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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" > < 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/implementation/fields.ht
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< h1 id = "elliptic-curves" > < a class = "header" href = "#elliptic-curves" > Elliptic Curves< / a > < / h1 >
< h2 id = "eccchip" > < a class = "header" href = "#eccchip" > < code > EccChip< / code > < / a > < / h2 >
< p > < code > halo2_gadgets< / code > provides a chip that implements < code > EccInstructions< / code > using 10 advice columns.
The chip is currently restricted to the Pallas curve, but will be extended to support the
< a href = "https://github.com/zcash/halo2/issues/578" > Vesta curve< / a > in the near future.< / p >
< h3 id = "chip-assumptions" > < a class = "header" href = "#chip-assumptions" > Chip assumptions< / a > < / h3 >
< p > A non-exhaustive list of assumptions made by < code > EccChip< / code > :< / 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:0.6444em;" > < / span > < span class = "mord" > 0< / span > < / span > < / span > < / span > is not an < 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 > -coordinate of a valid point on the curve.
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< ul >
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< li > Holds for Pallas because < span class = "katex" > < span class = "katex-html" aria-hidden = "true" > < span class = "base" > < span class = "strut" style = "height:0.6444em;" > < / span > < span class = "mord" > 5< / span > < / span > < / span > < / span > is not square in < span class = "katex" > < span class = "katex-html" aria-hidden = "true" > < span class = "base" > < span class = "strut" style = "height:0.975em;vertical-align:-0.2861em;" > < / span > < span class = "mord" > < span class = "mord mathbb" > F< / span > < span class = "msupsub" > < span class = "vlist-t vlist-t2" > < span class = "vlist-r" > < span class = "vlist" style = "height:0.1514em;" > < 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 mathnormal mtight" style = "margin-right:0.03588em;" > q< / span > < / span > < / span > < / span > < span class = "vlist-s" > < / span > < / span > < span class = "vlist-r" > < span class = "vlist" style = "height:0.2861em;" > < span > < / span > < / span > < / span > < / span > < / span > < / span > < / span > < / span > < / span > .< / li >
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< / ul >
< / li >
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< li > < span class = "katex" > < span class = "katex-html" aria-hidden = "true" > < span class = "base" > < span class = "strut" style = "height:0.6444em;" > < / span > < span class = "mord" > 0< / span > < / span > < / span > < / span > is not a < 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 > -coordinate of a valid point on the curve.
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< ul >
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< li > Holds for Pallas because < 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" > − < / span > < span class = "mord" > 5< / span > < / span > < / span > < / span > is not a cube in < span class = "katex" > < span class = "katex-html" aria-hidden = "true" > < span class = "base" > < span class = "strut" style = "height:0.975em;vertical-align:-0.2861em;" > < / span > < span class = "mord" > < span class = "mord mathbb" > F< / span > < span class = "msupsub" > < span class = "vlist-t vlist-t2" > < span class = "vlist-r" > < span class = "vlist" style = "height:0.1514em;" > < 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 mathnormal mtight" style = "margin-right:0.03588em;" > q< / span > < / span > < / span > < / span > < span class = "vlist-s" > < / span > < / span > < span class = "vlist-r" > < span class = "vlist" style = "height:0.2861em;" > < span > < / span > < / span > < / span > < / span > < / span > < / span > < / span > < / span > < / span > .< / li >
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< / ul >
< / li >
< / ul >
< h3 id = "layout" > < a class = "header" href = "#layout" > Layout< / a > < / h3 >
< p > The following table shows how columns are used by the gates for various chip sub-areas:< / 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:0.6833em;" > < / span > < span class = "mord mathnormal" style = "margin-right:0.13889em;" > W< / span > < / span > < / span > < / span > - witnessing points.< / li >
< li > < 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" > A< / span > < span class = "mord mathnormal" style = "margin-right:0.07847em;" > I< / span > < / span > < / span > < / span > - incomplete point addition.< / li >
< li > < 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" > A< / span > < span class = "mord mathnormal" style = "margin-right:0.07153em;" > C< / span > < / span > < / span > < / span > - complete point addition.< / li >
< li > < 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:0.13889em;" > MF< / span > < / span > < / span > < / span > - Fixed-base scalar multiplication.< / li >
< li > < 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:0.10903em;" > M< / span > < span class = "mord mathnormal" style = "margin-right:0.22222em;" > V< / span > < span class = "mord mathnormal" style = "margin-right:0.07847em;" > I< / span > < / span > < / span > < / span > - variable-base scalar multiplication, incomplete rounds.< / li >
< li > < 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:0.10903em;" > M< / span > < span class = "mord mathnormal" style = "margin-right:0.22222em;" > V< / span > < span class = "mord mathnormal" style = "margin-right:0.07153em;" > C< / span > < / span > < / span > < / span > - variable-base scalar multiplication, complete rounds.< / li >
< li > < 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:0.10903em;" > M< / span > < span class = "mord mathnormal" style = "margin-right:0.22222em;" > V< / span > < span class = "mord mathnormal" style = "margin-right:0.02778em;" > O< / span > < / span > < / span > < / span > - variable-base scalar multiplication, overflow check.< / li >
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