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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 > < / ol > < / li > < li class = "chapter-item expanded " > < a href = "../design.html" > < strong aria-hidden = "true" > 3.< / strong > Design< / a > < / li > < li > < ol class = "section" > < li class = "chapter-item expanded " > < a href = "../design/proving-system.html" class = "active" > < strong aria-hidden = "true" > 3.1.< / 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" > 3.1.1.< / strong > Lookup argument< / a > < / li > < li class = "chapter-item expanded " > < a href = "../design/proving-system/permutation.html" > < strong aria-hidden = "true" > 3.1.2.< / strong > Permutation argument< / a > < / li > < li class = "chapter-item expanded " > < a href = "../design/proving-system/circuit-commitments.html" > < strong aria-hidden = "true" > 3.1.3.< / strong > Circuit commitments< / a > < / li > < li class = "chapter-item expanded " > < a href = "../design/proving-system/vanishing.html" > < strong aria-hidden = "true" > 3.1.4.< / strong > Vanishing argument< / a > < / li > < li class = "chapter-item expanded " > < a href = "../design/proving-system/multipoint-opening.html" > < strong aria-hidden = "true" > 3.1.5.< / strong > Multipoint opening argument< / a > < / li > < li class = "chapter-item expanded " > < a href = "../design/proving-system/inner-product.html" > < strong aria-hidden = "true" > 3.1.6.< / strong > Inner product argument< / a > < / li > < li class = "chapter-item expanded " > < a href = "../design/proving-system/comparison.html" > < strong aria-hidden = "true" > 3.1.7.< / strong > Comparison to other work< / a > < / li > < / ol > < / li > < li class = "chapter-item expanded " > < a href = "../design/protocol.html" > < strong aria-hidden = "true" > 3.2.< / strong > Protocol Description< / a > < / li > < li class = "chapter-item expanded " > < a href = "../design/implementation.html" > < strong aria-hidden = "true" > 3.3.< / strong > Implementation< / a > < / li > < li > < ol class = "section" > < li class = "chapter-item expanded " > < a href = "../design/implementation/proofs.html" > < strong aria-hidden = "true" > 3.3.1.< / strong > Proofs< / a > < / li > < li class = "chapter-item expanded " > < a href = "../design/implementation/fields.html" > < strong aria-hidden = "true" > 3.3.2.< / strong > Fields< / a > < / li > < / ol > < / li > < li class = "chapter-item expanded " > < a href = "../design/gadgets.html" > < strong aria-hidden = "true" > 3.4.< / strong > Gadgets< / a > < / li > < li > < ol class = "section" > < li class = "chapter-item expanded " > < a href = "../design/gadgets/ecc.html" > < strong aria-hidden = "true" > 3.4.1.< / strong > Elliptic curve cryptography< / a > < / li > < li > < ol class = "section" > < li class = "chapter-item expanded " > < a href = "../design/gadgets/ecc/addition.html" > < strong aria-hidd
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< h1 class = "menu-title" > The halo2 Book< / h1 >
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< h1 id = "proving-system" > < a class = "header" href = "#proving-system" > Proving system< / a > < / h1 >
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< p > The Halo 2 proving system can be broken down into five stages:< / p >
< ol >
< li > Commit to polynomials encoding the main components of the circuit:
< ul >
< li > Cell assignments.< / li >
< li > Permuted values and products for each lookup argument.< / li >
< li > Equality constraint permutations.< / li >
< / ul >
< / li >
< li > Construct the vanishing argument to constrain all circuit relations to zero:
< ul >
< li > Standard and custom gates.< / li >
< li > Lookup argument rules.< / li >
< li > Equality constraint permutation rules.< / li >
< / ul >
< / li >
< li > Evaluate the above polynomials at all necessary points:
< ul >
< li > All relative rotations used by custom gates across all columns.< / li >
< li > Vanishing argument pieces.< / li >
< / ul >
< / li >
< li > Construct the multipoint opening argument to check that all evaluations are consistent
with their respective commitments.< / li >
< li > Run the inner product argument to create a polynomial commitment opening proof for the
multipoint opening argument polynomial.< / li >
< / ol >
< p > These stages are presented in turn across this section of the book.< / p >
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< h2 id = "example" > < a class = "header" href = "#example" > Example< / a > < / h2 >
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< p > To aid our explanations, we will at times refer to the following example constraint
system:< / p >
< ul >
< li > Four advice columns < span class = "katex" > < span class = "katex-html" aria-hidden = "true" > < span class = "base" > < span class = "strut" style = "height:0.8888799999999999em;vertical-align:-0.19444em;" > < / span > < span class = "mord mathnormal" > a< / span > < span class = "mpunct" > ,< / span > < span class = "mspace" style = "margin-right:0.16666666666666666em;" > < / span > < span class = "mord mathnormal" > b< / span > < span class = "mpunct" > ,< / span > < span class = "mspace" style = "margin-right:0.16666666666666666em;" > < / span > < span class = "mord mathnormal" > c< / span > < span class = "mpunct" > ,< / span > < span class = "mspace" style = "margin-right:0.16666666666666666em;" > < / span > < span class = "mord mathnormal" > d< / span > < / span > < / span > < / span > .< / li >
< li > One fixed column < span class = "katex" > < span class = "katex-html" aria-hidden = "true" > < span class = "base" > < span class = "strut" style = "height:0.8888799999999999em;vertical-align:-0.19444em;" > < / span > < span class = "mord mathnormal" style = "margin-right:0.10764em;" > f< / span > < / span > < / span > < / span > .< / li >
< li > Three custom gates:
< ul >
< li > < span class = "katex" > < span class = "katex-html" aria-hidden = "true" > < span class = "base" > < span class = "strut" style = "height:0.44445em;vertical-align:0em;" > < / span > < span class = "mord mathnormal" > a< / span > < span class = "mspace" style = "margin-right:0.2222222222222222em;" > < / span > < span class = "mbin" > ⋅< / span > < span class = "mspace" style = "margin-right:0.2222222222222222em;" > < / span > < / span > < span class = "base" > < span class = "strut" style = "height:0.69444em;vertical-align:0em;" > < / span > < span class = "mord mathnormal" > b< / span > < span class = "mspace" style = "margin-right:0.2222222222222222em;" > < / span > < span class = "mbin" > ⋅< / span > < span class = "mspace" style = "margin-right:0.2222222222222222em;" > < / span > < / span > < span class = "base" > < span class = "strut" style = "height:0.791661em;vertical-align:-0.208331em;" > < / span > < span class = "mord" > < span class = "mord mathnormal" > c< / span > < span class = "msupsub" > < span class = "vlist-t vlist-t2" > < span class = "vlist-r" > < span class = "vlist" style = "height:0.301108em;" > < span style = "top:-2.5500000000000003em;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 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:0.208331em;" > < span > < / span > < / span > < / span > < / span > < / span > < / span > < span class = "mspace" style = "margin-right:0.2222222222222222em;" > < / span > < span class = "mbin" > − < / span > < span class = "mspace" style = "margin-right:0.2222222222222222em;" > < / span > < / span > < span class = "base" > < span class = "strut" style = "height:0.69444em;vertical-align:0em;" > < / span > < span class = "mord mathnormal" > d< / span > < span class = "mspace" style = "margin-right:0.2777777777777778em;" > < / span > < span class = "mrel" > =< / span > < span class = "mspace" style = "margin-right:0.2777777777777778em;" > < / span > < / span > < span class = "base" > < span class = "strut" style = "height:0.64444em;vertical-align:0em;" > < / span > < span class = "mord" > 0< / span > < / span > < / span > < / span > < / li >
< li > < span class = "katex" > < span class = "katex-html" aria-hidden = "true" > < span class = "base" > < span class = "strut" style = "height:0.902771em;vertical-align:-0.208331em;" > < / 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:0.301108em;" > < span style = "top:-2.5500000000000003em;margin-left:-0.10764em;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 class = "vlist-s" > < / span > < / span > < span class = "vlist-r" > < span class = "vlist" style = "height:0.208331em;" > < span > < / span > < / span > < / span > < / span > < / span > < / span > < span class = "mspace" style = "margin-right:0.2222222222222222em;" > < / span > < span class = "mbin" > ⋅< / span > < span class = "mspace" style = "margin-right:0.2222222222222222em;" > < / span > < / span > < span class = "base" > < span class = "strut" style = "height:0.43056em;vertical-align:0em;" > < / span > < span class = "mord mathnormal" > c< / span > < span class = "mspace" style = "margin-right:0.2777777777777778em;" > < / span > < span class = "mrel" > =< / span > < span class = "mspace" style = "margin-right:0.2777777777777778em;" > < / span > < / span > < span class = "base" > < span class = "strut" style = "height:0.64444em;vertical-align:0em;" > < / span > < span class = "mord" > 0< / span > < / span > < / span > < / span > < / li >
< li > < span class = "katex" > < span class = "katex-html" aria-hidden = "true" > < span class = "base" > < span class = "strut" style = "height:0.8888799999999999em;vertical-align:-0.19444em;" > < / span > < span class = "mord mathnormal" style = "margin-right:0.10764em;" > f< / span > < span class = "mspace" style = "margin-right:0.2222222222222222em;" > < / span > < span class = "mbin" > ⋅< / span > < span class = "mspace" style = "margin-right:0.2222222222222222em;" > < / span > < / span > < span class = "base" > < span class = "strut" style = "height:0.69444em;vertical-align:0em;" > < / span > < span class = "mord mathnormal" > d< / span > < span class = "mspace" style = "margin-right:0.2222222222222222em;" > < / span > < span class = "mbin" > ⋅< / span > < span class = "mspace" style = "margin-right:0.2222222222222222em;" > < / span > < / span > < span class = "base" > < span class = "strut" style = "height:0.43056em;vertical-align:0em;" > < / span > < span class = "mord mathnormal" > a< / span > < span class = "mspace" style = "margin-right:0.2777777777777778em;" > < / span > < span class = "mrel" > =< / span > < span class = "mspace" style = "margin-right:0.2777777777777778em;" > < / span > < / span > < span class = "base" > < span class = "strut" style = "height:0.64444em;vertical-align:0em;" > < / span > < span class = "mord" > 0< / span > < / span > < / span > < / span > < / li >
< / ul >
< / li >
< / ul >
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< h2 id = "tldr" > < a class = "header" href = "#tldr" > tl;dr< / a > < / h2 >
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< p > The table below provides a (probably too) succinct description of the Halo 2 protocol.
This description will likely be replaced by the Halo 2 paper and security proof, but for
now serves as a summary of the following sub-sections.< / p >
< table > < thead > < tr > < th > Prover< / th > < th > < / th > < th > Verifier< / th > < / tr > < / thead > < tbody >
< tr > < td > < / td > < td > < span class = "katex" > < span class = "katex-html" aria-hidden = "true" > < span class = "base" > < span class = "strut" style = "height:0.36687em;vertical-align:0em;" > < / span > < span class = "mrel" > ←< / span > < / span > < / span > < / span > < / td > < td > < 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:0.2777777777777778em;" > < / span > < span class = "mrel" > =< / span > < span class = "mspace" style = "margin-right:0.2777777777777778em;" > < / 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:0.664392em;" > < 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:0.2222222222222222em;" > < / span > < span class = "mbin" > − < / span > < span class = "mspace" style = "margin-right:0.2222222222222222em;" > < / 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 > < / td > < / tr >
< tr > < td > < / td > < td > < span class = "katex" > < span class = "katex-html" aria-hidden = "true" > < span class = "base" > < span class = "strut" style = "height:0.36687em;vertical-align:0em;" > < / span > < span class = "mrel" > ←< / span > < / span > < / span > < / span > < / td > < td > < span class = "katex" > < span class = "katex-html" aria-hidden = "true" > < span class = "base" > < span class = "strut" style = "height:0.68333em;vertical-align:0em;" > < / span > < span class = "mord mathnormal" style = "margin-right:0.13889em;" > F< / span > < span class = "mspace" style = "margin-right:0.2777777777777778em;" > < / span > < span class = "mrel" > =< / span > < span class = "mspace" style = "margin-right:0.2777777777777778em;" > < / 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:0.13889em;" > F< / span > < span class = "msupsub" > < span class = "vlist-t vlist-t2" > < span class = "vlist-r" > < span class = "vlist" style = "height:0.30110799999999993em;" > < span style = "top:-2.5500000000000003em;margin-left:-0.13889em;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 = "mpunct" > ,< / span > < span class = "mspace" style = "margin-right:0.16666666666666666em;" > < / span > < span class = "mord" > < span class = "mord mathnormal" style = "margin-right:0.13889em;" > F< / span > < span class = "msupsub" > < span class = "vlist-t vlist-t2" > < span class = "vlist-r" > < span class = "vlist" style = "height:0.30110799999999993em;" > < span style = "top:-2.5500000000000003em;margin-left:-0.13889em;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 = "mpunct" > ,< / span > < span class = "mspace" style = "margin-right:0.16666666666666666em;" > < / span > < span class = "minner" > …< / span > < span class = "mspace" style = "margin-right:0.16666666666666666em;" > < / span > < span class = "mpunct" > ,< / span > < span class = "mspace" style = "margin-right:0.16666666666666666em;" > < / span > < span class = "mord" > < span class = "mord mathnormal" style = "margin-right:0.13889em;" > F< / span > < span class = "msupsub" > < span class = "vlist-t vlist-t2" > < span class = "vlist-r" > < span class = "vlist" style = "height:0.301108em;" > < span style = "top:-2.5500000000000003em;margin-left:-0.13889em;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" > m< / 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:0.208331em;" > < span > < / span > < / span > < / span > < / span > < / span > < / span > < span class = "mclose" > ]< / span > < / span > < / span > < / span > < / td > < / tr >
2021-05-03 15:55:00 -07:00
< tr > < td > < span class = "katex" > < span class = "katex-html" aria-hidden = "true" > < span class = "base" > < span class = "strut" style = "height:0.68611em;vertical-align:0em;" > < / span > < span class = "mord mathbf" > A< / span > < span class = "mspace" style = "margin-right:0.2777777777777778em;" > < / span > < span class = "mrel" > =< / span > < span class = "mspace" style = "margin-right:0.2777777777777778em;" > < / 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" > A< / span > < span class = "msupsub" > < span class = "vlist-t vlist-t2" > < span class = "vlist-r" > < span class = "vlist" style = "height:0.30110799999999993em;" > < span style = "top:-2.5500000000000003em;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 = "mpunct" > ,< / span > < span class = "mspace" style = "margin-right:0.16666666666666666em;" > < / 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:0.30110799999999993em;" > < span style = "top:-2.5500000000000003em;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 = "mpunct" > ,< / span > < span class = "mspace" style = "margin-right:0.16666666666666666em;" > < / span > < span class = "minner" > …< / span > < span class = "mspace" style = "margin-right:0.16666666666666666em;" > < / span > < span class = "mpunct" > ,< / span > < span class = "mspace" style = "margin-right:0.16666666666666666em;" > < / 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:0.301108em;" > < span style = "top:-2.5500000000000003em;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" > m< / 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:0.208331em;" > < span > < / span > < / span > < / span > < / span > < / span > < / span > < span class = "mclose" > ]< / span > < / span > < / span > < / span > < / td > < td > < span class = "katex" > < span class = "katex-html" aria-hidden = "true" > < span class = "base" > < span class = "strut" style = "height:0.36687em;vertical-align:0em;" > < / span > < span class = "mrel" > →< / span > < / span > < / span > < / span > < / td > < td > < / td > < / tr >
2021-02-12 08:17:18 -08:00
< tr > < td > < / td > < td > < span class = "katex" > < span class = "katex-html" aria-hidden = "true" > < span class = "base" > < span class = "strut" style = "height:0.36687em;vertical-align:0em;" > < / span > < span class = "mrel" > ←< / span > < / span > < / span > < / span > < / td > < td > < span class = "katex" > < span class = "katex-html" aria-hidden = "true" > < span class = "base" > < span class = "strut" style = "height:0.69444em;vertical-align:0em;" > < / span > < span class = "mord mathnormal" style = "margin-right:0.02778em;" > θ< / span > < / span > < / span > < / span > < / td > < / tr >
2021-05-03 15:55:00 -07:00
< tr > < td > < span class = "katex" > < span class = "katex-html" aria-hidden = "true" > < span class = "base" > < span class = "strut" style = "height:0.68611em;vertical-align:0em;" > < / span > < span class = "mord mathbf" > L< / span > < span class = "mspace" style = "margin-right:0.2777777777777778em;" > < / span > < span class = "mrel" > =< / span > < span class = "mspace" style = "margin-right:0.2777777777777778em;" > < / span > < / span > < span class = "base" > < span class = "strut" style = "height:1.058331em;vertical-align:-0.30643899999999996em;" > < / span > < span class = "mopen" > [(< / 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:0.751892em;" > < span style = "top:-2.4518920000000004em;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 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 > < / span > < / span > < / span > < span class = "vlist-s" > < / span > < / span > < span class = "vlist-r" > < span class = "vlist" style = "height:0.24810799999999997em;" > < span > < / span > < / span > < / span > < / span > < / span > < / span > < span class = "mpunct" > ,< / span > < span class = "mspace" style = "margin-right:0.16666666666666666em;" > < / span > < span class = "mord" > < span class = "mord mathnormal" style = "margin-right:0.05764em;" > S< / span > < span class = "msupsub" > < span class = "vlist-t vlist-t2" > < span class = "vlist-r" > < span class = "vlist" style = "height:0.751892em;" > < span style = "top:-2.4518920000000004em;margin-left:-0.05764em;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 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 > < / span > < / span > < / span > < span class = "vlist-s" > < / span > < / span > < span class = "vlist-r" > < span class = "vlist" style = "height:0.24810799999999997em;" > < span > < / span > < / span > < / span > < / span > < / span > < / span > < span class = "mclose" > )< / span > < span class = "mpunct" > ,< / span > < span class = "mspace" style = "margin-right:0.16666666666666666em;" > < / span > < span class = "minner" > …< / span > < span class = "mspace" style = "margin-right:0.16666666666666666em;" > < / span > < span class = "mpunct" > ,< / span > < span class = "mspace" style = "margin-right:0.16666666666666666em;" > < / span > < span class = "mopen" > (< / 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:0.751892em;" > < span style = "top:-2.451892em;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" > m< / span > < span class = "mbin mtight" > − < / span > < span class = "mord mtight" > 1< / span > < / span > < / span > < / span > < 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 > < / span > < / span > < / span > < span class = "vlist-s" > < / span > < / span > < span class = "vlist-r" > < span class = "vlist" style = "height:0.30643899999999996em;" > < span > < / span > < / span > < / span > < / span > < / span > < / span > < span class = "mpunct" > ,< / span > < span class = "mspace" style = "margin-right:0.16666666666666666em;" > < / span > < span class = "mord" > < span class = "mord mathnormal" style = "margin-right:0.05764em;" > S< / span > < span class = "msupsub" > < span class = "vlist-t vlist-t2" > < span class = "vlist-r" > < span class = "vlist" style = "height:0.751892em;" > < span style = "top:-2.451892em;margin-left:-0.05764em;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" > m< / span > < span class = "mbin mt
2021-02-12 08:17:18 -08:00
< tr > < td > < / td > < td > < span class = "katex" > < span class = "katex-html" aria-hidden = "true" > < span class = "base" > < span class = "strut" style = "height:0.36687em;vertical-align:0em;" > < / span > < span class = "mrel" > ←< / span > < / span > < / span > < / span > < / td > < td > < span class = "katex" > < span class = "katex-html" aria-hidden = "true" > < span class = "base" > < span class = "strut" style = "height:0.8888799999999999em;vertical-align:-0.19444em;" > < / span > < span class = "mord mathnormal" style = "margin-right:0.05278em;" > β< / span > < span class = "mpunct" > ,< / span > < span class = "mspace" style = "margin-right:0.16666666666666666em;" > < / span > < span class = "mord mathnormal" style = "margin-right:0.05556em;" > γ < / span > < / span > < / span > < / span > < / td > < / tr >
2021-05-03 15:55:00 -07:00
< tr > < td > < span class = "katex" > < span class = "katex-html" aria-hidden = "true" > < span class = "base" > < span class = "strut" style = "height:0.83611em;vertical-align:-0.15em;" > < / span > < span class = "mord" > < span class = "mord mathbf" > Z< / span > < span class = "msupsub" > < span class = "vlist-t vlist-t2" > < span class = "vlist-r" > < span class = "vlist" style = "height:0.33027699999999993em;" > < span style = "top:-2.5500000000000003em;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 mathbf mtight" > P< / 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 = "mspace" style = "margin-right:0.2777777777777778em;" > < / span > < span class = "mrel" > =< / span > < span class = "mspace" style = "margin-right:0.2777777777777778em;" > < / span > < / span > < span class = "base" > < span class = "strut" style = "height:1.036108em;vertical-align:-0.286108em;" > < / span > < span class = "mopen" > [< / span > < span class = "mord" > < span class = "mord mathnormal" style = "margin-right:0.07153em;" > Z< / span > < span class = "msupsub" > < span class = "vlist-t vlist-t2" > < span class = "vlist-r" > < span class = "vlist" style = "height:0.328331em;" > < span style = "top:-2.5500000000000003em;margin-left:-0.07153em;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" style = "margin-right:0.13889em;" > P< / span > < span class = "mpunct mtight" > ,< / span > < span class = "mord mtight" > 0< / span > < / span > < / span > < / span > < / span > < span class = "vlist-s" > < / span > < / span > < span class = "vlist-r" > < span class = "vlist" style = "height:0.286108em;" > < span > < / span > < / span > < / span > < / span > < / span > < / span > < span class = "mpunct" > ,< / span > < span class = "mspace" style = "margin-right:0.16666666666666666em;" > < / span > < span class = "mord" > < span class = "mord mathnormal" style = "margin-right:0.07153em;" > Z< / span > < span class = "msupsub" > < span class = "vlist-t vlist-t2" > < span class = "vlist-r" > < span class = "vlist" style = "height:0.328331em;" > < span style = "top:-2.5500000000000003em;margin-left:-0.07153em;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" style = "margin-right:0.13889em;" > P< / span > < span class = "mpunct 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:0.286108em;" > < span > < / span > < / span > < / span > < / span > < / span > < / span > < span class = "mpunct" > ,< / span > < span class = "mspace" style = "margin-right:0.16666666666666666em;" > < / span > < span class = "minner" > …< / span > < span class = "mclose" > ]< / span > < / span > < / span > < / span > < / td > < td > < span class = "katex" > < span class = "katex-html" aria-hidden = "true" > < span class = "base" > < span class = "strut" style = "height:0.36687em;vertical-align:0em;" > < / span > < span class = "mrel" > →< / span > < / span > < / span > < / span > < / td > < td > < / td > < / tr >
< tr > < td > < span class = "katex" > < span class = "katex-html" aria-hidden = "true" > < span class = "base" > < span class = "strut" style = "height:0.83611em;vertical-align:-0.15em;" > < / span > < span class = "mord" > < span class = "mord mathbf" > Z< / span > < span class = "msupsub" > < span class = "vlist-t vlist-t2" > < span class = "vlist-r" > < span class = "vlist" style = "height:0.33027699999999993em;" > < span style = "top:-2.5500000000000003em;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 mathbf mtight" > L< / 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 = "mspace" style = "margin-right:0.2777777777777778em;" > < / span > < span class = "mrel" > =< / span > < span class = "mspace" style = "margin-right:0.2777777777777778em;" > < / span > < / span > < span class = "base" > < span class = "strut" style = "height:1.036108em;vertical-align:-0.286108em;" > < / span > < span class = "mopen" > [< / span > < span class = "mord" > < span class = "mord mathnormal" style = "margin-right:0.07153em;" > Z< / span > < span class = "msupsub" > < span class = "vlist-t vlist-t2" > < span class = "vlist-r" > < span class = "vlist" style = "height:0.328331em;" > < span style = "top:-2.5500000000000003em;margin-left:-0.07153em;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" > L< / span > < span class = "mpunct mtight" > ,< / span > < span class = "mord mtight" > 0< / span > < / span > < / span > < / span > < / span > < span class = "vlist-s" > < / span > < / span > < span class = "vlist-r" > < span class = "vlist" style = "height:0.286108em;" > < span > < / span > < / span > < / span > < / span > < / span > < / span > < span class = "mpunct" > ,< / span > < span class = "mspace" style = "margin-right:0.16666666666666666em;" > < / span > < span class = "mord" > < span class = "mord mathnormal" style = "margin-right:0.07153em;" > Z< / span > < span class = "msupsub" > < span class = "vlist-t vlist-t2" > < span class = "vlist-r" > < span class = "vlist" style = "height:0.328331em;" > < span style = "top:-2.5500000000000003em;margin-left:-0.07153em;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" > L< / span > < span class = "mpunct 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:0.286108em;" > < span > < / span > < / span > < / span > < / span > < / span > < / span > < span class = "mpunct" > ,< / span > < span class = "mspace" style = "margin-right:0.16666666666666666em;" > < / span > < span class = "minner" > …< / span > < span class = "mclose" > ]< / span > < / span > < / span > < / span > < / td > < td > < span class = "katex" > < span class = "katex-html" aria-hidden = "true" > < span class = "base" > < span class = "strut" style = "height:0.36687em;vertical-align:0em;" > < / span > < span class = "mrel" > →< / span > < / span > < / span > < / span > < / td > < td > < / td > < / tr >
2021-02-12 08:17:18 -08:00
< tr > < td > < / td > < td > < span class = "katex" > < span class = "katex-html" aria-hidden = "true" > < span class = "base" > < span class = "strut" style = "height:0.36687em;vertical-align:0em;" > < / span > < span class = "mrel" > ←< / span > < / span > < / span > < / span > < / td > < td > < span class = "katex" > < span class = "katex-html" aria-hidden = "true" > < span class = "base" > < span class = "strut" style = "height:0.625em;vertical-align:-0.19444em;" > < / span > < span class = "mord mathnormal" style = "margin-right:0.03588em;" > y< / span > < / span > < / span > < / span > < / td > < / tr >
< tr > < td > < 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:0.2777777777777778em;" > < / span > < span class = "mrel" > =< / span > < span class = "mspace" style = "margin-right:0.2777777777777778em;" > < / span > < / span > < span class = "base" > < span class = "strut" style = "height:1.647568em;vertical-align:-0.52em;" > < / 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:1.127568em;" > < span style = "top:-2.655em;" > < span class = "pstrut" style = "height:3em;" > < / span > < span class = "sizing reset-size6 size3 mtight" > < span class = "mord mtight" > < span class = "mord mathnormal mtight" > t< / span > < span class = "mopen mtight" > (< / span > < span class = "mord mathnormal mtight" style = "margin-right:0.07847em;" > X< / span > < span class = "mclose mtight" > )< / span > < / 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:-3.496108em;" > < span class = "pstrut" style = "height:3em;" > < / span > < span class = "sizing reset-size6 size3 mtight" > < span class = "mord mtight" > < span class = "mord mtight" > < span class = "mord text mtight" > < span class = "mord mtight" > gate< / span > < / span > < span class = "msupsub" > < span class = "vlist-t vlist-t2" > < span class = "vlist-r" > < span class = "vlist" style = "height:0.19444571428571428em;" > < span style = "top:-2.2341314285714287em;margin-right:0.07142857142857144em;" > < span class = "pstrut" style = "height:2.5em;" > < / span > < span class = "sizing reset-size3 size1 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:0.26586857142857145em;" > < span > < / span > < / span > < / span > < / span > < / span > < / span > < span class = "mopen mtight" > (< / span > < span class = "mord mathnormal mtight" style = "margin-right:0.07847em;" > X< / span > < span class = "mclose mtight" > )< / span > < span class = "mbin mtight" > +< / span > < span class = "minner mtight" > ⋯< / span > < span class = "mbin mtight" > +< / span > < span class = "mord mtight" > < span class = "mord mathnormal mtight" style = "margin-right:0.03588em;" > y< / span > < span class = "msupsub" > < span class = "vlist-t" > < span class = "vlist-r" > < span class = "vlist" style = "height:0.9020857142857143em;" > < span style = "top:-2.931em;margin-right:0.07142857142857144em;" > < span class = "pstrut" style = "height:2.5em;" > < / span > < span class = "sizing reset-size3 size1 mtight" > < span class = "mord mathnormal mtight" > i< / span > < / span > < / span > < / span > < / span > < / span > < / span > < / span > < span class = "mbin mtight" > ⋅< / span > < span class = "mord mtight" > < span class = "mord text mtight" > < span class = "mord mtight" > gate< / span > < / span > < span class = "msupsub" > < span class = "vlist-t vlist-t2" > < span class = "vlist-r" > < span class = "vlist" style = "height:0.20521714285714282em;" > < span style = "top:-2.2341314285714287em;margin-right:0.07142857142857144em;" > < span class = "pstrut" style = "height:2.5em;" > < / span > < span class = "sizing reset-size3 size1 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:0.26586857142857145em;" > < span > < / span > < / span > < / span > < / span > < / span > < / span > < span class = "mopen mtight" > (< / span > < span class = "mord mathnormal mtight" style = "margin-right:0.07847em;" > X< / span > < span class = "mclose mtight" > )< / span > < / span > < / span > < / span > < / span > < span class = "vlist-s" > < / span > < / span > < span class = "vlist-r" > < span class = "vlist" style = "height:0.52em;" > < span > < / span > < / span > < / span > < / span > < / span > < span class = "mclose nulldelimiter" > < / span > < / span > < / span > < / span > < / span > < / td > < td > < / td > < td > < / td > < / tr >
< tr > < td > < 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:0.2777777777777778em;" > < / span > < span class = "mrel" > =< / span > < span class = "mspace" style = "margin-right:0.2777777777777778em;" > < / 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:0.30110799999999993em;" > < span style = "top:-2.5500000000000003em;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:0.2222222222222222em;" > < / span > < span class = "mbin" > +< / span > < span class = "mspace" style = "margin-right:0.2222222222222222em;" > < / span > < / span > < span class = "base" > < span class = "strut" style = "height:0.66666em;vertical-align:-0.08333em;" > < / span > < span class = "minner" > ⋯< / span > < span class = "mspace" style = "margin-right:0.2222222222222222em;" > < / span > < span class = "mbin" > +< / span > < span class = "mspace" style = "margin-right:0.2222222222222222em;" > < / span > < / span > < span class = "base" > < span class = "strut" style = "height:1.138em;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:0.8879999999999999em;" > < 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 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:0.3361079999999999em;" > < span style = "top:-2.5500000000000003em;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:0.208331em;" > < 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 > < / td > < td > < / td > < td > < / td > < / tr >
2021-05-03 15:55:00 -07:00
< tr > < td > < span class = "katex" > < span class = "katex-html" aria-hidden = "true" > < span class = "base" > < span class = "strut" style = "height:0.68611em;vertical-align:0em;" > < / span > < span class = "mord mathbf" > H< / span > < span class = "mspace" style = "margin-right:0.2777777777777778em;" > < / span > < span class = "mrel" > =< / span > < span class = "mspace" style = "margin-right:0.2777777777777778em;" > < / 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:0.08125em;" > H< / span > < span class = "msupsub" > < span class = "vlist-t vlist-t2" > < span class = "vlist-r" > < span class = "vlist" style = "height:0.30110799999999993em;" > < span style = "top:-2.5500000000000003em;margin-left:-0.08125em;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 = "mpunct" > ,< / span > < span class = "mspace" style = "margin-right:0.16666666666666666em;" > < / span > < span class = "mord" > < span class = "mord mathnormal" style = "margin-right:0.08125em;" > H< / span > < span class = "msupsub" > < span class = "vlist-t vlist-t2" > < span class = "vlist-r" > < span class = "vlist" style = "height:0.30110799999999993em;" > < span style = "top:-2.5500000000000003em;margin-left:-0.08125em;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 = "mpunct" > ,< / span > < span class = "mspace" style = "margin-right:0.16666666666666666em;" > < / span > < span class = "minner" > …< / span > < span class = "mspace" style = "margin-right:0.16666666666666666em;" > < / span > < span class = "mpunct" > ,< / span > < span class = "mspace" style = "margin-right:0.16666666666666666em;" > < / span > < span class = "mord" > < span class = "mord mathnormal" style = "margin-right:0.08125em;" > H< / span > < span class = "msupsub" > < span class = "vlist-t vlist-t2" > < span class = "vlist-r" > < span class = "vlist" style = "height:0.3361079999999999em;" > < span style = "top:-2.5500000000000003em;margin-left:-0.08125em;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:0.208331em;" > < span > < / span > < / span > < / span > < / span > < / span > < / span > < span class = "mclose" > ]< / span > < / span > < / span > < / span > < / td > < td > < span class = "katex" > < span class = "katex-html" aria-hidden = "true" > < span class = "base" > < span class = "strut" style = "height:0.36687em;vertical-align:0em;" > < / span > < span class = "mrel" > →< / span > < / span > < / span > < / span > < / td > < td > < / td > < / tr >
2021-02-12 08:17:18 -08:00
< tr > < td > < / td > < td > < span class = "katex" > < span class = "katex-html" aria-hidden = "true" > < span class = "base" > < span class = "strut" style = "height:0.36687em;vertical-align:0em;" > < / span > < span class = "mrel" > ←< / span > < / span > < / span > < / span > < / td > < td > < span class = "katex" > < span class = "katex-html" aria-hidden = "true" > < span class = "base" > < span class = "strut" style = "height:0.43056em;vertical-align:0em;" > < / span > < span class = "mord mathnormal" > x< / span > < / span > < / span > < / span > < / td > < / tr >
2021-05-03 15:55:00 -07:00
< tr > < td > < span class = "katex" > < span class = "katex-html" aria-hidden = "true" > < span class = "base" > < span class = "strut" style = "height:0.69444em;vertical-align:0em;" > < / span > < span class = "mord mathnormal" > e< / span > < span class = "mord mathnormal" style = "margin-right:0.03588em;" > v< / span > < span class = "mord mathnormal" > a< / span > < span class = "mord mathnormal" style = "margin-right:0.01968em;" > l< / span > < span class = "mord mathnormal" > s< / span > < span class = "mspace" style = "margin-right:0.2777777777777778em;" > < / span > < span class = "mrel" > =< / span > < span class = "mspace" style = "margin-right:0.2777777777777778em;" > < / 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" > A< / span > < span class = "msupsub" > < span class = "vlist-t vlist-t2" > < span class = "vlist-r" > < span class = "vlist" style = "height:0.30110799999999993em;" > < span style = "top:-2.5500000000000003em;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 class = "mpunct" > ,< / span > < span class = "mspace" style = "margin-right:0.16666666666666666em;" > < / span > < span class = "minner" > …< / span > < span class = "mspace" style = "margin-right:0.16666666666666666em;" > < / span > < span class = "mpunct" > ,< / span > < span class = "mspace" style = "margin-right:0.16666666666666666em;" > < / span > < span class = "mord" > < span class = "mord mathnormal" style = "margin-right:0.08125em;" > H< / span > < span class = "msupsub" > < span class = "vlist-t vlist-t2" > < span class = "vlist-r" > < span class = "vlist" style = "height:0.3361079999999999em;" > < span style = "top:-2.5500000000000003em;margin-left:-0.08125em;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:0.208331em;" > < span > < / span > < / span > < / span > < / span > < / span > < / span > < span class = "mopen" > (< / span > < span class = "mord mathnormal" > x< / span > < span class = "mclose" > )]< / span > < / span > < / span > < / span > < / td > < td > < span class = "katex" > < span class = "katex-html" aria-hidden = "true" > < span class = "base" > < span class = "strut" style = "height:0.36687em;vertical-align:0em;" > < / span > < span class = "mrel" > →< / span > < / span > < / span > < / span > < / td > < td > < / td > < / tr >
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< tr > < td > < / td > < td > < / td > < td > Checks < 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 > < / span > < / span > < / td > < / tr >
< tr > < td > < / td > < td > < span class = "katex" > < span class = "katex-html" aria-hidden = "true" > < span class = "base" > < span class = "strut" style = "height:0.36687em;vertical-align:0em;" > < / span > < span class = "mrel" > ←< / span > < / span > < / span > < / span > < / td > < td > < span class = "katex" > < span class = "katex-html" aria-hidden = "true" > < span class = "base" > < span class = "strut" style = "height:0.625em;vertical-align:-0.19444em;" > < / span > < span class = "mord" > < span class = "mord mathnormal" > x< / span > < span class = "msupsub" > < span class = "vlist-t vlist-t2" > < span class = "vlist-r" > < span class = "vlist" style = "height:0.30110799999999993em;" > < span style = "top:-2.5500000000000003em;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 = "mpunct" > ,< / span > < span class = "mspace" style = "margin-right:0.16666666666666666em;" > < / span > < span class = "mord" > < span class = "mord mathnormal" > x< / span > < span class = "msupsub" > < span class = "vlist-t vlist-t2" > < span class = "vlist-r" > < span class = "vlist" style = "height:0.30110799999999993em;" > < span style = "top:-2.5500000000000003em;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 > < / span > < / span > < / td > < / tr >
< tr > < td > Constructs < span class = "katex" > < span class = "katex-html" aria-hidden = "true" > < span class = "base" > < span class = "strut" style = "height:1.001892em;vertical-align:-0.25em;" > < / span > < span class = "mord" > < span class = "mord mathnormal" > h< / span > < span class = "msupsub" > < span class = "vlist-t" > < span class = "vlist-r" > < span class = "vlist" style = "height:0.751892em;" > < 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 > < / 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 > multipoint opening poly< / td > < td > < / td > < td > < / td > < / tr >
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< tr > < td > < span class = "katex" > < span class = "katex-html" aria-hidden = "true" > < span class = "base" > < span class = "strut" style = "height:0.68333em;vertical-align:0em;" > < / span > < span class = "mord mathnormal" style = "margin-right:0.10903em;" > U< / span > < span class = "mspace" style = "margin-right:0.2777777777777778em;" > < / span > < span class = "mrel" > =< / span > < span class = "mspace" style = "margin-right:0.2777777777777778em;" > < / span > < / span > < span class = "base" > < span class = "strut" style = "height:1.001892em;vertical-align:-0.25em;" > < / 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" > < span class = "vlist-r" > < span class = "vlist" style = "height:0.751892em;" > < 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 > < / 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 > < / td > < td > < span class = "katex" > < span class = "katex-html" aria-hidden = "true" > < span class = "base" > < span class = "strut" style = "height:0.36687em;vertical-align:0em;" > < / span > < span class = "mrel" > →< / span > < / span > < / span > < / span > < / td > < td > < / td > < / tr >
2021-02-12 08:17:18 -08:00
< tr > < td > < / td > < td > < span class = "katex" > < span class = "katex-html" aria-hidden = "true" > < span class = "base" > < span class = "strut" style = "height:0.36687em;vertical-align:0em;" > < / span > < span class = "mrel" > ←< / span > < / span > < / span > < / span > < / td > < td > < span class = "katex" > < span class = "katex-html" aria-hidden = "true" > < span class = "base" > < span class = "strut" style = "height:0.58056em;vertical-align:-0.15em;" > < / span > < span class = "mord" > < span class = "mord mathnormal" > x< / span > < span class = "msupsub" > < span class = "vlist-t vlist-t2" > < span class = "vlist-r" > < span class = "vlist" style = "height:0.30110799999999993em;" > < span style = "top:-2.5500000000000003em;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 > < / span > < / span > < / td > < / tr >
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< tr > < td > < span class = "katex" > < span class = "katex-html" aria-hidden = "true" > < span class = "base" > < span class = "strut" style = "height:0.63888em;vertical-align:-0.19444em;" > < / span > < span class = "mord" > < span class = "mord mathbf" > q< / span > < span class = "msupsub" > < span class = "vlist-t vlist-t2" > < span class = "vlist-r" > < span class = "vlist" style = "height:0.33610799999999996em;" > < span style = "top:-2.5500000000000003em;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 text mtight" > < span class = "mord mtight" > evals< / span > < / 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 = "mspace" style = "margin-right:0.2777777777777778em;" > < / span > < span class = "mrel" > =< / span > < span class = "mspace" style = "margin-right:0.2777777777777778em;" > < / 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" > Q< / span > < span class = "msupsub" > < span class = "vlist-t vlist-t2" > < span class = "vlist-r" > < span class = "vlist" style = "height:0.30110799999999993em;" > < span style = "top:-2.5500000000000003em;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" > < span class = "mord mathnormal" > x< / span > < span class = "msupsub" > < span class = "vlist-t vlist-t2" > < span class = "vlist-r" > < span class = "vlist" style = "height:0.30110799999999993em;" > < span style = "top:-2.5500000000000003em;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 = "mclose" > )< / span > < span class = "mpunct" > ,< / span > < span class = "mspace" style = "margin-right:0.16666666666666666em;" > < / span > < span class = "mord" > < span class = "mord mathnormal" > Q< / span > < span class = "msupsub" > < span class = "vlist-t vlist-t2" > < span class = "vlist-r" > < span class = "vlist" style = "height:0.30110799999999993em;" > < span style = "top:-2.5500000000000003em;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" > < span class = "mord mathnormal" > x< / span > < span class = "msupsub" > < span class = "vlist-t vlist-t2" > < span class = "vlist-r" > < span class = "vlist" style = "height:0.30110799999999993em;" > < span style = "top:-2.5500000000000003em;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 = "mclose" > )< / span > < span class = "mpunct" > ,< / span > < span class = "mspace" style = "margin-right:0.16666666666666666em;" > < / span > < span class = "minner" > …< / span > < span class = "mspace" style = "margin-right:0.16666666666666666em;" > < / span > < span class = "mclose" > ]< / span > < / span > < / span > < / span > < / td > < td > < span class = "katex" > < span class = "katex-html" aria-hidden = "true" > < span class = "base" > < span class = "strut" style = "height:0.36687em;vertical-align:0em;" > < / span > < span class = "mrel" > →< / span > < / span > < / span > < / span > < / td > < td > < / td > < / tr >
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< tr > < td > < span class = "katex" > < span class = "katex-html" aria-hidden = "true" > < span class = "base" > < span class = "strut" style = "height:0.58056em;vertical-align:-0.15em;" > < / span > < span class = "mord" > < span class = "mord mathnormal" > u< / span > < span class = "msupsub" > < span class = "vlist-t vlist-t2" > < span class = "vlist-r" > < span class = "vlist" style = "height:0.33610799999999996em;" > < span style = "top:-2.5500000000000003em;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 text mtight" > < span class = "mord mtight" > eval< / span > < / 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 = "mspace" style = "margin-right:0.2777777777777778em;" > < / span > < span class = "mrel" > =< / span > < span class = "mspace" style = "margin-right:0.2777777777777778em;" > < / span > < / span > < span class = "base" > < span class = "strut" style = "height:1em;vertical-align:-0.25em;" > < / span > < span class = "mord mathnormal" style = "margin-right:0.10903em;" > U< / span > < span class = "mopen" > (< / span > < span class = "mord" > < span class = "mord mathnormal" > x< / span > < span class = "msupsub" > < span class = "vlist-t vlist-t2" > < span class = "vlist-r" > < span class = "vlist" style = "height:0.30110799999999993em;" > < span style = "top:-2.5500000000000003em;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 = "mclose" > )< / span > < / span > < / span > < / span > < / td > < td > < span class = "katex" > < span class = "katex-html" aria-hidden = "true" > < span class = "base" > < span class = "strut" style = "height:0.36687em;vertical-align:0em;" > < / span > < span class = "mrel" > →< / span > < / span > < / span > < / span > < / td > < td > < / td > < / tr >
< tr > < td > < / td > < td > < span class = "katex" > < span class = "katex-html" aria-hidden = "true" > < span class = "base" > < span class = "strut" style = "height:0.36687em;vertical-align:0em;" > < / span > < span class = "mrel" > ←< / span > < / span > < / span > < / span > < / td > < td > < span class = "katex" > < span class = "katex-html" aria-hidden = "true" > < span class = "base" > < span class = "strut" style = "height:0.58056em;vertical-align:-0.15em;" > < / span > < span class = "mord" > < span class = "mord mathnormal" > x< / span > < span class = "msupsub" > < span class = "vlist-t vlist-t2" > < span class = "vlist-r" > < span class = "vlist" style = "height:0.30110799999999993em;" > < span style = "top:-2.5500000000000003em;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" > 4< / 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 > < / span > < / span > < / td > < / tr >
< / tbody > < / table >
< p > Then the prover and verifier:< / p >
< ul >
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< li > Construct < 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 text" > < span class = "mord" > finalPoly< / span > < / span > < span class = "mopen" > (< / span > < span class = "mord mathnormal" style = "margin-right:0.07847em;" > X< / span > < span class = "mclose" > )< / span > < / span > < / span > < / span > as a linear combination of < span class = "katex" > < span class = "katex-html" aria-hidden = "true" > < span class = "base" > < span class = "strut" style = "height:0.8805499999999999em;vertical-align:-0.19444em;" > < / span > < span class = "mord mathbf" > Q< / span > < / span > < / span > < / span > and < span class = "katex" > < span class = "katex-html" aria-hidden = "true" > < span class = "base" > < span class = "strut" style = "height:0.68333em;vertical-align:0em;" > < / span > < span class = "mord mathnormal" style = "margin-right:0.10903em;" > U< / span > < / span > < / span > < / span > using
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powers of < span class = "katex" > < span class = "katex-html" aria-hidden = "true" > < span class = "base" > < span class = "strut" style = "height:0.58056em;vertical-align:-0.15em;" > < / span > < span class = "mord" > < span class = "mord mathnormal" > x< / span > < span class = "msupsub" > < span class = "vlist-t vlist-t2" > < span class = "vlist-r" > < span class = "vlist" style = "height:0.30110799999999993em;" > < span style = "top:-2.5500000000000003em;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" > 4< / 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 > < / span > < / span > ;< / li >
< li > Construct < span class = "katex" > < span class = "katex-html" aria-hidden = "true" > < span class = "base" > < span class = "strut" style = "height:0.8888799999999999em;vertical-align:-0.19444em;" > < / span > < span class = "mord text" > < span class = "mord" > finalPolyEval< / span > < / span > < / span > < / span > < / span > as the equivalent linear combination of
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< span class = "katex" > < span class = "katex-html" aria-hidden = "true" > < span class = "base" > < span class = "strut" style = "height:0.63888em;vertical-align:-0.19444em;" > < / span > < span class = "mord" > < span class = "mord mathbf" > q< / span > < span class = "msupsub" > < span class = "vlist-t vlist-t2" > < span class = "vlist-r" > < span class = "vlist" style = "height:0.33610799999999996em;" > < span style = "top:-2.5500000000000003em;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 text mtight" > < span class = "mord mtight" > evals< / span > < / 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 > < / span > < / span > and < span class = "katex" > < span class = "katex-html" aria-hidden = "true" > < span class = "base" > < span class = "strut" style = "height:0.58056em;vertical-align:-0.15em;" > < / span > < span class = "mord" > < span class = "mord mathnormal" > u< / span > < span class = "msupsub" > < span class = "vlist-t vlist-t2" > < span class = "vlist-r" > < span class = "vlist" style = "height:0.33610799999999996em;" > < span style = "top:-2.5500000000000003em;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 text mtight" > < span class = "mord mtight" > eval< / span > < / 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 > < / span > < / span > ; and< / li >
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< li > Perform < 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 text" > < span class = "mord" > InnerProduct< / span > < / span > < span class = "mopen" > (< / span > < span class = "mord text" > < span class = "mord" > finalPoly< / 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:0.16666666666666666em;" > < / span > < span class = "mord" > < span class = "mord mathnormal" > x< / span > < span class = "msupsub" > < span class = "vlist-t vlist-t2" > < span class = "vlist-r" > < span class = "vlist" style = "height:0.30110799999999993em;" > < span style = "top:-2.5500000000000003em;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 = "mpunct" > ,< / span > < span class = "mspace" style = "margin-right:0.16666666666666666em;" > < / span > < span class = "mord text" > < span class = "mord" > finalPolyEval< / span > < / span > < span class = "mclose" > )< / span > < span class = "mord" > .< / span > < / span > < / span > < / span > < / li >
< / ul >
< blockquote >
< p > TODO: Write up protocol components that provide zero-knowledge.< / p >
< / blockquote >
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