > < / span > < / span > < / span > < / span > < span class = "mclose" > )< / span > < span class = "mspace
< li > < span class = "katex" > < span class = "katex-html" aria-hidden = "true" > < span class = "base" > < span class = "strut" style = "height:1em;vertical-align:-0.25em;" > < / span > < span class = "mord" > < span class = "mord text" > < span class = "mord" > gate< / span > < / span > < span class = "msupsub" > < span class = "vlist-t vlist-t2" > < span class = "vlist-r" > < span class = "vlist" style = "height:0.207em;" > < span style = "top:-2.4559em;margin-right:0.05em;" > < span class = "pstrut" style = "height:2.7em;" > < / span > < span class = "sizing reset-size6 size3 mtight" > < span class = "mord mtight" > 1< / span > < / span > < / span > < / span > < span class = "vlist-s" > < / span > < / span > < span class = "vlist-r" > < span class = "vlist" style = "height:0.2441em;" > < span > < / span > < / span > < / span > < / span > < / span > < / span > < span class = "mopen" > (< / span > < span class = "mord mathnormal" style = "margin-right:0.07847em;" > X< / span > < span class = "mclose" > )< / span > < span class = "mspace" style = "margin-right:0.2778em;" > < / span > < span class = "mrel" > =< / span > < span class = "mspace" style = "margin-right:0.2778em;" > < / span > < / span > < span class = "base" > < span class = "strut" style = "height:1.0641em;vertical-align:-0.25em;" > < / span > < span class = "mord" > < span class = "mord mathnormal" style = "margin-right:0.10764em;" > f< / span > < span class = "msupsub" > < span class = "vlist-t vlist-t2" > < span class = "vlist-r" > < span class = "vlist" style = "height:0.3011em;" > < span style = "top:-2.55em;margin-left:-0.1076em;margin-right:0.05em;" > < span class = "pstrut" style = "height:2.7em;" > < / span > < span class = "sizing reset-size6 size3 mtight" > < span class = "mord mtight" > 0< / span > < / span > < / span > < / span > < span class = "vlist-s" > < / span > < / span > < span class = "vlist-r" > < span class = "vlist" style = "height:0.15em;" > < span > < / span > < / span > < / span > < / span > < / span > < / span > < span class = "mopen" > (< / span > < span class = "mord mathnormal" style = "margin-right:0.07847em;" > X< / span > < span class = "mord" > < span class = "mord mathnormal" style = "margin-right:0.03588em;" > ω< / span > < span class = "msupsub" > < span class = "vlist-t" > < span class = "vlist-r" > < span class = "vlist" style = "height:0.8141em;" > < span style = "top:-3.063em;margin-right:0.05em;" > < span class = "pstrut" style = "height:2.7em;" > < / span > < span class = "sizing reset-size6 size3 mtight" > < span class = "mord mtight" > < span class = "mord mtight" > − < / span > < span class = "mord mtight" > 1< / span > < / span > < / span > < / span > < / span > < / span > < / span > < / span > < / span > < span class = "mclose" > )< / span > < span class = "mspace" style = "margin-right:0.2222em;" > < / span > < span class = "mbin" > ⋅< / span > < span class = "mspace" style = "margin-right:0.2222em;" > < / span > < / span > < span class = "base" > < span class = "strut" style = "height:1em;vertical-align:-0.25em;" > < / span > < span class = "mord" > < span class = "mord mathnormal" > a< / span > < span class = "msupsub" > < span class = "vlist-t vlist-t2" > < span class = "vlist-r" > < span class = "vlist" style = "height:0.3011em;" > < span style = "top:-2.55em;margin-left:0em;margin-right:0.05em;" > < span class = "pstrut" style = "height:2.7em;" > < / span > < span class = "sizing reset-size6 size3 mtight" > < span class = "mord mtight" > 2< / span > < / span > < / span > < / span > < span class = "vlist-s" > < / span > < / span > < span class = "vlist-r" > < span class = "vlist" style = "height:0.15em;" > < span > < / span > < / span > < / span > < / span > < / span > < / span > < span class = "mopen" > (< / span > < span class = "mord mathnormal" style = "margin-right:0.07847em;" > X< / span > < span class = "mclose" > )< / span > < / span > < / span > < / span > < / li >
< li > < span class = "katex" > < span class = "katex-html" aria-hidden = "true" > < span class = "base" > < span class = "strut" style = "height:1em;vertical-align:-0.25em;" > < / span > < span class = "mord" > < span class = "mord text" > < span class = "mord" > gate< / span > < / span > < span class = "msupsub" > < span class = "vlist-t vlist-t2" > < span class = "vlist-r" > < span class = "vlist" style = "height:0.207em;" > < span style = "top:-2.4559em;margin-right:0.05em;" > < span class = "pstrut" style = "height:2.7em;" > < / span > < span class = "sizing reset-size6 size3 mtight" > < span class = "mord mtight" > 2< / span > < / span > < / span > < / span > < span class = "vlist-s" > < / span > < / span > < span class = "vlist-r" > < span class = "vlist" style = "height:0.2441em;" > < span > < / span > < / span > < / span > < / span > < / span > < / span > < span class = "mopen" > (< / span > < span class = "mord mathnormal" style = "margin-right:0.07847em;" > X< / span > < span class = "mclose" > )< / span > < span class = "mspace" style = "margin-right:0.2778em;" > < / span > < span class = "mrel" > =< / span > < span class = "mspace" style = "margin-right:0.2778em;" > < / span > < / span > < span class = "base" > < span class = "strut" style = "height:1em;vertical-align:-0.25em;" > < / span > < span class = "mord" > < span class = "mord mathnormal" style = "margin-right:0.10764em;" > f< / span > < span class = "msupsub" > < span class = "vlist-t vlist-t2" > < span class = "vlist-r" > < span class = "vlist" style = "height:0.3011em;" > < span style = "top:-2.55em;margin-left:-0.1076em;margin-right:0.05em;" > < span class = "pstrut" style = "height:2.7em;" > < / span > < span class = "sizing reset-size6 size3 mtight" > < span class = "mord mtight" > 0< / span > < / span > < / span > < / span > < span class = "vlist-s" > < / span > < / span > < span class = "vlist-r" > < span class = "vlist" style = "height:0.15em;" > < span > < / span > < / span > < / span > < / span > < / span > < / span > < span class = "mopen" > (< / span > < span class = "mord mathnormal" style = "margin-right:0.07847em;" > X< / span > < span class = "mclose" > )< / span > < span class = "mspace" style = "margin-right:0.2222em;" > < / span > < span class = "mbin" > ⋅< / span > < span class = "mspace" style = "margin-right:0.2222em;" > < / span > < / span > < span class = "base" > < span class = "strut" style = "height:1em;vertical-align:-0.25em;" > < / span > < span class = "mord" > < span class = "mord mathnormal" > a< / span > < span class = "msupsub" > < span class = "vlist-t vlist-t2" > < span class = "vlist-r" > < span class = "vlist" style = "height:0.3011em;" > < span style = "top:-2.55em;margin-left:0em;margin-right:0.05em;" > < span class = "pstrut" style = "height:2.7em;" > < / span > < span class = "sizing reset-size6 size3 mtight" > < span class = "mord mtight" > 3< / span > < / span > < / span > < / span > < span class = "vlist-s" > < / span > < / span > < span class = "vlist-r" > < span class = "vlist" style = "height:0.15em;" > < span > < / span > < / span > < / span > < / span > < / span > < / span > < span class = "mopen" > (< / span > < span class = "mord mathnormal" style = "margin-right:0.07847em;" > X< / span > < span class = "mclose" > )< / span > < span class = "mspace" style = "margin-right:0.2222em;" > < / span > < span class = "mbin" > ⋅< / span > < span class = "mspace" style = "margin-right:0.2222em;" > < / span > < / span > < span class = "base" > < span class = "strut" style = "height:1em;vertical-align:-0.25em;" > < / span > < span class = "mord" > < span class = "mord mathnormal" > a< / span > < span class = "msupsub" > < span class = "vlist-t vlist-t2" > < span class = "vlist-r" > < span class = "vlist" style = "height:0.3011em;" > < span style = "top:-2.55em;margin-left:0em;margin-right:0.05em;" > < span class = "pstrut" style = "height:2.7em;" > < / span > < span class = "sizing reset-size6 size3 mtight" > < span class = "mord mtight" > 0< / span > < / span > < / span > < / span > < span class = "vlist-s" > < / span > < / span > < span class = "vlist-r" > < span class = "vlist" style = "height:0.15em;" > < span > < / span > < / span > < / span > < / span > < / span > < / span > < span class = "mopen" > (< / span > < span class = "mord mathnormal" style = "margin-right:0.07847em;" > X< / span > < span class = "mclose" > )< / span > < / span > < / span > < / span > < / li >
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< / ul >
< / blockquote >
< p > A relation is satisfied if its polynomial is equal to zero. One way to demonstrate this is
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to divide each polynomial relation by the vanishing polynomial < span class = "katex" > < span class = "katex-html" aria-hidden = "true" > < span class = "base" > < span class = "strut" style = "height:1em;vertical-align:-0.25em;" > < / span > < span class = "mord mathnormal" > t< / span > < span class = "mopen" > (< / span > < span class = "mord mathnormal" style = "margin-right:0.07847em;" > X< / span > < span class = "mclose" > )< / span > < span class = "mspace" style = "margin-right:0.2778em;" > < / span > < span class = "mrel" > =< / span > < span class = "mspace" style = "margin-right:0.2778em;" > < / span > < / span > < span class = "base" > < span class = "strut" style = "height:1em;vertical-align:-0.25em;" > < / span > < span class = "mopen" > (< / span > < span class = "mord" > < span class = "mord mathnormal" style = "margin-right:0.07847em;" > X< / span > < span class = "msupsub" > < span class = "vlist-t" > < span class = "vlist-r" > < span class = "vlist" style = "height:0.6644em;" > < span style = "top:-3.063em;margin-right:0.05em;" > < span class = "pstrut" style = "height:2.7em;" > < / span > < span class = "sizing reset-size6 size3 mtight" > < span class = "mord mathnormal mtight" > n< / span > < / span > < / span > < / span > < / span > < / span > < / span > < / span > < span class = "mspace" style = "margin-right:0.2222em;" > < / span > < span class = "mbin" > − < / span > < span class = "mspace" style = "margin-right:0.2222em;" > < / span > < / span > < span class = "base" > < span class = "strut" style = "height:1em;vertical-align:-0.25em;" > < / span > < span class = "mord" > 1< / span > < span class = "mclose" > )< / span > < / span > < / span > < / span > , which
is the lowest-degree polynomial that has roots at every < span class = "katex" > < span class = "katex-html" aria-hidden = "true" > < span class = "base" > < span class = "strut" style = "height:0.8247em;" > < / span > < span class = "mord" > < span class = "mord mathnormal" style = "margin-right:0.03588em;" > ω< / span > < span class = "msupsub" > < span class = "vlist-t" > < span class = "vlist-r" > < span class = "vlist" style = "height:0.8247em;" > < span style = "top:-3.063em;margin-right:0.05em;" > < span class = "pstrut" style = "height:2.7em;" > < / span > < span class = "sizing reset-size6 size3 mtight" > < span class = "mord mathnormal mtight" > i< / span > < / span > < / span > < / span > < / span > < / span > < / span > < / span > < / span > < / span > < / span > . If relation's polynomial
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is perfectly divisible by < span class = "katex" > < span class = "katex-html" aria-hidden = "true" > < span class = "base" > < span class = "strut" style = "height:1em;vertical-align:-0.25em;" > < / span > < span class = "mord mathnormal" > t< / span > < span class = "mopen" > (< / span > < span class = "mord mathnormal" style = "margin-right:0.07847em;" > X< / span > < span class = "mclose" > )< / span > < / span > < / span > < / span > , it is equal to zero over the domain (as desired).< / p >
< p > This simple construction would require a polynomial commitment per relation. Instead, we
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commit to all of the circuit relations simultaneously: the verifier samples < span class = "katex" > < span class = "katex-html" aria-hidden = "true" > < span class = "base" > < span class = "strut" style = "height:0.625em;vertical-align:-0.1944em;" > < / span > < span class = "mord mathnormal" style = "margin-right:0.03588em;" > y< / span > < / span > < / span > < / span > , and then
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the prover constructs the quotient polynomial< / p >
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< p > < span class = "katex-display" > < span class = "katex" > < span class = "katex-html" aria-hidden = "true" > < span class = "base" > < span class = "strut" style = "height:1em;vertical-align:-0.25em;" > < / span > < span class = "mord mathnormal" > h< / span > < span class = "mopen" > (< / span > < span class = "mord mathnormal" style = "margin-right:0.07847em;" > X< / span > < span class = "mclose" > )< / span > < span class = "mspace" style = "margin-right:0.2778em;" > < / span > < span class = "mrel" > =< / span > < span class = "mspace" style = "margin-right:0.2778em;" > < / span > < / span > < span class = "base" > < span class = "strut" style = "height:2.4377em;vertical-align:-0.936em;" > < / span > < span class = "mord" > < span class = "mopen nulldelimiter" > < / span > < span class = "mfrac" > < span class = "vlist-t vlist-t2" > < span class = "vlist-r" > < span class = "vlist" style = "height:1.5017em;" > < span style = "top:-2.314em;" > < span class = "pstrut" style = "height:3em;" > < / span > < span class = "mord" > < span class = "mord mathnormal" > t< / span > < span class = "mopen" > (< / span > < span class = "mord mathnormal" style = "margin-right:0.07847em;" > X< / span > < span class = "mclose" > )< / span > < / span > < / span > < span style = "top:-3.23em;" > < span class = "pstrut" style = "height:3em;" > < / span > < span class = "frac-line" style = "border-bottom-width:0.04em;" > < / span > < / span > < span style = "top:-3.677em;" > < span class = "pstrut" style = "height:3em;" > < / span > < span class = "mord" > < span class = "mord" > < span class = "mord text" > < span class = "mord" > gate< / span > < / span > < span class = "msupsub" > < span class = "vlist-t vlist-t2" > < span class = "vlist-r" > < span class = "vlist" style = "height:0.207em;" > < span style = "top:-2.4559em;margin-right:0.05em;" > < span class = "pstrut" style = "height:2.7em;" > < / span > < span class = "sizing reset-size6 size3 mtight" > < span class = "mord mtight" > 0< / span > < / span > < / span > < / span > < span class = "vlist-s" > < / span > < / span > < span class = "vlist-r" > < span class = "vlist" style = "height:0.2441em;" > < span > < / span > < / span > < / span > < / span > < / span > < / span > < span class = "mopen" > (< / span > < span class = "mord mathnormal" style = "margin-right:0.07847em;" > X< / span > < span class = "mclose" > )< / span > < span class = "mspace" style = "margin-right:0.2222em;" > < / span > < span class = "mbin" > +< / span > < span class = "mspace" style = "margin-right:0.2222em;" > < / span > < span class = "mord mathnormal" style = "margin-right:0.03588em;" > y< / span > < span class = "mspace" style = "margin-right:0.2222em;" > < / span > < span class = "mbin" > ⋅< / span > < span class = "mspace" style = "margin-right:0.2222em;" > < / span > < span class = "mord" > < span class = "mord text" > < span class = "mord" > gate< / span > < / span > < span class = "msupsub" > < span class = "vlist-t vlist-t2" > < span class = "vlist-r" > < span class = "vlist" style = "height:0.207em;" > < span style = "top:-2.4559em;margin-right:0.05em;" > < span class = "pstrut" style = "height:2.7em;" > < / span > < span class = "sizing reset-size6 size3 mtight" > < span class = "mord mtight" > 1< / span > < / span > < / span > < / span > < span class = "vlist-s" > < / span > < / span > < span class = "vlist-r" > < span class = "vlist" style = "height:0.2441em;" > < span > < / span > < / span > < / span > < / span > < / span > < / span > < span class = "mopen" > (< / span > < span class = "mord mathnormal" style = "margin-right:0.07847em;" > X< / span > < span class = "mclose" > )< / span > < span class = "mspace" style = "margin-right:0.2222em;" > < / span > < span class = "mbin" > +< / span > < span class = "mspace" style = "margin-right:0.2222em;" > < / span > < span class = "minner" > ⋯< / span > < span class = "mspace" style = "margin-right:0.2222em;" > < / span > < span class = "mbin" > +< / span > < span class = "mspace" style = "margin-right:0.2222em;" > < / span > < span class = "mord" > < span class = "mord mathnormal" style = "margin-right:0.03588em;" > y< / span > < span class = "msupsub" > < span class = "vlist-t" > < span class = "vlist-r" > < span class = "vlist" style = "height:0.8247em;" > < span style = "top:-3.063em;margin-right:0.05em;" > < span class = "pstrut" style = "height:2.7em;" > < / span > < span class = "sizing reset-size6 size3 mtight" > < span class = "mord mathnormal mtight" > i< / span > < / span > < / span > < / span > < / span > < / span > < / span > < / span > < span class = "mspace" style = "margin-right:0.2222em;" > < / span > < span class = "mbin" > ⋅< / span > < span class = "mspace" style = "margin-right:0.2222em;" > < / span > < span class = "mord" > < span class = "mord text" > < span class = "mord" > gate< / span > < / span > < span class = "msupsub" > < span class = "vlist-t vlist-t2" > < span class = "vlist-r" > < span class = "vlist"
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< p > where the numerator is a random (the prover commits to the cell assignments before the
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verifier samples < span class = "katex" > < span class = "katex-html" aria-hidden = "true" > < span class = "base" > < span class = "strut" style = "height:0.625em;vertical-align:-0.1944em;" > < / span > < span class = "mord mathnormal" style = "margin-right:0.03588em;" > y< / span > < / span > < / span > < / span > ) linear combination of the circuit relations.< / p >
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< ul >
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< li > If the numerator polynomial (in formal indeterminate < span class = "katex" > < span class = "katex-html" aria-hidden = "true" > < span class = "base" > < span class = "strut" style = "height:0.6833em;" > < / span > < span class = "mord mathnormal" style = "margin-right:0.07847em;" > X< / span > < / span > < / span > < / span > ) is perfectly divisible by
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< span class = "katex" > < span class = "katex-html" aria-hidden = "true" > < span class = "base" > < span class = "strut" style = "height:1em;vertical-align:-0.25em;" > < / span > < span class = "mord mathnormal" > t< / span > < span class = "mopen" > (< / span > < span class = "mord mathnormal" style = "margin-right:0.07847em;" > X< / span > < span class = "mclose" > )< / span > < / span > < / span > < / span > , then with high probability all relations are satisfied.< / li >
< li > Conversely, if at least one relation is not satisfied, then with high probability
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< span class = "katex" > < span class = "katex-html" aria-hidden = "true" > < span class = "base" > < span class = "strut" style = "height:1em;vertical-align:-0.25em;" > < / span > < span class = "mord mathnormal" > h< / span > < span class = "mopen" > (< / span > < span class = "mord mathnormal" > x< / span > < span class = "mclose" > )< / span > < span class = "mspace" style = "margin-right:0.2222em;" > < / span > < span class = "mbin" > ⋅< / span > < span class = "mspace" style = "margin-right:0.2222em;" > < / span > < / span > < span class = "base" > < span class = "strut" style = "height:1em;vertical-align:-0.25em;" > < / span > < span class = "mord mathnormal" > t< / span > < span class = "mopen" > (< / span > < span class = "mord mathnormal" > x< / span > < span class = "mclose" > )< / span > < / span > < / span > < / span > will not equal the evaluation of the numerator at < span class = "katex" > < span class = "katex-html" aria-hidden = "true" > < span class = "base" > < span class = "strut" style = "height:0.4306em;" > < / span > < span class = "mord mathnormal" > x< / span > < / span > < / span > < / span > . In this case,
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the numerator polynomial would not be perfectly divisible by < span class = "katex" > < span class = "katex-html" aria-hidden = "true" > < span class = "base" > < span class = "strut" style = "height:1em;vertical-align:-0.25em;" > < / span > < span class = "mord mathnormal" > t< / span > < span class = "mopen" > (< / span > < span class = "mord mathnormal" style = "margin-right:0.07847em;" > X< / span > < span class = "mclose" > )< / span > < / span > < / span > < / span > .< / li >
< / ul >
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< h2 id = "committing-to-hx" > < a class = "header" href = "#committing-to-hx" > Committing to < span class = "katex" > < span class = "katex-html" aria-hidden = "true" > < span class = "base" > < span class = "strut" style = "height:1em;vertical-align:-0.25em;" > < / span > < span class = "mord mathnormal" > h< / span > < span class = "mopen" > (< / span > < span class = "mord mathnormal" style = "margin-right:0.07847em;" > X< / span > < span class = "mclose" > )< / span > < / span > < / span > < / span > < / a > < / h2 >
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< p > < span class = "katex" > < span class = "katex-html" aria-hidden = "true" > < span class = "base" > < span class = "strut" style = "height:1em;vertical-align:-0.25em;" > < / span > < span class = "mord mathnormal" > h< / span > < span class = "mopen" > (< / span > < span class = "mord mathnormal" style = "margin-right:0.07847em;" > X< / span > < span class = "mclose" > )< / span > < / span > < / span > < / span > has degree < span class = "katex" > < span class = "katex-html" aria-hidden = "true" > < span class = "base" > < span class = "strut" style = "height:1em;vertical-align:-0.25em;" > < / span > < span class = "mord mathnormal" > d< / span > < span class = "mopen" > (< / span > < span class = "mord mathnormal" > n< / span > < span class = "mspace" style = "margin-right:0.2222em;" > < / span > < span class = "mbin" > − < / span > < span class = "mspace" style = "margin-right:0.2222em;" > < / span > < / span > < span class = "base" > < span class = "strut" style = "height:1em;vertical-align:-0.25em;" > < / span > < span class = "mord" > 1< / span > < span class = "mclose" > )< / span > < span class = "mspace" style = "margin-right:0.2222em;" > < / span > < span class = "mbin" > − < / span > < span class = "mspace" style = "margin-right:0.2222em;" > < / span > < / span > < span class = "base" > < span class = "strut" style = "height:0.4306em;" > < / span > < span class = "mord mathnormal" > n< / span > < / span > < / span > < / span > (because the divisor < span class = "katex" > < span class = "katex-html" aria-hidden = "true" > < span class = "base" > < span class = "strut" style = "height:1em;vertical-align:-0.25em;" > < / span > < span class = "mord mathnormal" > t< / span > < span class = "mopen" > (< / span > < span class = "mord mathnormal" style = "margin-right:0.07847em;" > X< / span > < span class = "mclose" > )< / span > < / span > < / span > < / span > has degree < span class = "katex" > < span class = "katex-html" aria-hidden = "true" > < span class = "base" > < span class = "strut" style = "height:0.4306em;" > < / span > < span class = "mord mathnormal" > n< / span > < / span > < / span > < / span > ). However, the
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polynomial commitment scheme we use for Halo 2 only supports committing to polynomials of
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degree < span class = "katex" > < span class = "katex-html" aria-hidden = "true" > < span class = "base" > < span class = "strut" style = "height:0.6667em;vertical-align:-0.0833em;" > < / span > < span class = "mord mathnormal" > n< / span > < span class = "mspace" style = "margin-right:0.2222em;" > < / span > < span class = "mbin" > − < / span > < span class = "mspace" style = "margin-right:0.2222em;" > < / span > < / span > < span class = "base" > < span class = "strut" style = "height:0.6444em;" > < / span > < span class = "mord" > 1< / span > < / span > < / span > < / span > (which is the maximum degree that the rest of the protocol needs to commit
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to). Instead of increasing the cost of the polynomial commitment scheme, the prover split
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< span class = "katex" > < span class = "katex-html" aria-hidden = "true" > < span class = "base" > < span class = "strut" style = "height:1em;vertical-align:-0.25em;" > < / span > < span class = "mord mathnormal" > h< / span > < span class = "mopen" > (< / span > < span class = "mord mathnormal" style = "margin-right:0.07847em;" > X< / span > < span class = "mclose" > )< / span > < / span > < / span > < / span > into pieces of degree < span class = "katex" > < span class = "katex-html" aria-hidden = "true" > < span class = "base" > < span class = "strut" style = "height:0.6667em;vertical-align:-0.0833em;" > < / span > < span class = "mord mathnormal" > n< / span > < span class = "mspace" style = "margin-right:0.2222em;" > < / span > < span class = "mbin" > − < / span > < span class = "mspace" style = "margin-right:0.2222em;" > < / span > < / span > < span class = "base" > < span class = "strut" style = "height:0.6444em;" > < / span > < span class = "mord" > 1< / span > < / span > < / span > < / span > < / p >
< p > < span class = "katex-display" > < span class = "katex" > < span class = "katex-html" aria-hidden = "true" > < span class = "base" > < span class = "strut" style = "height:1em;vertical-align:-0.25em;" > < / span > < span class = "mord" > < span class = "mord mathnormal" > h< / span > < span class = "msupsub" > < span class = "vlist-t vlist-t2" > < span class = "vlist-r" > < span class = "vlist" style = "height:0.3011em;" > < span style = "top:-2.55em;margin-left:0em;margin-right:0.05em;" > < span class = "pstrut" style = "height:2.7em;" > < / span > < span class = "sizing reset-size6 size3 mtight" > < span class = "mord mtight" > 0< / span > < / span > < / span > < / span > < span class = "vlist-s" > < / span > < / span > < span class = "vlist-r" > < span class = "vlist" style = "height:0.15em;" > < span > < / span > < / span > < / span > < / span > < / span > < / span > < span class = "mopen" > (< / span > < span class = "mord mathnormal" style = "margin-right:0.07847em;" > X< / span > < span class = "mclose" > )< / span > < span class = "mspace" style = "margin-right:0.2222em;" > < / span > < span class = "mbin" > +< / span > < span class = "mspace" style = "margin-right:0.2222em;" > < / span > < / span > < span class = "base" > < span class = "strut" style = "height:1em;vertical-align:-0.25em;" > < / span > < span class = "mord" > < span class = "mord mathnormal" style = "margin-right:0.07847em;" > X< / span > < span class = "msupsub" > < span class = "vlist-t" > < span class = "vlist-r" > < span class = "vlist" style = "height:0.7144em;" > < span style = "top:-3.113em;margin-right:0.05em;" > < span class = "pstrut" style = "height:2.7em;" > < / span > < span class = "sizing reset-size6 size3 mtight" > < span class = "mord mathnormal mtight" > n< / span > < / span > < / span > < / span > < / span > < / span > < / span > < / span > < span class = "mord" > < span class = "mord mathnormal" > h< / span > < span class = "msupsub" > < span class = "vlist-t vlist-t2" > < span class = "vlist-r" > < span class = "vlist" style = "height:0.3011em;" > < span style = "top:-2.55em;margin-left:0em;margin-right:0.05em;" > < span class = "pstrut" style = "height:2.7em;" > < / span > < span class = "sizing reset-size6 size3 mtight" > < span class = "mord mtight" > 1< / span > < / span > < / span > < / span > < span class = "vlist-s" > < / span > < / span > < span class = "vlist-r" > < span class = "vlist" style = "height:0.15em;" > < span > < / span > < / span > < / span > < / span > < / span > < / span > < span class = "mopen" > (< / span > < span class = "mord mathnormal" style = "margin-right:0.07847em;" > X< / span > < span class = "mclose" > )< / span > < span class = "mspace" style = "margin-right:0.2222em;" > < / span > < span class = "mbin" > +< / span > < span class = "mspace" style = "margin-right:0.2222em;" > < / span > < / span > < span class = "base" > < span class = "strut" style = "height:0.6667em;vertical-align:-0.0833em;" > < / span > < span class = "minner" > ⋯< / span > < span class = "mspace" style = "margin-right:0.2222em;" > < / span > < span class = "mbin" > +< / span > < span class = "mspace" style = "margin-right:0.2222em;" > < / span > < / span > < span class = "base" > < span class = "strut" style = "height:1.188em;vertical-align:-0.25em;" > < / span > < span class = "mord" > < span class = "mord mathnormal" style = "margin-right:0.07847em;" > X< / span > < span class = "msupsub" > < span class = "vlist-t" > < span class = "vlist-r" > < span class = "vlist" style = "height:0.938em;" > < span style = "top:-3.113em;margin-right:0.05em;" > < span class = "pstrut" style = "height:2.7em;" > < / span > < span class = "sizing reset-size6 size3 mtight" > < span class = "mord mtight" > < span class = "mord mathnormal mtight" > n< / span > < span class = "mopen mtight" > (< / span > < span class = "mord mathnormal mtight" > d< / span > < span class = "mbin mtight" > − < / span > < span class = "mord mtight" > 1< / span > < span class = "mclose mtight" > )< / span > < / span > < / span > < / span > < / span > < / span > < / span > < / span > < / span > < span class = "mord" > < span class = "mord mathnormal" > h< / span > < span class = "msupsub" > < span class = "vlist-t vlist-t2" > < span class = "vlist-r" > < span class = "vlist" style = "height:0.3361em;" > < span style = "top:-2.55em;margin-left:0em;margin-right:0.05em;" > < span class = "pstrut" style = "height:2.7em;" > < / span > < span class = "sizing reset-size6 size3 mtight" > < span class = "mord mtight" > < span class = "mord mathnormal mtight" > d< / span > < span class = "mbin mtight" > − < / span > < span class = "mord mtight" > 1< / span > < / span > < / span > < / span > < / span > < span class = "vlist-s" > < / span > < / span > < span class = "vlist-r" > < span class = "vlist" style = "height:0.2083em;" > < span > < / span > < / span > < / span > < / span > < / span > < / span > < span class = "mopen" > (< / span > < span class = "mord mathnor
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< p > and produces blinding commitments to each piece< / p >
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< p > < span class = "katex-display" > < span class = "katex" > < span class = "katex-html" aria-hidden = "true" > < span class = "base" > < span class = "strut" style = "height:0.6861em;" > < / span > < span class = "mord mathbf" > H< / span > < span class = "mspace" style = "margin-right:0.2778em;" > < / span > < span class = "mrel" > =< / span > < span class = "mspace" style = "margin-right:0.2778em;" > < / span > < / span > < span class = "base" > < span class = "strut" style = "height:1em;vertical-align:-0.25em;" > < / span > < span class = "mopen" > [< / span > < span class = "mord text" > < span class = "mord" > Commit< / span > < / span > < span class = "mopen" > (< / span > < span class = "mord" > < span class = "mord mathnormal" > h< / span > < span class = "msupsub" > < span class = "vlist-t vlist-t2" > < span class = "vlist-r" > < span class = "vlist" style = "height:0.3011em;" > < span style = "top:-2.55em;margin-left:0em;margin-right:0.05em;" > < span class = "pstrut" style = "height:2.7em;" > < / span > < span class = "sizing reset-size6 size3 mtight" > < span class = "mord mtight" > 0< / span > < / span > < / span > < / span > < span class = "vlist-s" > < / span > < / span > < span class = "vlist-r" > < span class = "vlist" style = "height:0.15em;" > < span > < / span > < / span > < / span > < / span > < / span > < / span > < span class = "mopen" > (< / span > < span class = "mord mathnormal" style = "margin-right:0.07847em;" > X< / span > < span class = "mclose" > ))< / span > < span class = "mpunct" > ,< / span > < span class = "mspace" style = "margin-right:0.1667em;" > < / span > < span class = "mord text" > < span class = "mord" > Commit< / span > < / span > < span class = "mopen" > (< / span > < span class = "mord" > < span class = "mord mathnormal" > h< / span > < span class = "msupsub" > < span class = "vlist-t vlist-t2" > < span class = "vlist-r" > < span class = "vlist" style = "height:0.3011em;" > < span style = "top:-2.55em;margin-left:0em;margin-right:0.05em;" > < span class = "pstrut" style = "height:2.7em;" > < / span > < span class = "sizing reset-size6 size3 mtight" > < span class = "mord mtight" > 1< / span > < / span > < / span > < / span > < span class = "vlist-s" > < / span > < / span > < span class = "vlist-r" > < span class = "vlist" style = "height:0.15em;" > < span > < / span > < / span > < / span > < / span > < / span > < / span > < span class = "mopen" > (< / span > < span class = "mord mathnormal" style = "margin-right:0.07847em;" > X< / span > < span class = "mclose" > ))< / span > < span class = "mpunct" > ,< / span > < span class = "mspace" style = "margin-right:0.1667em;" > < / span > < span class = "minner" > …< / span > < span class = "mspace" style = "margin-right:0.1667em;" > < / span > < span class = "mpunct" > ,< / span > < span class = "mspace" style = "margin-right:0.1667em;" > < / span > < span class = "mord text" > < span class = "mord" > Commit< / span > < / span > < span class = "mopen" > (< / span > < span class = "mord" > < span class = "mord mathnormal" > h< / span > < span class = "msupsub" > < span class = "vlist-t vlist-t2" > < span class = "vlist-r" > < span class = "vlist" style = "height:0.3361em;" > < span style = "top:-2.55em;margin-left:0em;margin-right:0.05em;" > < span class = "pstrut" style = "height:2.7em;" > < / span > < span class = "sizing reset-size6 size3 mtight" > < span class = "mord mtight" > < span class = "mord mathnormal mtight" > d< / span > < span class = "mbin mtight" > − < / span > < span class = "mord mtight" > 1< / span > < / span > < / span > < / span > < / span > < span class = "vlist-s" > < / span > < / span > < span class = "vlist-r" > < span class = "vlist" style = "height:0.2083em;" > < span > < / span > < / span > < / span > < / span > < / span > < / span > < span class = "mopen" > (< / span > < span class = "mord mathnormal" style = "margin-right:0.07847em;" > X< / span > < span class = "mclose" > ))]< / span > < span class = "mord" > .< / span > < / span > < / span > < / span > < / span > < / p >
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< h2 id = "evaluating-the-polynomials" > < a class = "header" href = "#evaluating-the-polynomials" > Evaluating the polynomials< / a > < / h2 >
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< p > At this point, we have committed to all properties of the circuit. The verifier now
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wants to see if the prover committed to the correct < span class = "katex" > < span class = "katex-html" aria-hidden = "true" > < span class = "base" > < span class = "strut" style = "height:1em;vertical-align:-0.25em;" > < / span > < span class = "mord mathnormal" > h< / span > < span class = "mopen" > (< / span > < span class = "mord mathnormal" style = "margin-right:0.07847em;" > X< / span > < span class = "mclose" > )< / span > < / span > < / span > < / span > polynomial. The verifier
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samples < span class = "katex" > < span class = "katex-html" aria-hidden = "true" > < span class = "base" > < span class = "strut" style = "height:0.4306em;" > < / span > < span class = "mord mathnormal" > x< / span > < / span > < / span > < / span > , and the prover produces the purported evaluations of the various polynomials
at < span class = "katex" > < span class = "katex-html" aria-hidden = "true" > < span class = "base" > < span class = "strut" style = "height:0.4306em;" > < / span > < span class = "mord mathnormal" > x< / span > < / span > < / span > < / span > , for all the relative offsets used in the circuit, as well as < span class = "katex" > < span class = "katex-html" aria-hidden = "true" > < span class = "base" > < span class = "strut" style = "height:1em;vertical-align:-0.25em;" > < / span > < span class = "mord mathnormal" > h< / span > < span class = "mopen" > (< / span > < span class = "mord mathnormal" style = "margin-right:0.07847em;" > X< / span > < span class = "mclose" > )< / span > < / span > < / span > < / span > .< / p >
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< blockquote >
< p > In our < a href = "../proving-system.html#example" > example< / a > , this would be:< / p >
< ul >
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< li > < span class = "katex" > < span class = "katex-html" aria-hidden = "true" > < span class = "base" > < span class = "strut" style = "height:1em;vertical-align:-0.25em;" > < / span > < span class = "mord" > < span class = "mord mathnormal" > a< / span > < span class = "msupsub" > < span class = "vlist-t vlist-t2" > < span class = "vlist-r" > < span class = "vlist" style = "height:0.3011em;" > < span style = "top:-2.55em;margin-left:0em;margin-right:0.05em;" > < span class = "pstrut" style = "height:2.7em;" > < / span > < span class = "sizing reset-size6 size3 mtight" > < span class = "mord mtight" > 0< / span > < / span > < / span > < / span > < span class = "vlist-s" > < / span > < / span > < span class = "vlist-r" > < span class = "vlist" style = "height:0.15em;" > < span > < / span > < / span > < / span > < / span > < / span > < / span > < span class = "mopen" > (< / span > < span class = "mord mathnormal" > x< / span > < span class = "mclose" > )< / span > < / span > < / span > < / span > < / li >
< li > < span class = "katex" > < span class = "katex-html" aria-hidden = "true" > < span class = "base" > < span class = "strut" style = "height:1em;vertical-align:-0.25em;" > < / span > < span class = "mord" > < span class = "mord mathnormal" > a< / span > < span class = "msupsub" > < span class = "vlist-t vlist-t2" > < span class = "vlist-r" > < span class = "vlist" style = "height:0.3011em;" > < span style = "top:-2.55em;margin-left:0em;margin-right:0.05em;" > < span class = "pstrut" style = "height:2.7em;" > < / span > < span class = "sizing reset-size6 size3 mtight" > < span class = "mord mtight" > 1< / span > < / span > < / span > < / span > < span class = "vlist-s" > < / span > < / span > < span class = "vlist-r" > < span class = "vlist" style = "height:0.15em;" > < span > < / span > < / span > < / span > < / span > < / span > < / span > < span class = "mopen" > (< / span > < span class = "mord mathnormal" > x< / span > < span class = "mclose" > )< / span > < / span > < / span > < / span > < / li >
< li > < span class = "katex" > < span class = "katex-html" aria-hidden = "true" > < span class = "base" > < span class = "strut" style = "height:1em;vertical-align:-0.25em;" > < / span > < span class = "mord" > < span class = "mord mathnormal" > a< / span > < span class = "msupsub" > < span class = "vlist-t vlist-t2" > < span class = "vlist-r" > < span class = "vlist" style = "height:0.3011em;" > < span style = "top:-2.55em;margin-left:0em;margin-right:0.05em;" > < span class = "pstrut" style = "height:2.7em;" > < / span > < span class = "sizing reset-size6 size3 mtight" > < span class = "mord mtight" > 2< / span > < / span > < / span > < / span > < span class = "vlist-s" > < / span > < / span > < span class = "vlist-r" > < span class = "vlist" style = "height:0.15em;" > < span > < / span > < / span > < / span > < / span > < / span > < / span > < span class = "mopen" > (< / span > < span class = "mord mathnormal" > x< / span > < span class = "mclose" > )< / span > < / span > < / span > < / span > , < span class = "katex" > < span class = "katex-html" aria-hidden = "true" > < span class = "base" > < span class = "strut" style = "height:1.0641em;vertical-align:-0.25em;" > < / span > < span class = "mord" > < span class = "mord mathnormal" > a< / span > < span class = "msupsub" > < span class = "vlist-t vlist-t2" > < span class = "vlist-r" > < span class = "vlist" style = "height:0.3011em;" > < span style = "top:-2.55em;margin-left:0em;margin-right:0.05em;" > < span class = "pstrut" style = "height:2.7em;" > < / span > < span class = "sizing reset-size6 size3 mtight" > < span class = "mord mtight" > 2< / span > < / span > < / span > < / span > < span class = "vlist-s" > < / span > < / span > < span class = "vlist-r" > < span class = "vlist" style = "height:0.15em;" > < span > < / span > < / span > < / span > < / span > < / span > < / span > < span class = "mopen" > (< / span > < span class = "mord mathnormal" > x< / span > < span class = "mord" > < span class = "mord mathnormal" style = "margin-right:0.03588em;" > ω< / span > < span class = "msupsub" > < span class = "vlist-t" > < span class = "vlist-r" > < span class = "vlist" style = "height:0.8141em;" > < span style = "top:-3.063em;margin-right:0.05em;" > < span class = "pstrut" style = "height:2.7em;" > < / span > < span class = "sizing reset-size6 size3 mtight" > < span class = "mord mtight" > < span class = "mord mtight" > − < / span > < span class = "mord mtight" > 1< / span > < / span > < / span > < / span > < / span > < / span > < / span > < / span > < / span > < span class = "mclose" > )< / span > < / span > < / span > < / span > < / li >
< li > < span class = "katex" > < span class = "katex-html" aria-hidden = "true" > < span class = "base" > < span class = "strut" style = "height:1em;vertical-align:-0.25em;" > < / span > < span class = "mord" > < span class = "mord mathnormal" > a< / span > < span class = "msupsub" > < span class = "vlist-t vlist-t2" > < span class = "vlist-r" > < span class = "vlist" style = "height:0.3011em;" > < span style = "top:-2.55em;margin-left:0em;margin-right:0.05em;" > < span class = "pstrut" style = "height:2.7em;" > < / span > < span class = "sizing reset-size6 size3 mtight" > < span class = "mord mtight" > 3< / span > < / span > < / span > < / span > < span class = "vlist-s" > < / span > < / span > < span class = "vlist-r" > < span class = "vlist" style = "height:0.15em;" > < span > < / span > < / span > < / span > < / span > < / span > < / span > < span class = "mopen" > (< / span > < span class = "mord mathnormal" > x< / span > < span class = "mclose" > )< / span > < / span > < / span > < / span > < / li >
< li > < span class = "katex" > < span class = "katex-html" aria-hidden = "true" > < span class = "base" > < span class = "strut" style = "height:1em;vertical-align:-0.25em;" > < / span > < span class = "mord" > < span class = "mord mathnormal" style = "margin-right:0.10764em;" > f< / span > < span class = "msupsub" > < span class = "vlist-t vlist-t2" > < span class = "vlist-r" > < span class = "vlist" style = "height:0.3011em;" > < span style = "top:-2.55em;margin-left:-0.1076em;margin-right:0.05em;" > < span class = "pstrut" style = "height:2.7em;" > < / span > < span class = "sizing reset-size6 size3 mtight" > < span class = "mord mtight" > 0< / span > < / span > < / span > < / span > < span class = "vlist-s" > < / span > < / span > < span class = "vlist-r" > < span class = "vlist" style = "height:0.15em;" > < span > < / span > < / span > < / span > < / span > < / span > < / span > < span class = "mopen" > (< / span > < span class = "mord mathnormal" > x< / span > < span class = "mclose" > )< / span > < / span > < / span > < / span > , < span class = "katex" > < span class = "katex-html" aria-hidden = "true" > < span class = "base" > < span class = "strut" style = "height:1.0641em;vertical-align:-0.25em;" > < / span > < span class = "mord" > < span class = "mord mathnormal" style = "margin-right:0.10764em;" > f< / span > < span class = "msupsub" > < span class = "vlist-t vlist-t2" > < span class = "vlist-r" > < span class = "vlist" style = "height:0.3011em;" > < span style = "top:-2.55em;margin-left:-0.1076em;margin-right:0.05em;" > < span class = "pstrut" style = "height:2.7em;" > < / span > < span class = "sizing reset-size6 size3 mtight" > < span class = "mord mtight" > 0< / span > < / span > < / span > < / span > < span class = "vlist-s" > < / span > < / span > < span class = "vlist-r" > < span class = "vlist" style = "height:0.15em;" > < span > < / span > < / span > < / span > < / span > < / span > < / span > < span class = "mopen" > (< / span > < span class = "mord mathnormal" > x< / span > < span class = "mord" > < span class = "mord mathnormal" style = "margin-right:0.03588em;" > ω< / span > < span class = "msupsub" > < span class = "vlist-t" > < span class = "vlist-r" > < span class = "vlist" style = "height:0.8141em;" > < span style = "top:-3.063em;margin-right:0.05em;" > < span class = "pstrut" style = "height:2.7em;" > < / span > < span class = "sizing reset-size6 size3 mtight" > < span class = "mord mtight" > < span class = "mord mtight" > − < / span > < span class = "mord mtight" > 1< / span > < / span > < / span > < / span > < / span > < / span > < / span > < / span > < / span > < span class = "mclose" > )< / span > < / span > < / span > < / span > < / li >
< li > < span class = "katex" > < span class = "katex-html" aria-hidden = "true" > < span class = "base" > < span class = "strut" style = "height:1em;vertical-align:-0.25em;" > < / span > < span class = "mord" > < span class = "mord mathnormal" > h< / span > < span class = "msupsub" > < span class = "vlist-t vlist-t2" > < span class = "vlist-r" > < span class = "vlist" style = "height:0.3011em;" > < span style = "top:-2.55em;margin-left:0em;margin-right:0.05em;" > < span class = "pstrut" style = "height:2.7em;" > < / span > < span class = "sizing reset-size6 size3 mtight" > < span class = "mord mtight" > 0< / span > < / span > < / span > < / span > < span class = "vlist-s" > < / span > < / span > < span class = "vlist-r" > < span class = "vlist" style = "height:0.15em;" > < span > < / span > < / span > < / span > < / span > < / span > < / span > < span class = "mopen" > (< / span > < span class = "mord mathnormal" > x< / span > < span class = "mclose" > )< / span > < / span > < / span > < / span > , ..., < span class = "katex" > < span class = "katex-html" aria-hidden = "true" > < span class = "base" > < span class = "strut" style = "height:1em;vertical-align:-0.25em;" > < / span > < span class = "mord" > < span class = "mord mathnormal" > h< / span > < span class = "msupsub" > < span class = "vlist-t vlist-t2" > < span class = "vlist-r" > < span class = "vlist" style = "height:0.3361em;" > < span style = "top:-2.55em;margin-left:0em;margin-right:0.05em;" > < span class = "pstrut" style = "height:2.7em;" > < / span > < span class = "sizing reset-size6 size3 mtight" > < span class = "mord mtight" > < span class = "mord mathnormal mtight" > d< / span > < span class = "mbin mtight" > − < / span > < span class = "mord mtight" > 1< / span > < / span > < / span > < / span > < / span > < span class = "vlist-s" > < / span > < / span > < span class = "vlist-r" > < span class = "vlist" style = "height:0.2083em;" > < span > < / span > < / span > < / span > < / span > < / span > < / span > < span class = "mopen" > (< / span > < span class = "mord mathnormal" > x< / span > < span class = "mclose" > )< / span > < / span > < / span > < / span > < / li >
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< / ul >
< / blockquote >
< p > The verifier checks that these evaluations satisfy the form of < span class = "katex" > < span class = "katex-html" aria-hidden = "true" > < span class = "base" > < span class = "strut" style = "height:1em;vertical-align:-0.25em;" > < / span > < span class = "mord mathnormal" > h< / span > < span class = "mopen" > (< / span > < span class = "mord mathnormal" style = "margin-right:0.07847em;" > X< / span > < span class = "mclose" > )< / span > < / span > < / span > < / span > :< / p >
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< p > < span class = "katex-display" > < span class = "katex" > < span class = "katex-html" aria-hidden = "true" > < span class = "base" > < span class = "strut" style = "height:2.4377em;vertical-align:-0.936em;" > < / span > < span class = "mord" > < span class = "mopen nulldelimiter" > < / span > < span class = "mfrac" > < span class = "vlist-t vlist-t2" > < span class = "vlist-r" > < span class = "vlist" style = "height:1.5017em;" > < span style = "top:-2.314em;" > < span class = "pstrut" style = "height:3em;" > < / span > < span class = "mord" > < span class = "mord mathnormal" > t< / span > < span class = "mopen" > (< / span > < span class = "mord mathnormal" > x< / span > < span class = "mclose" > )< / span > < / span > < / span > < span style = "top:-3.23em;" > < span class = "pstrut" style = "height:3em;" > < / span > < span class = "frac-line" style = "border-bottom-width:0.04em;" > < / span > < / span > < span style = "top:-3.677em;" > < span class = "pstrut" style = "height:3em;" > < / span > < span class = "mord" > < span class = "mord" > < span class = "mord text" > < span class = "mord" > gate< / span > < / span > < span class = "msupsub" > < span class = "vlist-t vlist-t2" > < span class = "vlist-r" > < span class = "vlist" style = "height:0.207em;" > < span style = "top:-2.4559em;margin-right:0.05em;" > < span class = "pstrut" style = "height:2.7em;" > < / span > < span class = "sizing reset-size6 size3 mtight" > < span class = "mord mtight" > 0< / span > < / span > < / span > < / span > < span class = "vlist-s" > < / span > < / span > < span class = "vlist-r" > < span class = "vlist" style = "height:0.2441em;" > < span > < / span > < / span > < / span > < / span > < / span > < / span > < span class = "mopen" > (< / span > < span class = "mord mathnormal" > x< / span > < span class = "mclose" > )< / span > < span class = "mspace" style = "margin-right:0.2222em;" > < / span > < span class = "mbin" > +< / span > < span class = "mspace" style = "margin-right:0.2222em;" > < / span > < span class = "minner" > ⋯< / span > < span class = "mspace" style = "margin-right:0.2222em;" > < / span > < span class = "mbin" > +< / span > < span class = "mspace" style = "margin-right:0.2222em;" > < / span > < span class = "mord" > < span class = "mord mathnormal" style = "margin-right:0.03588em;" > y< / span > < span class = "msupsub" > < span class = "vlist-t" > < span class = "vlist-r" > < span class = "vlist" style = "height:0.8247em;" > < span style = "top:-3.063em;margin-right:0.05em;" > < span class = "pstrut" style = "height:2.7em;" > < / span > < span class = "sizing reset-size6 size3 mtight" > < span class = "mord mathnormal mtight" > i< / span > < / span > < / span > < / span > < / span > < / span > < / span > < / span > < span class = "mspace" style = "margin-right:0.2222em;" > < / span > < span class = "mbin" > ⋅< / span > < span class = "mspace" style = "margin-right:0.2222em;" > < / span > < span class = "mord" > < span class = "mord text" > < span class = "mord" > gate< / span > < / span > < span class = "msupsub" > < span class = "vlist-t vlist-t2" > < span class = "vlist-r" > < span class = "vlist" style = "height:0.2175em;" > < span style = "top:-2.4559em;margin-right:0.05em;" > < span class = "pstrut" style = "height:2.7em;" > < / span > < span class = "sizing reset-size6 size3 mtight" > < span class = "mord mathnormal mtight" > i< / span > < / span > < / span > < / span > < span class = "vlist-s" > < / span > < / span > < span class = "vlist-r" > < span class = "vlist" style = "height:0.2441em;" > < span > < / span > < / span > < / span > < / span > < / span > < / span > < span class = "mopen" > (< / span > < span class = "mord mathnormal" > x< / span > < span class = "mclose" > )< / span > < span class = "mspace" style = "margin-right:0.2222em;" > < / span > < span class = "mbin" > +< / span > < span class = "mspace" style = "margin-right:0.2222em;" > < / span > < span class = "minner" > …< / span > < / span > < / span > < / span > < span class = "vlist-s" > < / span > < / span > < span class = "vlist-r" > < span class = "vlist" style = "height:0.936em;" > < span > < / span > < / span > < / span > < / span > < / span > < span class = "mclose nulldelimiter" > < / span > < / span > < span class = "mspace" style = "margin-right:0.2778em;" > < / span > < span class = "mrel" > =< / span > < span class = "mspace" style = "margin-right:0.2778em;" > < / span > < / span > < span class = "base" > < span class = "strut" style = "height:1em;vertical-align:-0.25em;" > < / span > < span class = "mord" > < span class = "mord mathnormal" > h< / span > < span class = "msupsub" > < span class = "vlist-t vlist-t2" > < span class = "vlist-r" > < span class = "vlist" style = "height:0.3011em;" > < span style = "top:-2.55em;margin-left:0em;margin-right:0.05em;" > < span class = "pstrut" style = "height:2.7em;" > < / span > < span class = "sizing reset-size6 size3 mtight" > < span class = "mord mtight" > 0< / span > < / span > < / span > < / span > < sp
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< p > Now content that the evaluations collectively satisfy the gate constraints, the verifier
needs to check that the evaluations themselves are consistent with the original
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< a href = "circuit-commitments.html" > circuit commitments< / a > , as well as < span class = "katex" > < span class = "katex-html" aria-hidden = "true" > < span class = "base" > < span class = "strut" style = "height:0.6861em;" > < / span > < span class = "mord mathbf" > H< / span > < / span > < / span > < / span > . To implement this
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efficiently, we use a < a href = "multipoint-opening.html" > multipoint opening argument< / a > .< / p >
< / main >
< nav class = "nav-wrapper" aria-label = "Page navigation" >
<!-- Mobile navigation buttons -->
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< a rel = "prev" href = "../../design/proving-system/circuit-commitments.html" class = "mobile-nav-chapters previous" title = "Previous chapter" aria-label = "Previous chapter" aria-keyshortcuts = "Left" >
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< i class = "fa fa-angle-left" > < / i >
< / a >
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< a rel = "next prefetch" href = "../../design/proving-system/multipoint-opening.html" class = "mobile-nav-chapters next" title = "Next chapter" aria-label = "Next chapter" aria-keyshortcuts = "Right" >
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< i class = "fa fa-angle-right" > < / i >
< / a >
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< div style = "clear: both" > < / div >
< / nav >
< / div >
< / div >
< nav class = "nav-wide-wrapper" aria-label = "Page navigation" >
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< a rel = "prev" href = "../../design/proving-system/circuit-commitments.html" class = "nav-chapters previous" title = "Previous chapter" aria-label = "Previous chapter" aria-keyshortcuts = "Left" >
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< i class = "fa fa-angle-left" > < / i >
< / a >
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< a rel = "next prefetch" href = "../../design/proving-system/multipoint-opening.html" class = "nav-chapters next" title = "Next chapter" aria-label = "Next chapter" aria-keyshortcuts = "Right" >
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< i class = "fa fa-angle-right" > < / i >
< / a >
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< / nav >
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< / div >
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< script >
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window.playground_copyable = true;
< / script >
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< script src = "../../elasticlunr.min.js" > < / script >
< script src = "../../mark.min.js" > < / script >
< script src = "../../searcher.js" > < / script >
< script src = "../../clipboard.min.js" > < / script >
< script src = "../../highlight.js" > < / script >
< script src = "../../book.js" > < / script >
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<!-- Custom JS scripts -->
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< / div >
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< / body >
< / html >
