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1,children:(0,n.jsxs)(e.span,{style:{top:"-3.413em",marginRight:"0.0347em"},children:[(0,n.jsx)(e.span,{className:"pstrut",style:{height:"3em"}}),(0,n.jsx)(e.span,{className:"sizing reset-size8 size6 mtight",children:(0,n.jsxs)(e.span,{className:"mord mtight",children:[(0,n.jsx)(e.span,{className:"mord mathnormal mtight",style:{marginRight:"0.07153em"},children:"Z"}),(0,n.jsx)(e.span,{className:"mord mathnormal mtight",style:{marginRight:"0.07153em"},children:"K"})]})})]})})})})})]}),(0,n.jsx)(e.span,{className:"mopen",children:"("}),(0,n.jsx)(e.span,{className:"mord text",children:(0,n.jsx)(e.span,{className:"mord",children:"public"})}),(0,n.jsx)(e.span,{className:"mspace",children:"\xa0"}),(0,n.jsx)(e.span,{className:"mord mathnormal",children:"x"}),(0,n.jsx)(e.span,{className:"mpunct",children:","}),(0,n.jsx)(e.span,{className:"mspace",style:{marginRight:"0.1667em"}}),(0,n.jsx)(e.span,{className:"mord 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",(0,n.jsxs)(e.span,{className:"katex",children:[(0,n.jsx)(e.span,{className:"katex-mathml",children:(0,n.jsx)(e.math,{xmlns:"http://www.w3.org/1998/Math/MathML",children:(0,n.jsxs)(e.semantics,{children:[(0,n.jsxs)(e.mrow,{children:[(0,n.jsx)(e.mi,{children:"d"}),(0,n.jsx)(e.mi,{children:"o"}),(0,n.jsx)(e.mi,{children:"S"}),(0,n.jsx)(e.mi,{children:"o"}),(0,n.jsx)(e.mi,{children:"m"}),(0,n.jsx)(e.mi,{children:"e"}),(0,n.jsx)(e.mi,{children:"t"}),(0,n.jsx)(e.mi,{children:"h"}),(0,n.jsx)(e.mi,{children:"i"}),(0,n.jsx)(e.mi,{children:"n"}),(0,n.jsxs)(e.msup,{children:[(0,n.jsx)(e.mi,{children:"g"}),(0,n.jsxs)(e.mrow,{children:[(0,n.jsx)(e.mi,{children:"Z"}),(0,n.jsx)(e.mi,{children:"K"})]})]})]}),(0,n.jsx)(e.annotation,{encoding:"application/x-tex",children:"doSomething^{ZK}"})]})})}),(0,n.jsx)(e.span,{className:"katex-html","aria-hidden":"true",children:(0,n.jsxs)(e.span,{className:"base",children:[(0,n.jsx)(e.span,{className:"strut",style:{height:"1.0358em",verticalAlign:"-0.1944em"}}),(0,n.jsx)(e.span,{className:"mord 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mathnormal",style:{marginRight:"0.03588em"},children:"g"}),(0,n.jsx)(e.span,{className:"msupsub",children:(0,n.jsx)(e.span,{className:"vlist-t",children:(0,n.jsx)(e.span,{className:"vlist-r",children:(0,n.jsx)(e.span,{className:"vlist",style:{height:"0.8413em"},children:(0,n.jsxs)(e.span,{style:{top:"-3.063em",marginRight:"0.05em"},children:[(0,n.jsx)(e.span,{className:"pstrut",style:{height:"2.7em"}}),(0,n.jsx)(e.span,{className:"sizing reset-size6 size3 mtight",children:(0,n.jsxs)(e.span,{className:"mord mtight",children:[(0,n.jsx)(e.span,{className:"mord mathnormal mtight",style:{marginRight:"0.07153em"},children:"Z"}),(0,n.jsx)(e.span,{className:"mord mathnormal mtight",style:{marginRight:"0.07153em"},children:"K"})]})})]})})})})})]})]})})]})," takes much longer to execute.\nResearch surrounding the subject of Zero-Knowledge Proofs has proliferated in the last few years.\nEvery work is attempting to optimize at least one of the following:"]}),"\n",(0,n.jsxs)(e.ol,{children:["\n",(0,n.jsxs)(e.li,{children:["The runtime of ",(0,n.jsxs)(e.span,{className:"katex",children:[(0,n.jsx)(e.span,{className:"katex-mathml",children:(0,n.jsx)(e.math,{xmlns:"http://www.w3.org/1998/Math/MathML",children:(0,n.jsxs)(e.semantics,{children:[(0,n.jsxs)(e.mrow,{children:[(0,n.jsx)(e.mi,{children:"d"}),(0,n.jsx)(e.mi,{children:"o"}),(0,n.jsx)(e.mi,{children:"S"}),(0,n.jsx)(e.mi,{children:"o"}),(0,n.jsx)(e.mi,{children:"m"}),(0,n.jsx)(e.mi,{children:"e"}),(0,n.jsx)(e.mi,{children:"t"}),(0,n.jsx)(e.mi,{children:"h"}),(0,n.jsx)(e.mi,{children:"i"}),(0,n.jsx)(e.mi,{children:"n"}),(0,n.jsxs)(e.msup,{children:[(0,n.jsx)(e.mi,{children:"g"}),(0,n.jsxs)(e.mrow,{children:[(0,n.jsx)(e.mi,{children:"Z"}),(0,n.jsx)(e.mi,{children:"K"})]})]})]}),(0,n.jsx)(e.annotation,{encoding:"application/x-tex",children:"doSomething^{ZK}"})]})})}),(0,n.jsx)(e.span,{className:"katex-html","aria-hidden":"true",children:(0,n.jsxs)(e.span,{className:"base",children:[(0,n.jsx)(e.span,{className:"strut",style:{height:"1.0358em",verticalAlign:"-0.1944em"}}),(0,n.jsx)(e.span,{className:"mord mathnormal",children:"d"}),(0,n.jsx)(e.span,{className:"mord mathnormal",children:"o"}),(0,n.jsx)(e.span,{className:"mord mathnormal",style:{marginRight:"0.05764em"},children:"S"}),(0,n.jsx)(e.span,{className:"mord mathnormal",children:"o"}),(0,n.jsx)(e.span,{className:"mord mathnormal",children:"m"}),(0,n.jsx)(e.span,{className:"mord mathnormal",children:"e"}),(0,n.jsx)(e.span,{className:"mord mathnormal",children:"t"}),(0,n.jsx)(e.span,{className:"mord mathnormal",children:"hin"}),(0,n.jsxs)(e.span,{className:"mord",children:[(0,n.jsx)(e.span,{className:"mord mathnormal",style:{marginRight:"0.03588em"},children:"g"}),(0,n.jsx)(e.span,{className:"msupsub",children:(0,n.jsx)(e.span,{className:"vlist-t",children:(0,n.jsx)(e.span,{className:"vlist-r",children:(0,n.jsx)(e.span,{className:"vlist",style:{height:"0.8413em"}
1,children:(0,n.jsxs)(e.span,{style:{top:"-3.063em",marginRight:"0.05em"},children:[(0,n.jsx)(e.span,{className:"pstrut",style:{height:"2.7em"}}),(0,n.jsx)(e.span,{className:"sizing reset-size6 size3 mtight",children:(0,n.jsxs)(e.span,{className:"mord mtight",children:[(0,n.jsx)(e.span,{className:"mord mathnormal mtight",style:{marginRight:"0.07153em"},children:"Z"}),(0,n.jsx)(e.span,{className:"mord mathnormal mtight",style:{marginRight:"0.07153em"},children:"K"})]})})]})})})})})]})]})})]})]}),"\n",(0,n.jsxs)(e.li,{children:["The verification time given ",(0,n.jsx)(e.code,{children:"result"})," and ",(0,n.jsx)(e.code,{children:"proof"})]}),"\n",(0,n.jsxs)(e.li,{children:["The size of ",(0,n.jsx)(e.code,{children:"proof"})]}),"\n"]}),"\n",(0,n.jsx)(e.h2,{id:"proofs",children:"Proofs"}),"\n",(0,n.jsxs)(e.p,{children:["In this section, we present a high level overview of the ",(0,n.jsx)(e.a,{href:"https://link.springer.com/chapter/10.1007/978-3-642-17373-8_11",children:"KZG polynomial commitment scheme"})," in order to explain why ",(0,n.jsxs)(e.span,{className:"katex",children:[(0,n.jsx)(e.span,{className:"katex-mathml",children:(0,n.jsx)(e.math,{xmlns:"http://www.w3.org/1998/Math/MathML",children:(0,n.jsxs)(e.semantics,{children:[(0,n.jsxs)(e.mrow,{children:[(0,n.jsx)(e.mi,{children:"d"}),(0,n.jsx)(e.mi,{children:"o"}),(0,n.jsx)(e.mi,{children:"S"}),(0,n.jsx)(e.mi,{children:"o"}),(0,n.jsx)(e.mi,{children:"m"}),(0,n.jsx)(e.mi,{children:"e"}),(0,n.jsx)(e.mi,{children:"t"}),(0,n.jsx)(e.mi,{children:"h"}),(0,n.jsx)(e.mi,{children:"i"}),(0,n.jsx)(e.mi,{children:"n"}),(0,n.jsxs)(e.msup,{children:[(0,n.jsx)(e.mi,{children:"g"}),(0,n.jsxs)(e.mrow,{children:[(0,n.jsx)(e.mi,{children:"Z"}),(0,n.jsx)(e.mi,{children:"K"})]})]})]}),(0,n.jsx)(e.annotation,{encoding:"application/x-tex",children:"doSomething^{ZK}"})]})})}),(0,n.jsx)(e.span,{className:"katex-html","aria-hidden":"true",children:(0,n.jsxs)(e.span,{className:"base",children:[(0,n.jsx)(e.span,{className:"strut",style:{height:"1.0358em",verticalAlign:"-0.1944em"}}),(0,n.jsx)(e.span,{className:"mord mathnormal",children:"d"}),(0,n.jsx)(e.span,{className:"mord mathnormal",children:"o"}),(0,n.jsx)(e.span,{className:"mord mathnormal",style:{marginRight:"0.05764em"},children:"S"}),(0,n.jsx)(e.span,{className:"mord mathnormal",children:"o"}),(0,n.jsx)(e.span,{className:"mord mathnormal",children:"m"}),(0,n.jsx)(e.span,{className:"mord mathnormal",children:"e"}),(0,n.jsx)(e.span,{className:"mord mathnormal",children:"t"}),(0,n.jsx)(e.span,{className:"mord mathnormal",children:"hin"}),(0,n.jsxs)(e.span,{className:"mord",children:[(0,n.jsx)(e.span,{className:"mord mathnormal",style:{marginRight:"0.03588em"},children:"g"}),(0,n.jsx)(e.span,{className:"msupsub",children:(0,n.jsx)(e.span,{className:"vlist-t",children:(0,n.jsx)(e.span,{className:"vlist-r",children:(0,n.jsx)(e.span,{className:"vlist",style:{height:"0.8413em"},children:(0,n.jsxs)(e.span,{style:{top:"-3.063em",marginRight:"0.05em"},children:[(0,n.jsx)(e.span,{className:"pstrut",style:{height:"2.7em"}}),(0,n.jsx)(e.span,{className:"sizing reset-size6 size3 mtight",children:(0,n.jsxs)(e.span,{className:"mord mtight",children:[(0,n.jsx)(e.span,{className:"mord mathnormal mtight",style:{marginRight:"0.07153em"},children:"Z"}),(0,n.jsx)(e.span,{className:"mord mathnormal mtight",style:{marginRight:"0.07153em"},children:"K"})]})})]})})})})})]})]})})]})," generally takes so long.\nKZG is a popular protocol choice for blockchain developers because the proof size is small and the verification work is minimized.\nFurthermore, while proving still needs to happen off-chain, verification is enabled on-chain."]}),"\n",(0,n.jsx)(e.p,{children:"One way to convince someone that you\u2019ve run code is to show your work by publishing the execution trace, i.e. all of the intermediate steps and variable values.\nYou could convince any verifier that you ran the program at the price of giving the verifier full knowledge of private inputs.\nAdditionally, the size of such a proof is unappealing.\nHowever, this transformation is still a useful first step towards a zero-knowledge proof."}),"\n",(0,n.jsx)(l.A,{img:a(1920)}),"\n",(0,n.jsxs)(e.p,{children:["The diagram above depicts the process of transforming pseudocode into an execution trace, which is sometimes called arithmetization.\nWe will call the right hand side a circuit: it is a matrix with each entry given by a formula representing some intermediate value of the program.\nFor a concrete input pair ",(0,n.jsx)(e.code,{children:"x"})," and ",(0,n.jsx)(e.code,{children:"y"}),", filling in the actual values of the circuit is known as witness generation."]}),"\n",(0,n.jsxs)(e.p,{children:["The next step is going to turn each column of the circuit into a polynomial.\nThis improvement is not initially obvious.\nIn the diagram below, we have a vector of length ",(0,n.jsx)(e.code,{children:"n"}),", which will turn into a polynomial of degree ",(0,n.jsx)(e.code,{children:"n"}),", and it takes the same amount of data to describe both the vector and the polynomial.\nHowever, we do not intend to publish the entire polynomial for a column.\nRather, we will c
1onvince a verifier that we know the global structure of the polynomial and that we can predict how it will behave in any local neighborhood."]}),"\n",(0,n.jsx)(l.A,{img:a(7323)}),"\n",(0,n.jsx)(e.p,{children:"The first phase of the polynomial argument, the global structure, functions similarly to a hash.\nRandom hash collisions are unlikely, meaning the hash output is a sufficiently identifying property of a polynomial.\nAdditionally, the hash is irreversible.\nSo, an adversary cannot learn much about the pre-image given only the hash output."}),"\n",(0,n.jsx)(l.A,{img:a(69344)}),"\n",(0,n.jsxs)(e.p,{children:["The second phase of the polynomial argument, the local structure, opens up the behavior of the polynomial around any random point (note that the prover does not get to choose this).\nConsider first the claim: \u201cI have a polynomial and when I plug ",(0,n.jsxs)(e.span,{className:"katex",children:[(0,n.jsx)(e.span,{className:"katex-mathml",children:(0,n.jsx)(e.math,{xmlns:"http://www.w3.org/1998/Math/MathML",children:(0,n.jsxs)(e.semantics,{children:[(0,n.jsx)(e.mrow,{children:(0,n.jsx)(e.mn,{children:"12"})}),(0,n.jsx)(e.annotation,{encoding:"application/x-tex",children:"12"})]})})}),(0,n.jsx)(e.span,{className:"katex-html","aria-hidden":"true",children:(0,n.jsxs)(e.span,{className:"base",children:[(0,n.jsx)(e.span,{className:"strut",style:{height:"0.6444em"}}),(0,n.jsx)(e.span,{className:"mord",children:"12"})]})})]})," into it, I get ",(0,n.jsxs)(e.span,{className:"katex",children:[(0,n.jsx)(e.span,{className:"katex-mathml",children:(0,n.jsx)(e.math,{xmlns:"http://www.w3.org/1998/Math/MathML",children:(0,n.jsxs)(e.semantics,{children:[(0,n.jsx)(e.mrow,{children:(0,n.jsx)(e.mn,{children:"42"})}),(0,n.jsx)(e.annotation,{encoding:"application/x-tex",children:"42"})]})})}),(0,n.jsx)(e.span,{className:"katex-html","aria-hidden":"true",children:(0,n.jsxs)(e.span,{className:"base",children:[(0,n.jsx)(e.span,{className:"strut",style:{height:"0.6444em"}}),(0,n.jsx)(e.span,{className:"mord",children:"42"})]})})]})," back\u201d.\nIn order to convince a hostile listener that I have full knowledge of this polynomial, I\u2019ll need to supply more information.\nLike the diagram below indicates, if I know ",(0,n.jsxs)(e.span,{className:"katex",children:[(0,n.jsx)(e.span,{className:"katex-mathml",children:(0,n.jsx)(e.math,{xmlns:"http://www.w3.org/1998/Math/MathML",children:(0,n.jsxs)(e.semantics,{children:[(0,n.jsxs)(e.mrow,{children:[(0,n.jsx)(e.mi,{children:"P"}),(0,n.jsx)(e.mo,{stretchy:"false",children:"("}),(0,n.jsx)(e.mn,{children:"12"}),(0,n.jsx)(e.mo,{stretchy:"false",children:")"}),(0,n.jsx)(e.mo,{children:"="}),(0,n.jsx)(e.mn,{children:"42"})]}),(0,n.jsx)(e.annotation,{encoding:"application/x-tex",children:"P(12)=42"})]})})}),(0,n.jsxs)(e.span,{className:"katex-html","aria-hidden":"true",children:[(0,n.jsxs)(e.span,{className:"base",children:[(0,n.jsx)(e.span,{className:"strut",style:{height:"1em",verticalAlign:"-0.25em"}}),(0,n.jsx)(e.span,{className:"mord 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",(0,n.jsxs)(e.span,{className:"katex",children:[(0,n.jsx)(e.span,{className:"katex-mathml",children:(0,n.jsx)(e.math,{xmlns:"http://www.w3.org/1998/Math/MathML",children:(0,n.jsxs)(e.semantics,{children:[(0,n.jsxs)(e.mrow,{children:[(0,n.jsx)(e.mi,{children:"P"}),(0,n.jsx)(e.mo,{stretchy:"false",children:"("}),(0,n.jsx)(e.mi,{children:"x"}),(0,n.jsx)(e.mo,{stretchy:"false",children:")"}),(0,n.jsx)(e.mo,{children:"\u2212"}),(0,n.jsx)(e.mn,{children:"42"})]}),(0,n.jsx)(e.annotation,{encoding:"application/x-tex",children:"P(x)-42"})]})})}),(0,n.jsxs)(e.span,{className:"katex-html","aria-hidden":"true",children:[(0,n.jsxs)(e.span,{className:"base",children:[(0,n.jsx)(e.span,{className:"strut",style:{height:"1em",verticalAlign:"-0.25em"}}),(0,n.jsx)(e.span,{className:"mord mathnormal",style:{marginRight:"0.13889em"},children:"P"}),(0,n.jsx)(e.span,{className:"mopen",children:"("}),(0,n.jsx)(e.span,{className:"mord mathnormal",children:"x"}),(0,n.jsx)(e.span,{className:"mclose",children:")"}),(0,n.jsx)(e.span,{className:"mspace",style:{marginRight:"0.2222em"}}),(0,n.jsx)(e.span,{className:"mbin",children:"\u2212"}),(0,n.jsx)(e.span,{className:"mspace",style:{marginRight:"0.2222em"}})]}),(0,n.jsxs)(e.span,{className:"base",children:[(0,n.jsx)(e.span,{className:"strut",style:{height:"0.6444em"}}),(0,n.jsx)(e.span,{className:"mord",children:"42"})]})]})]})," will cross the x-axis at ",(0,n.jsxs)(e.span,{className:"katex",children:[(0,n.jsx)(e.span,{className:"katex-mathml",children:(0,n.jsx)(e.math,{xmlns:"http://www.w3.org/1998/Math/MathML",children:(0,n.jsxs)(e.semantics,{children:[(0,n.jsx)(e.mrow,{children:(0,n.jsx)(e.mn,{children:"12"})}),(0,n.jsx)(e.annotation,{encoding:"application/x-tex",children:"12"})]})})}),(0,n.jsx)(e.span,{className:"katex-html","aria-hidden":"true",children:(0,n.jsxs)(e.span,{className:"base",children:[(0,n.jsx)(e.span,{className:"strut",style:{height:"0.6444em"}}),(0,n.jsx)(e.span,{className:"mord",children:"12"})]})})]}),".\nThere is a programmatic way (polynomial division) to retrieve a polynomial of degree ",(0,n.jsxs)(e.span,{className:"katex",children:[(0,n.jsx)(e.span,{className:"katex-mathml",children:(0,n.jsx)(e.math,{xmlns:"http://www.w3.org/1998/Math/MathML",children:(0,n.jsxs)(e.semantics,{children:[(0,n.jsxs)(e.mrow,{children:[(0,n.jsx)(e.mi,{children:"n"}),(0,n.jsx)(e.mo,{children:"\u2212"}),(0,n.jsx)(e.mn,{children:"1"})]}),(0,n.jsx)(e.annotation,{encoding:"application/x-tex",children:"n-1"})]})})}),(0,n.jsxs)(e.span,{className:"katex-html","aria-hidden":"true",children:[(0,n.jsxs)(e.span,{className:"base",children:[(0,n.jsx)(e.span,{className:"strut",style:{height:"0.6667em",verticalAlign:"-0.0833em"}}),(0,n.jsx)(e.span,{className:"mord mathnormal",children:"n"}),(0,n.jsx)(e.span,{className:"mspace",style:{marginRight:"0.2222em"}}),(0,n.jsx)(e.span,{className:"mbin",children:"\u2212"}),(0,n.jsx)(e.span,{className:"mspace",style:{marginRight:"0.2222em"}})]}),(0,n.jsxs)(e.span,{className:"base",children:[(0,n.jsx)(e.span,{className:"strut",style:{height:"0.6444em"}}),(0,n.jsx)(e.span,{className:"mord",children:"1"})]})]})]})," that crosses the x-axis in all the same places as ",(0,n.jsxs)(e.span,{className:"katex",children:[(0,n.jsx)(e.span,{className:"katex-mathml",children:(0,n.jsx)(e.math,{xmlns:"http://www.w3.org/1998/Math/MathML",children:(0,n.jsxs)(e.semantics,{children:[(0,n.jsxs)(e.mrow,{children:[(0,n.jsx)(e.mi,{children:"P"}),(0,n.jsx)(e.mo,{stretchy:"false",children:"("}),(0,n.jsx)(e.mi,{children:"x"}),(0,n.jsx)(e.mo,{stretchy:"false",children:")"}),(0,n.jsx)(e.mo,{children:"\u2212"}),(0,n.jsx)(e.mn,{children:"42"})]}),(0,n.jsx)(e.annotation,{encoding:"application/x-tex",children:"P(x)-42"})]})})}),(0,n.jsxs)(e.span,{className:"katex-html","aria-hidden":"true",children:[(0,n.jsxs)(e.span,{className:"base",children:[(0,n.jsx)(e.span,{className:"strut",style:{height:"1em",verticalAlign:"-0.25em"}}),(0,n.jsx)(e.span,{className:"mord mathnormal",style:{marginRight:"0.13889em"},children:"P"}),(0,n.jsx)(e.span,{className:"mopen",children:"("}),(0,n.jsx)(e.span,{className:"mord mathnormal",children:"x"}),(0,n.jsx)(e.span,{className:"mclose",children:")"}),(0,n.jsx)(e.span,{className:"mspace",style:{marginRight:"0.2222em"}}),(0,n.jsx)(e.span,{className:"mbin",children:"\u2212"}),(0,n.jsx)(e.span,{className:"mspace",style:{marginRight:"0.2222em"}})]}),(0,n.jsxs)(e.span,{className:"base",children:[(0,n.jsx)(e.span,{className:"strut",style:{height:"0.6444em"}}),(0,n.jsx)(e.span,{className:"mord",children:"42"})]})]})]}),", except at the root ",(0,n.jsxs)(e.span,{className:"katex",children:[(0,n.jsx)(e.span,{className:"katex-mathml",children:(0,n.jsx)(e.math,{xmlns:"http://www.w3.org/1998/Math/MathML",children:(0,n.jsxs)(e.semantics,{children:[(0,n.jsx)(e.mrow,{children:(0,n.jsx)(e.mn,{children:"12"})}),(0,n.jsx)(e.annotation,{encoding:"application/x-tex",children:"12"})]})})}),(0,n.jsx)(e.span,{className:"katex-html","aria-hidden":"true",children:(0,n.jsxs)(e.span,{className:"base",children:[(0,n.jsx)(e.span,{className:"strut",style:{height:"0.6444em"}}),(0,n.jsx)(e.span,{className:"mord",children:"12"})]})})]}),".\nIn short, in my role as the prover I need to supply ",(0,n.jsxs)(e.span,{className:"katex",children:[(0,n.jsx)(e.span,{className:"katex-mathml",children:(0,n.jsx)(e.math,{xmlns:"http://www.w3.org/1998/Math/MathML",children:(0,n.jsxs)(e.semantics,{children:[(0,n.jsx)(e.mrow,{children:(0,n.jsx)(e.mi,{children:"Q"})}),(0,n.jsx)(e.annotation,{encoding:"application/x-tex",children:"Q"})]})})}),(0,n.jsx)(e.span,{className:"katex-html","aria-hidden":"true",children:(0,n.jsxs)(e.span,{className:"base",children:[(0,n.jsx)(e.span,{className:"strut",style:{height:"0.8778em",verticalAlign:"-0.1944em"}}),(0,n.jsx)(e.span,{className:"mord mathnormal",children:"Q"})]})})]})," so that ",(0,n.jsxs)(e.span,{className:"katex",children:[(0,n.jsx)(e.span,{className:"katex-mathml",children:(0,n.jsx)(e.math,{xmlns:"http://www.w3.org/1998/Math/MathML",children:(0,n.jsxs)(e.semantics,{children:[(0,n.jsxs)(e.mrow,{children:[(0,n.jsx)(e.mi,{children:"Q"}),(0,n.jsx)(e.mo,{stretchy:"false",children:"("}),(0,n.jsx)(e.mi,{children:"x"}),(0,n.jsx)(e.mo,{stretchy:"false",children:")"}),(0,n.jsx)(e.mo,{stretchy:"false",children:"("}),(0,n.jsx)(e.mi,{children:"x"}),(0,n.jsx)(e.mo,{children:"\u2212"}),(0,n.jsx)(e.mn,{children:"12"}),(0,n.jsx)(e.mo,{stretchy:"false",children:")"}),(0,n.jsx)(e.mo,{children:"="}),(0,n.jsx)(e.mi,{children:"P"}),(0,n.jsx)(e.mo,{stretchy:"false",children:"("}),(0,n.jsx)(e.mi,{children:"x"}),(0,n.jsx)(e.mo,{stretchy:"false",children:")"}),(0,n.jsx)(e.mo,{children:"\u2212"}),(0,n.jsx)(e.mn,{children:"42"})]}),(0,n.jsx)(e.annotation,{encoding:"application/x-tex",children:"Q(x)(x-12) = P(x)-42"})]})})}),(0,n.jsxs)(e.span,{className:"katex-html","aria-hidden":"true",children:[(0,n.jsxs)(e.span,{className:"base",children:[(0,n.jsx)(e.span,{className:"strut",style:{height:"1em",verticalAlign:"-0.25em"}}),(0,n.jsx)(e.span,{className:"mord mathnormal",children:"Q"}),(0,n.jsx)(e.span,{className:"mopen",children:"("}),(0,n.jsx)(e.span,{className:"mord mathnormal",children:"x"}),(0,n.jsx)(e.span,{className:"mclose",children:")"}),(0,n.jsx)(e.span,{className:"mopen",children:"("}),(0,n.jsx)(e.span,{className:"mord mathnormal",children:"x"}),(0,n.jsx)(e.span,{className:"mspace",style:{marginRight:"0.2222em"}}),(0,n.jsx)(e.span,{className:"mbin",children:"\u2212"}),(0,n.jsx)(e.span,{className:"mspace",style:{marginRight:"0.2222em"}})]}),(0,n.jsxs)(e.span,{className:"base",children:[(0,n.jsx)(e.span,{className:"strut",style:{height:"1em",verticalAlign:"-0.25em"}}),(0,n.jsx)(e.span,{className:"mord",children:"12"}),(0,n.jsx)(e.span,{className:"mclose",children:")"}),(0,n.jsx)(e.span,{className:"mspace",style:{marginRight:"0.2778em"}}),(0,n.jsx)(e.span,{className:"mrel",children:"="}),(0,n.jsx)(e.span,{className:"mspace",style:{marginRight:"0.2778em"}})]}),(0,n.jsxs)(e.span,{className:"base",children:[(0,n.jsx)(e.span,{className:"strut",style:{height:"1em",verticalAlign:"-0.25em"}}),(0,n.jsx)(e.span,{className:"mord mathnormal",style:{marginRight:"0.13889em"},children:"P"}),(0,n.jsx)(e.span,{className:"mopen",children:"("}),(0,n.jsx)(e.span,{className:"mord mathnormal",children:"x"}),(0,n.jsx)(e.span,{className:"mclose",children:")"}),(0,n.jsx)(e.span,{className:"mspace",style:{marginRight:"0.2222em"}}),(0,n.jsx)(e.span,{className:"mbin",children:"\u2212"}),(0,n.jsx)(e.span,{className:"mspace",style:{marginRight:"0.2222em"}})]}),(0,n.jsxs)(e.span,{className:"base",children:[(0,n.jsx)(e.span,{className:"strut",style:{height:"0.6444em"}}),(0,n.jsx)(e.span,{className:"mord",children:"42"})]})]})]}),".\nThis is a very convincing argument to the verifier that I know the polynomial ",(0,n.jsxs)(e.span,{className:"katex",children:[(0,n.jsx)(e.span,{className:"katex-mathml",children:(0,n.jsx)(e.math,{xmlns:"http://www.w3.org/1998/Math/MathML",children:(0,n.jsxs)(e.semantics,{children:[(0,n.jsxs)(e.mrow,{children:[(0,n.jsx)(e.mi,{children:"P"}),(0,n.jsx)(e.mo,{stretchy:"false",children:"("}),(0,n.jsx)(e.mi,{children:"x"}),(0,n.jsx)(e.mo,{stretchy:"false",children:")"})]}),(0,n.jsx)(e.annotation,{encoding:"application/x-tex",children:"P(x)"})]})})}),(0,n.jsx)(e.span,{className:"katex-html","aria-hidden":"true",children:(0,n.jsxs)(e.span,{className:"base",children:[(0,n.jsx)(e.span,{className:"strut",style:{height:"1em",verticalAlign:"-0.25em"}}),(0,n.jsx)(e.span,{className:"mord mathnormal",style:{marginRight:"0.13889em"},children:"P"}),(0,n.jsx)(e.span,{className:"mopen",children:"("}),(0,n.jsx)(e.span,{className:"mord mathnormal",children:"x"}),(0,n.jsx)(e.span,{className:"mclose",children:")"})]})})]}),".\nBut, to stick to the principles of ZK, I need to withhold both ",(0,n.jsxs)(e.span,{className:"katex",children:[(0,n.jsx)(e.span,{className:"katex-mathml",children:(0,n.jsx)(e.math,{xmlns:"http://www.w3.org/1998/Math/MathML",children:(0,n.jsxs)(e.semantics,{children:[(0,n.jsxs)(e.mrow,{children:[(0,n.jsx)(e.mi,{children:"P"}),(0,n.jsx)(e.mo,{stretchy:"false",children:"("}),(0,n.jsx)(e.mi,{children:"x"}),(0,n.jsx)(e.mo,{stretchy:"false",children:")"})]}),(0,n.jsx)(e.annotation,{encoding:"application/x-tex",children:"P(x)"})]})})}),(0,n.jsx)(e.span,{className:"katex-html","aria-hidden":"true",children:(0,n.jsxs)(e.span,{className:"base",children:[(0,n.jsx)(e.span,{className:"strut",style:{height:"1em",verticalAlign:"-0.25em"}}),(0,n.jsx)(e.span,{className:"mord mathnormal",style:{marginRight:"0.13889em"},children:"P"}),(0,n.jsx)(e.span,{className:"mopen",children:"("}),(0,n.jsx)(e.span,{className:"mord mathnormal",children:"x"}),(0,n.jsx)(e.span,{className:"mclose",children:")"})]})})]})," and ",(0,n.jsxs)(e.span,{className:"katex",children:[(0,n.jsx)(e.span,{className:"katex-mathml",children:(0,n.jsx)(e.math,{xmlns:"http://www.w3.org/1998/Math/MathML",children:(0,n.jsxs)(e.semantics,{children:[(0,n.jsxs)(e.mrow,{children:[(0,n.jsx)(e.mi,{children:"Q"}),(0,n.jsx)(e.mo,{stretchy:"false",children:"("}),(0,n.jsx)(e.mi,{children:"x"}),(0,n.jsx)(e.mo,{stretchy:"false",children:")"})]}),(0,n.jsx)(e.annotation,{encoding:"application/x-tex",children:"Q(x)"})]})})}),(0,n.jsx)(e.span,{className:"katex-html","aria-hidden":"true",children:(0,n.jsxs)(e.span,{className:"base",children:[(0,n.jsx)(e.span,{className:"strut",style:{height:"1em",verticalAlign:"-0.25em"}}),(0,n.jsx)(e.span,{className:"mord mathnormal",children:"Q"}),(0,n.jsx)(e.span,{className:"mopen",children:"("}),(0,n.jsx)(e.span,{className:"mord mathnormal",children:"x"}),(0,n.jsx)(e.span,{className:"mclose",children:")"})]})})]}),".\nWe already have a hashed version of ",(0,n.jsxs)(e.span,{className:"katex",children:[(0,n.jsx)(e.span,{className:"katex-mathml",children:(0,n.jsx)(e.math,{xmlns:"http://www.w3.org/1998/Math/MathML",children:(0,n.jsxs)(e.semantics,{children:[(0,n.jsx)(e.mrow,{children:(0,n.jsx)(e.mi,{children:"P"})}),(0,n.jsx)(e.annotation,{encoding:"application/x-tex",children:"P"})]})})}),(0,n.jsx)(e.span,{className:"katex-html","aria-hidden":"true",children:(0,n.jsxs)(e.span,{className:"base",children:[(0,n.jsx)(e.span,{className:"strut",style:{height:"0.6833em"}}),(0,n.jsx)(e.span,{className:"mord mathnormal",style:{marginRight:"0.13889em"},children:"P"})]})})]}),", so we\u2019ll supply the hashed version of ",(0,n.jsxs)(e.span,{className:"katex",children:[(0,n.jsx)(e.span,{className:"katex-mathml",children:(0,n.jsx)(e.math,{xmlns:"http://www.w3.org/1998/Math/MathML",children:(0,n.jsxs)(e.semantics,{children:[(0,n.jsx)(e.mrow,{children:(0,n.jsx)(e.mi,{children:"Q"})}),(0,n.jsx)(e.annotation,{encoding:"application/x-tex",children:"Q"})]})})}),(0,n.jsx)(e.span,{className:"katex-html","aria-hidden":"true",children:(0,n.jsxs)(e.span,{className:"base",childr
1en:[(0,n.jsx)(e.span,{className:"strut",style:{height:"0.8778em",verticalAlign:"-0.1944em"}}),(0,n.jsx)(e.span,{className:"mord mathnormal",children:"Q"})]})})]})," as well."]}),"\n",(0,n.jsx)(l.A,{img:a(27743)}),"\n",(0,n.jsxs)(e.p,{children:["The final act is for the verifier to evaluate the ",(0,n.jsx)(e.code,{children:"proof"})," they have received.\nThe prover sent along commitments for the two polynomials ",(0,n.jsxs)(e.span,{className:"katex",children:[(0,n.jsx)(e.span,{className:"katex-mathml",children:(0,n.jsx)(e.math,{xmlns:"http://www.w3.org/1998/Math/MathML",children:(0,n.jsxs)(e.semantics,{children:[(0,n.jsx)(e.mrow,{children:(0,n.jsx)(e.mi,{children:"P"})}),(0,n.jsx)(e.annotation,{encoding:"application/x-tex",children:"P"})]})})}),(0,n.jsx)(e.span,{className:"katex-html","aria-hidden":"true",children:(0,n.jsxs)(e.span,{className:"base",children:[(0,n.jsx)(e.span,{className:"strut",style:{height:"0.6833em"}}),(0,n.jsx)(e.span,{className:"mord mathnormal",style:{marginRight:"0.13889em"},children:"P"})]})})]})," and ",(0,n.jsxs)(e.span,{className:"katex",children:[(0,n.jsx)(e.span,{className:"katex-mathml",children:(0,n.jsx)(e.math,{xmlns:"http://www.w3.org/1998/Math/MathML",children:(0,n.jsxs)(e.semantics,{children:[(0,n.jsx)(e.mrow,{children:(0,n.jsx)(e.mi,{children:"Q"})}),(0,n.jsx)(e.annotation,{encoding:"application/x-tex",children:"Q"})]})})}),(0,n.jsx)(e.span,{className:"katex-html","aria-hidden":"true",children:(0,n.jsxs)(e.span,{className:"base",children:[(0,n.jsx)(e.span,{className:"strut",style:{height:"0.8778em",verticalAlign:"-0.1944em"}}),(0,n.jsx)(e.span,{className:"mord mathnormal",children:"Q"})]})})]})," in addition to the claim that ",(0,n.jsxs)(e.span,{className:"katex",children:[(0,n.jsx)(e.span,{className:"katex-mathml",children:(0,n.jsx)(e.math,{xmlns:"http://www.w3.org/1998/Math/MathML",children:(0,n.jsxs)(e.semantics,{children:[(0,n.jsx)(e.mrow,{children:(0,n.jsx)(e.mi,{children:"P"})}),(0,n.jsx)(e.annotation,{encoding:"application/x-tex",children:"P"})]})})}),(0,n.jsx)(e.span,{className:"katex-html","aria-hidden":"true",children:(0,n.jsxs)(e.span,{className:"base",children:[(0,n.jsx)(e.span,{className:"strut",style:{height:"0.6833em"}}),(0,n.jsx)(e.span,{className:"mord mathnormal",style:{marginRight:"0.13889em"},children:"P"})]})})]})," evaluates to ",(0,n.jsxs)(e.span,{className:"katex",children:[(0,n.jsx)(e.span,{className:"katex-mathml",children:(0,n.jsx)(e.math,{xmlns:"http://www.w3.org/1998/Math/MathML",children:(0,n.jsxs)(e.semantics,{children:[(0,n.jsx)(e.mrow,{children:(0,n.jsx)(e.mn,{children:"42"})}),(0,n.jsx)(e.annotation,{encoding:"application/x-tex",children:"42"})]})})}),(0,n.jsx)(e.span,{className:"katex-html","aria-hidden":"true",children:(0,n.jsxs)(e.span,{className:"base",children:[(0,n.jsx)(e.span,{className:"strut",style:{height:"0.6444em"}}),(0,n.jsx)(e.span,{className:"mord",children:"42"})]})})]})," when you plug in ",(0,n.jsxs)(e.span,{className:"katex",children:[(0,n.jsx)(e.span,{className:"katex-mathml",children:(0,n.jsx)(e.math,{xmlns:"http://www.w3.org/1998/Math/MathML",children:(0,n.jsxs)(e.semantics,{children:[(0,n.jsx)(e.mrow,{children:(0,n.jsx)(e.mn,{children:"12"})}),(0,n.jsx)(e.annotation,{encoding:"application/x-tex",children:"12"})]})})}),(0,n.jsx)(e.span,{className:"katex-html","aria-hidden":"true",children:(0,n.jsxs)(e.span,{className:"base",children:[(0,n.jsx)(e.span,{className:"strut",style:{height:"0.6444em"}}),(0,n.jsx)(e.span,{className:"mord",children:"12"})]})})]}),".\nThe verifier is not able to to directly check that the polynomial equation ",(0,n.jsxs)(e.span,{className:"katex",children:[(0,n.jsx)(e.span,{className:"katex-mathml",children:(0,n.jsx)(e.math,{xmlns:"http://www.w3.org/1998/Math/MathML",children:(0,n.jsxs)(e.semantics,{children:[(0,n.jsxs)(e.mrow,{children:[(0,n.jsx)(e.mi,{children:"Q"}),(0,n.jsx)(e.mo,{stretchy:"false",children:"("}),(0,n.jsx)(e.mi,{children:"x"}),(0,n.jsx)(e.mo,{stretchy:"false",children:")"}),(0,n.jsx)(e.mo,{stretchy:"false",children:"("}),(0,n.jsx)(e.mi,{children:"x"}),(0,n.jsx)(e.mo,{children:"\u2212"}),(0,n.jsx)(e.mn,{children:"12"}),(0,n.jsx)(e.mo,{stretchy:"false",children:")"}),(0,n.jsx)(e.mo,{children:"="}),(0,n.jsx)(e.mi,{children:"P"}),(0,n.jsx)(e.mo,{stretchy:"false",children:"("}),(0,n.jsx)(e.mi,{children:"x"}),(0,n.jsx)(e.mo,{stretchy:"false",children:")"}),(0,n.jsx)(e.mo,{children:"\u2212"}),(0,n.jsx)(e.mn,{children:"42"})]}),(0,n.jsx)(e.annotation,{encoding:"application/x-tex",children:"Q(x)(x-12) = 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mathnormal",children:"x"}),(0,n.jsx)(e.span,{className:"mclose",children:")"}),(0,n.jsx)(e.span,{className:"mspace",style:{marginRight:"0.2222em"}}),(0,n.jsx)(e.span,{className:"mbin",children:"\u2212"}),(0,n.jsx)(e.span,{className:"mspace",style:{marginRight:"0.2222em"}})]}),(0,n.jsxs)(e.span,{className:"base",children:[(0,n.jsx)(e.span,{className:"strut",style:{height:"0.6444em"}}),(0,n.jsx)(e.span,{className:"mord",children:"42"})]})]})]})," holds but they can check a \u201chashed\u201d version of this equality, very similar to the way the commitments for ",(0,n.jsxs)(e.span,{className:"katex",children:[(0,n.jsx)(e.span,{className:"katex-mathml",children:(0,n.jsx)(e.math,{xmlns:"http://www.w3.org/1998/Math/MathML",children:(0,n.jsxs)(e.semantics,{children:[(0,n.jsx)(e.mrow,{children:(0,n.jsx)(e.mi,{children:"P"})}),(0,n.jsx)(e.annotation,{encoding:"application/x-tex",children:"P"})]})})}),(0,n.jsx)(e.span,{className:"katex-html","aria-hidden":"true",children:(0,n.jsxs)(e.span,{className:"base",children:[(0,n.jsx)(e.span,{className:"strut",style:{height:"0.6833em"}}),(0,n.jsx)(e.span,{className:"mord mathnormal",style:{marginRight:"0.13889em"},children:"P"})]})})]})," and ",(0,n.jsxs)(e.span,{className:"katex",children:[(0,n.jsx)(e.span,{className:"katex-mathml",children:(0,n.jsx)(e.math,{xmlns:"http://www.w3.org/1998/Math/MathML",children:(0,n.jsxs)(e.semantics,{children:[(0,n.jsx)(e.mrow,{children:(0,n.jsx)(e.mi,{children:"Q"})}),(0,n.jsx)(e.annotation,{encoding:"application/x-tex",children:"Q"})]})})}),(0,n.jsx)(e.span,{className:"katex-html","aria-hidden":"true",children:(0,n.jsxs)(e.span,{className:"base",children:[(0,n.jsx)(e.span,{className:"strut",style:{height:"0.8778em",verticalAlign:"-0.1944em"}}),(0,n.jsx)(e.span,{className:"mord mathnormal",children:"Q"})]})})]})," were obtained."]}),"\n",(0,n.jsx)(l.A,{img:a(80648)}),"\n",(0,n.jsxs)(e.p,{children:["That concludes the steps in a ZK proof.\nThe \u201cnot quite zero-knowledge\u201d comment in the diagram above deserves elaboration.\nIf you were to rerun 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Here are some more specific comments:"}),"\n",(0,n.jsxs)(e.ol,{children:["\n",(0,n.jsxs)(e.li,{children:["The commitment function, or hash, is not fast.\nAnother name for this operation is \u2018multi-scalar multiplication\u2019 or MSM.\nEach of the coefficients of your polynomial is represented with around 250 bits and the ",(0,n.jsxs)(e.span,{className:"katex",children:[(0,n.jsx)(e.span,{className:"katex-mathml",children:(0,n.jsx)(e.math,{xmlns:"http://www.w3.org/1998/Math/MathML",children:(0,n.jsxs)(e.semantics,{children:[(0,n.jsx)(e.mrow,{children:(0,n.jsx)(e.mi,{children:"\u03c4"})}),(0,n.jsx)(e.annotation,{encoding:"application/x-tex",children:"\\tau"})]})})}),(0,n.jsx)(e.span,{className:"katex-html","aria-hidden":"true",children:(0,n.jsxs)(e.span,{className:"base",children:[(0,n.jsx)(e.span,{className:"strut",style:{height:"0.4306em"}}),(0,n.jsx)(e.span,{className:"mord mathnormal",style:{marginRight:"0.1132em"},children:"\u03c4"})]})})]})," values are elliptic curve points which require around 380 bits to represent.\nWhile MSM is O(n/log(n)), n is already large enough for this to be a heavy computational burden.\nAdditionally, KZG requires more calls to MSM than the operation described next, so this is the bottleneck."]}),"\n",(0,n.jsx)(e.li,{children:"Calculating Q(x) requires polynomial division.\nFast fourier transform (FFT) techniques place this computation in O(n*log(n)).\nIn addition to the complexity introduced by large n (the degree of your polynomial), recall that the coefficients are each around 250 bits."}),"\n"]}),"\n",(0,n.jsx)(e.h2,{id:"sindris-api",children:"Sindri's API"}),"\n",(0,n.jsxs)(e.p,{children:["Hopefully the discussion above has convinced you of the difficulties of producing ZK proofs.\nIf you have your own 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