https://www.binius.xyz/assets/polynomials-C6hdPcF6.js

1import{u as n,j as s}from"./index-Bpb9-9J-.js";const t={title:"Shifted Witness Polynomials",description:"Expressing bitshifting algebraically"};function l(a){const e={a:"a",annotation:"annotation",aside:"aside",div:"div",em:"em",h1:"h1",h2:"h2",header:"header",li:"li",math:"math",mi:"mi",mn:"mn",mo:"mo",mover:"mover",mrow:"mrow",mstyle:"mstyle",msub:"msub",msup:"msup",mtable:"mtable",mtd:"mtd",mtext:"mtext",mtr:"mtr",munder:"munder",p:"p",path:"path",semantics:"semantics",span:"span",svg:"svg",ul:"ul",...n(),...a.components};return s.jsxs(s.Fragment,{children:[s.jsxs(e.header,{children:[s.jsxs(e.h1,{id:"shifted-witness-polynomials",children:["Shifted Witness Polynomials",s.jsx(e.a,{"aria-hidden":"true",tabIndex:"-1",href:"#shifted-witness-polynomials",children:s.jsx(e.div,{"data-autolink-icon":!0})})]}),s.jsx(e.div,{role:"doc-subtitle",children:"Expressing bitshifting algebraically"})]}),`
2`,s.jsxs(e.p,{children:["Let's say we have a length-",s.jsxs(e.span,{className:"katex",children:[s.jsx(e.span,{className:"katex-mathml",children:s.jsx(e.math,{xmlns:"http://www.w3.org/1998/Math/MathML",children:s.jsxs(e.semantics,{children:[s.jsx(e.mrow,{children:s.jsxs(e.msub,{children:[s.jsx(e.mi,{children:"n"}),s.jsx(e.mtext,{children:"words"})]})}),s.jsx(e.annotation,{encoding:"application/x-tex",children:"n_\\text{words}"})]})})}),s.jsx(e.span,{className:"katex-html","aria-hidden":"true",children:s.jsxs(e.span,{className:"base",children:[s.jsx(e.span,{className:"strut",style:{height:"0.5806em",verticalAlign:"-0.15em"}}),s.jsxs(e.span,{className:"mord",children:[s.jsx(e.span,{className:"mord mathnormal",children:"n"}),s.jsx(e.span,{className:"msupsub",children:s.jsxs(e.span,{className:"vlist-t vlist-t2",children:[s.jsxs(e.span,{className:"vlist-r",children:[s.jsx(e.span,{className:"vlist",style:{height:"0.3361em"},children:s.jsxs(e.span,{style:{top:"-2.55em",marginLeft:"0em",marginRight:"0.05em"},children:[s.jsx(e.span,{className:"pstrut",style:{height:"2.7em"}}),s.jsx(e.span,{className:"sizing reset-size6 size3 mtight",children:s.jsx(e.span,{className:"mord text mtight",children:s.jsx(e.span,{className:"mord mtight",children:"words"})})})]})}),s.jsx(e.span,{className:"vlist-s",children:"​"})]}),s.jsx(e.span,{className:"vlist-r",children:s.jsx(e.span,{className:"vlist",style:{height:"0.15em"},children:s.jsx(e.span,{})})})]})})]})]})})]})," vector ",s.jsxs(e.span,{className:"katex",children:[s.jsx(e.span,{className:"katex-mathml",children:s.jsx(e.math,{xmlns:"http://www.w3.org/1998/Math/MathML",children:s.jsxs(e.semantics,{children:[s.jsx(e.mrow,{children:s.jsx(e.mi,{children:"w"})}),s.jsx(e.annotation,{encoding:"application/x-tex",children:"w"})]})})}),s.jsx(e.span,{className:"katex-html","aria-hidden":"true",children:s.jsxs(e.span,{className:"base",children:[s.jsx(e.span,{className:"strut",style:{height:"0.4306em"}}),s.jsx(e.span,{className:"mord mathnormal",style:{marginRight:"0.02691em"},children:"w"})]})})]}),", whose components ",s.jsxs(e.span,{className:"katex",children:[s.jsx(e.span,{className:"katex-mathml",children:s.jsx(e.math,{xmlns:"http://www.w3.org/1998/Math/MathML",children:s.jsxs(e.semantics,{children:[s.jsxs(e.mrow,{children:[s.jsx(e.mi,{children:"w"}),s.jsx(e.mo,{stretchy:"false",children:"["}),s.jsx(e.mi,{children:"y"}),s.jsx(e.mo,{stretchy:"false",children:"]"})]}),s.jsx(e.annotation,{encoding:"application/x-tex",children:"w[y]"})]})})}),s.jsx(e.span,{className:"katex-html","aria-hidden":"true",children:s.jsxs(e.span,{className:"base",children:[s.jsx(e.span,{className:"strut",style:{height:"1em",verticalAlign:"-0.25em"}}),s.jsx(e.span,{className:"mord mathnormal",style:{marginRight:"0.02691em"},children:"w"}),s.jsx(e.span,{className:"mopen",children:"["}),s.jsx(e.span,{className:"mord mathnormal",style:{marginRight:"0.03588em"},children:"y"}),s.jsx(e.span,{className:"mclose",children:"]"})]})})]})," are 64-bit words."]}),`
3`,s.jsxs(e.p,{children:["Let's now fix some ",s.jsx(e.a,{href:"/blueprint/constraints/indices",children:"shift type"})," ",s.jsxs(e.span,{className:"katex",children:[s.jsx(e.span,{className:"katex-mathml",children:s.jsx(e.math,{xmlns:"http://www.w3.org/1998/Math/MathML",children:s.jsxs(e.semantics,{children:[s.jsxs(e.mrow,{children:[s.jsx(e.mtext,{children:"op"}),s.jsx(e.mo,{children:"∈"}),s.jsx(e.mo,{stretchy:"false",children:"{"}),s.jsxs(e.mrow,{children:[s.jsx(e.mi,{mathvariant:"sans-serif",children:"s"}),s.jsx(e.mi,{mathvariant:"sans-serif",children:"l"}),s.jsx(e.mi,{mathvariant:"sans-serif",children:"l"})]}),s.jsx(e.mo,{separator:"true",children:","}),s.jsxs(e.mrow,{children:[s.jsx(e.mi,{mathvariant:"sans-serif",children:"s"}),s.jsx(e.mi,{mathvariant:"sans-serif",children:"r"}),s.jsx(e.mi,{mathvariant:"sans-serif",children:"l"})]}),s.jsx(e.mo,{separator:"true",children:","}),s.jsxs(e.mrow,{children:[s.jsx(e.mi,{mathvariant:"sans-serif",children:"s"}),s.jsx(e.mi,{mathvariant:"sans-serif",children:"r"}),s.jsx(e.mi,{mathvariant:"sans-serif",children:"a"})]}),s.jsx(e.mo,{stretchy:"false",children:"}"})]}),s.jsx(e.annotation,{encoding:"application/x-tex",children:"\\text{op} \\in \\{\\mathsf{sll}, \\mathsf{srl}, \\mathsf{sra}\\}"})]})})}),s.jsxs(e.span,{className:"katex-html","aria-hidden":"true",children:[s.jsxs(e.span,{className:"base",children:[s.jsx(e.span,{className:"strut",style:{height:"0.7335em",verticalAlign:"-0.1944em"}}),s.jsx(e.span,{className:"mord text",children:s.jsx(e.span,{className:"mord",children:"op"})}),s.jsx(e.span,{className:"mspace",style:{marginRight:"0.2778em"}}),s.jsx(e.span,{className:"mrel",children:"∈"}),s.jsx(e.span,{className:"mspace",style:{marginRight:"0.2778em"}})]}),s.jsxs(e.span,{className:"base",children:[s.jsx(e.span,{className:"strut",style:{height:"1em",verticalAlign:"-0.25em"}}),s.jsx(e.span,{className:"mopen",children:"{"}),s.jsx(e.span,{className:"mord",children:s.jsx(e.span,{className:"mord mathsf",children:"sll"})}),s.jsx(e.span,{className:"mpunct",children:","}),s.jsx(e.span,{className:"mspace",style:{marginRight:"0.1667em"}}),s.jsx(e.span,{className:"mord",children:s.jsx(e.span,{className:"mord mathsf",children:"srl"})}),s.jsx(e.span,{className:"mpunct",children:","}),s.jsx(e.span,{className:"mspace",style:{marginRight:"0.1667em"}}),s.jsx(e.span,{className:"mord",children:s.jsx(e.span,{className:"mord mathsf",children:"sra"})}),s.jsx(e.span,{className:"mclose",children:"}"})]})]})]})," and a shift amount ",s.jsxs(e.span,{className:"katex",children:[s.jsx(e.span,{className:"katex-mathml",children:s.jsx(e.math,{xmlns:"http://www.w3.org/1998/Math/MathML",children:s.jsxs(e.semantics,{children:[s.jsxs(e.mrow,{children:[s.jsx(e.mi,{children:"s"}),s.jsx(e.mo,{children:"∈"}),s.jsx(e.mo,{stretchy:"false",children:"{"}),s.jsx(e.mn,{children:"0"}),s.jsx(e.mo,{separator:"true",children:","}),s.jsx(e.mo,{children:"…"}),s.jsx(e.mo,{separator:"true",children:","}),s.jsx(e.mn,{children:"63"}),s.jsx(e.mo,{stretchy:"false",children:"}"})]}),s.jsx(e.annotation,{encoding:"application/x-tex",children:"s \\in \\{0, \\ldots , 63\\}"})]})})}),s.jsxs(e.span,{className:"katex-html","aria-hidden":"true",children:[s.jsxs(e.span,{className:"base",children:[s.jsx(e.span,{className:"strut",style:{height:"0.5782em",verticalAlign:"-0.0391em"}}),s.jsx(e.span,{className:"mord mathnormal",children:"s"}),s.jsx(e.span,{className:"mspace",style:{marginRight:"0.2778em"}}),s.jsx(e.span,{className:"mrel",children:"∈"}),s.jsx(e.span,{className:"mspace",style:{marginRight:"0.2778em"}})]}),s.jsxs(e.span,{className:"base",children:[s.jsx(e.span,{className:"strut",style:{height:"1em",verticalAlign:"-0.25em"}}),s.jsx(e.span,{className:"mopen",children:"{"}),s.jsx(e.span,{className:"mord",children:"0"}),s.jsx(e.span,{className:"mpunct",children:","}),s.jsx(e.span,{className:"mspace",style:{marginRight:"0.1667em"}}),s.jsx(e.span,{className:"minner",children:"…"}),s.jsx(e.span,{className:"mspace",style:{marginRight:"0.1667em"}}),s.jsx(e.span,{className:"mpunct",children:","}),s.jsx(e.span,{className:"mspace",style:{marginRight:"0.1667em"}}),s.jsx(e.span,{className:"mord",children:"63"}),s.jsx(e.span,{className:"mclose",children:"}"})]})]})]}),". Let's say we want to shift each of ",s.jsxs(e.span,{className:"katex",children:[s.jsx(e.span,{className:"katex-mathml",children:s.jsx(e.math,{xmlns:"http://www.w3.org/1998/Math/MathML",children:s.jsxs(e.semantics,{children:[s.jsx(e.mrow,{children:s.jsx(e.mi,{children:"w"})}),s.jsx(e.annotation,{encoding:"application/x-tex",children:"w"})]})})}),s.jsx(e.span,{className:"katex-html","aria-hidden":"true",children:s.jsxs(e.span,{className:"base",children:[s.jsx(e.span,{className:"strut",style:{height:"0.4306em"}}),s.jsx(e.span,{className:"mord mathnormal",style:{marginRight:"0.02691em"},children:"w"})]})})]}),"'s ",s.jsxs(e.span,{className:"katex",children:[s.jsx(e.span,{className:"katex-mathml",children:s.jsx(e.math,{xmlns:"http://www.w3.org/1998/Math/MathML",children:s.jsxs(e.semantics,{children:[s.jsx(e.mrow,{children:s.jsxs(e.msub,{children:[s.jsx(e.mi,{children:"n"}),s.jsx(e.mtext,{children:"words"})]})}),s.jsx(e.annotation,{encoding:"application/x-tex",children:"n_\\text{words}"})]})})}),s.jsx(e.span,{className:"katex-html","aria-hidden":"true",children:s.jsxs(e.span,{className:"base",children:[s.jsx(e.span,{className:"strut",style:{height:"0.5806em",verticalAlign:"-0.15em"}}),s.jsxs(e.span,{className:"mord",children:[s.jsx(e.span,{className:"mord mathnormal",children:"n"}),s.jsx(e.span,{className:"msupsub",children:s.jsxs(e.span,{className:"vlist-t vlist-t2",children:[s.jsxs(e.span,{className:"vlist-r",children:[s.jsx(e.span,{className:"vlist",style:{height:"0.3361em"}

3,children:s.jsxs(e.span,{style:{top:"-2.55em",marginLeft:"0em",marginRight:"0.05em"},children:[s.jsx(e.span,{className:"pstrut",style:{height:"2.7em"}}),s.jsx(e.span,{className:"sizing reset-size6 size3 mtight",children:s.jsx(e.span,{className:"mord text mtight",children:s.jsx(e.span,{className:"mord mtight",children:"words"})})})]})}),s.jsx(e.span,{className:"vlist-s",children:"​"})]}),s.jsx(e.span,{className:"vlist-r",children:s.jsx(e.span,{className:"vlist",style:{height:"0.15em"},children:s.jsx(e.span,{})})})]})})]})]})})]})," components by ",s.jsxs(e.span,{className:"katex",children:[s.jsx(e.span,{className:"katex-mathml",children:s.jsx(e.math,{xmlns:"http://www.w3.org/1998/Math/MathML",children:s.jsxs(e.semantics,{children:[s.jsx(e.mrow,{children:s.jsx(e.mi,{children:"s"})}),s.jsx(e.annotation,{encoding:"application/x-tex",children:"s"})]})})}),s.jsx(e.span,{className:"katex-html","aria-hidden":"true",children:s.jsxs(e.span,{className:"base",children:[s.jsx(e.span,{className:"strut",style:{height:"0.4306em"}}),s.jsx(e.span,{className:"mord mathnormal",children:"s"})]})})]})," using ",s.jsxs(e.span,{className:"katex",children:[s.jsx(e.span,{className:"katex-mathml",children:s.jsx(e.math,{xmlns:"http://www.w3.org/1998/Math/MathML",children:s.jsxs(e.semantics,{children:[s.jsx(e.mrow,{children:s.jsx(e.mtext,{children:"op"})}),s.jsx(e.annotation,{encoding:"application/x-tex",children:"\\text{op}"})]})})}),s.jsx(e.span,{className:"katex-html","aria-hidden":"true",children:s.jsxs(e.span,{className:"base",children:[s.jsx(e.span,{className:"strut",style:{height:"0.625em",verticalAlign:"-0.1944em"}}),s.jsx(e.span,{className:"mord text",children:s.jsx(e.span,{className:"mord",children:"op"})})]})})]}),". This gives us a new ",s.jsxs(e.span,{className:"katex",children:[s.jsx(e.span,{className:"katex-mathml",children:s.jsx(e.math,{xmlns:"http://www.w3.org/1998/Math/MathML",children:s.jsxs(e.semantics,{children:[s.jsx(e.mrow,{children:s.jsxs(e.msub,{children:[s.jsx(e.mi,{children:"n"}),s.jsx(e.mtext,{children:"words"})]})}),s.jsx(e.annotation,{encoding:"application/x-tex",children:"n_\\text{words}"})]})})}),s.jsx(e.span,{className:"katex-html","aria-hidden":"true",children:s.jsxs(e.span,{className:"base",children:[s.jsx(e.span,{className:"strut",style:{height:"0.5806em",verticalAlign:"-0.15em"}}),s.jsxs(e.span,{className:"mord",children:[s.jsx(e.span,{className:"mord mathnormal",children:"n"}),s.jsx(e.span,{className:"msupsub",children:s.jsxs(e.span,{className:"vlist-t vlist-t2",children:[s.jsxs(e.span,{className:"vlist-r",children:[s.jsx(e.span,{className:"vlist",style:{height:"0.3361em"},children:s.jsxs(e.span,{style:{top:"-2.55em",marginLeft:"0em",marginRight:"0.05em"},children:[s.jsx(e.span,{className:"pstrut",style:{height:"2.7em"}}),s.jsx(e.span,{className:"sizing reset-size6 size3 mtight",children:s.jsx(e.span,{className:"mord text mtight",children:s.jsx(e.span,{className:"mord mtight",children:"words"})})})]})}),s.jsx(e.span,{className:"vlist-s",children:"​"})]}),s.jsx(e.span,{className:"vlist-r",children:s.jsx(e.span,{className:"vlist",style:{height:"0.15em"},children:s.jsx(e.span,{})})})]})})]})]})})]}),"-word list. Let's give it a name: for each ",s.jsxs(e.span,{className:"katex",children:[s.jsx(e.span,{className:"katex-mathml",children:s.jsx(e.math,{xmlns:"http://www.w3.org/1998/Math/MathML",children:s.jsxs(e.semantics,{children:[s.jsxs(e.mrow,{children:[s.jsx(e.mi,{children:"y"}),s.jsx(e.mo,{children:"∈"}),s.jsx(e.mo,{stretchy:"false",children:"{"}),s.jsx(e.mn,{children:"0"}),s.jsx(e.mo,{separator:"true",children:","}),s.jsx(e.mo,{children:"…"}),s.jsx(e.mo,{separator:"true",children:","}),s.jsxs(e.msub,{children:[s.jsx(e.mi,{children:"n"}),s.jsx(e.mtext,{children:"words"})]}),s.jsx(e.mo,{children:"−"}),s.jsx(e.mn,{children:"1"}),s.jsx(e.mo,{stretchy:"false",children:"}"})]}),s.jsx(e.annotation,{encoding:"application/x-tex",children:"y \\in \\{0, \\ldots , n_\\text{words} - 1\\}"})]})})}),s.jsxs(e.span,{className:"katex-html","aria-hidden":"true",children:[s.jsxs(e.span,{className:"base",children:[s.jsx(e.span,{className:"strut",style:{height:"0.7335em",verticalAlign:"-0.1944em"}}),s.jsx(e.span,{className:"mord mathnormal",style:{marginRight:"0.03588em"},children:"y"}),s.jsx(e.span,{className:"mspace",style:{marginRight:"0.2778em"}}),s.jsx(e.span,{className:"mrel",children:"∈"}),s.jsx(e.span,{className:"mspace",style:{marginRight:"0.2778em"}})]}),s.jsxs(e.span,{className:"base",children:[s.jsx(e.span,{className:"strut",style:{height:"1em",verticalAlign:"-0.25em"}}),s.jsx(e.span,{className:"mopen",children:"{"}),s.jsx(e.span,{className:"mord",children:"0"}),s.jsx(e.span,{className:"mpunct",children:","}),s.jsx(e.span,{className:"mspace",style:{marginRight:"0.1667em"}}),s.jsx(e.span,{className:"minner",children:"…"}),s.jsx(e.span,{className:"mspace",style:{marginRight:"0.1667em"}}),s.jsx(e.span,{className:"mpunct",children:","}),s.jsx(e.span,{className:"mspace",style:{marginRight:"0.1667em"}}),s.jsxs(e.span,{className:"mord",children:[s.jsx(e.span,{className:"mord mathnormal",children:"n"}),s.jsx(e.span,{className:"msupsub",children:s.jsxs(e.span,{className:"vlist-t vlist-t2",children:[s.jsxs(e.span,{className:"vlist-r",children:[s.jsx(e.span,{className:"vlist",style:{height:"0.3361em"}

3,children:s.jsxs(e.span,{style:{top:"-2.55em",marginLeft:"0em",marginRight:"0.05em"},children:[s.jsx(e.span,{className:"pstrut",style:{height:"2.7em"}}),s.jsx(e.span,{className:"sizing reset-size6 size3 mtight",children:s.jsx(e.span,{className:"mord text mtight",children:s.jsx(e.span,{className:"mord mtight",children:"words"})})})]})}),s.jsx(e.span,{className:"vlist-s",children:"​"})]}),s.jsx(e.span,{className:"vlist-r",children:s.jsx(e.span,{className:"vlist",style:{height:"0.15em"},children:s.jsx(e.span,{})})})]})})]}),s.jsx(e.span,{className:"mspace",style:{marginRight:"0.2222em"}}),s.jsx(e.span,{className:"mbin",children:"−"}),s.jsx(e.span,{className:"mspace",style:{marginRight:"0.2222em"}})]}),s.jsxs(e.span,{className:"base",children:[s.jsx(e.span,{className:"strut",style:{height:"1em",verticalAlign:"-0.25em"}}),s.jsx(e.span,{className:"mord",children:"1"}),s.jsx(e.span,{className:"mclose",children:"}"})]})]})]}),", we let"]}),`
4`,s.jsx(e.span,{className:"katex-display",children:s.jsxs(e.span,{className:"katex",children:[s.jsx(e.span,{className:"katex-mathml",children:s.jsx(e.math,{xmlns:"http://www.w3.org/1998/Math/MathML",display:"block",children:s.jsxs(e.semantics,{children:[s.jsx(e.mtable,{rowspacing:"0.16em",columnspacing:"1em",children:s.jsx(e.mtr,{children:s.jsx(e.mtd,{children:s.jsx(e.mstyle,{scriptlevel:"0",displaystyle:"true",children:s.jsxs(e.mrow,{children:[s.jsxs(e.msub,{children:[s.jsx(e.mtext,{mathvariant:"sans-serif",children:"shift"}),s.jsx(e.mtext,{children:"op"})]}),s.jsx(e.mo,{stretchy:"false",children:"("}),s.jsx(e.mi,{children:"w"}),s.jsx(e.mo,{stretchy:"false",children:")"}),s.jsx(e.mo,{stretchy:"false",children:"("}),s.jsx(e.mi,{children:"y"}),s.jsx(e.mo,{separator:"true",children:","}),s.jsx(e.mi,{children:"s"}),s.jsx(e.mo,{stretchy:"false",children:")"}),s.jsx(e.mo,{children:s.jsx(e.mi,{mathvariant:"normal",children:"≔"})}),s.jsx(e.mtext,{children:"op"}),s.jsx(e.mo,{stretchy:"false",children:"("}),s.jsx(e.mi,{children:"w"}),s.jsx(e.mo,{stretchy:"false",children:"["}),s.jsx(e.mi,{children:"y"}),s.jsx(e.mo,{stretchy:"false",children:"]"}),s.jsx(e.mo,{separator:"true",children:","}),s.jsx(e.mi,{children:"s"}),s.jsx(e.mo,{stretchy:"false",children:")"}),s.jsx(e.mi,{mathvariant:"normal",children:"."})]})})})})}),s.jsx(e.annotation,{encoding:"application/x-tex",children:"\\begin{equation*}\\textsf{shift}_{\\text{op}}(w)(y, s) \\coloneqq \\text{op}(w[y], s).\\end{equation*}"})]})})}),s.jsx(e.span,{className:"katex-html","aria-hidden":"true",children:s.jsxs(e.span,{className:"base",children:[s.jsx(e.span,{className:"strut",style:{height:"1.2em",verticalAlign:"-0.35em"}}),s.jsx(e.span,{className:"mord",children:s.jsx(e.span,{className:"mtable",children:s.jsx(e.span,{className:"col-align-c",children:s.jsxs(e.span,{className:"vlist-t vlist-t2",children:[s.jsxs(e.span,{className:"vlist-r",children:[s.jsx(e.span,{className:"vlist",style:{height:"0.85em"},children:s.jsxs(e.span,{style:{top:"-3.01em"},children:[s.jsx(e.span,{className:"pstrut",style:{height:"3em"}}),s.jsxs(e.span,{className:"mord",children:[s.jsxs(e.span,{className:"mord",children:[s.jsx(e.span,{className:"mord text",children:s.jsx(e.span,{className:"mord textsf",children:"shift"})}),s.jsx(e.span,{className:"msupsub",children:s.jsxs(e.span,{className:"vlist-t vlist-t2",children:[s.jsxs(e.span,{className:"vlist-r",children:[s.jsx(e.span,{className:"vlist",style:{height:"0.1514em"},children:s.jsxs(e.span,{style:{top:"-2.55em",marginRight:"0.05em"},children:[s.jsx(e.span,{className:"pstrut",style:{height:"2.7em"}}),s.jsx(e.span,{className:"sizing reset-size6 size3 mtight",children:s.jsx(e.span,{className:"mord mtight",children:s.jsx(e.span,{className:"mord text mtight",children:s.jsx(e.span,{className:"mord mtight",children:"op"})})})})]})}),s.jsx(e.span,{className:"vlist-s",children:"​"})]}),s.jsx(e.span,{className:"vlist-r",children:s.jsx(e.span,{className:"vlist",style:{height:"0.2861em"},children:s.jsx(e.span,{})})})]})})]}),s.jsx(e.span,{className:"mopen",children:"("}),s.jsx(e.span,{className:"mord mathnormal",style:{marginRight:"0.02691em"},children:"w"}),s.jsx(e.span,{className:"mclose",children:")"}),s.jsx(e.span,{className:"mopen",children:"("}),s.jsx(e.span,{className:"mord mathnormal",style:{marginRight:"0.03588em"},children:"y"}),s.jsx(e.span,{className:"mpunct",children:","}),s.jsx(e.span,{className:"mspace",style:{marginRight:"0.1667em"}}),s.jsx(e.span,{className:"mord mathnormal",children:"s"}),s.jsx(e.span,{className:"mclose",children:")"}),s.jsx(e.span,{className:"mspace",style:{marginRight:"0.2778em"}}),s.jsxs(e.span,{className:"mrel",children:[s.jsx(e.span,{className:"mrel",children:s.jsx(e.span,{className:"mop",style:{position:"relative",top:"-0.0347em"},children:":"})}),s.jsx(e.span,{className:"mrel",children:s.jsx(e.span,{className:"mspace",style:{marginRight:"-0.0667em"}})}),s.jsx(e.span,{className:"mrel",children:"="})]}),s.jsx(e.span,{className:"mspace",style:{marginRight:"0.2778em"}}),s.jsx(e.span,{className:"mord text",children:s.jsx(e.span,{className:"mord",children:"op"})}),s.jsx(e.span,{className:"mopen",children:"("}),s.jsx(e.span,{className:"mord mathnormal",style:{marginRight:"0.02691em"},children:"w"}),s.jsx(e.span,{className:"mopen",children:"["}),s.jsx(e.span,{className:"mord mathnormal",style:{marginRight:"0.03588em"},children:"y"}),s.jsx(e.span,{className:"mclose",children:"]"}),s.jsx(e.span,{className:"mpunct",children:","}),s.jsx(e.span,{className:"mspace",style:{marginRight:"0.1667em"}}),s.jsx(e.span,{className:"mord mathnormal",children:"s"}),s.jsx(e.span,{className:"mclose",children:")"}),s.jsx(e.span,{className:"mord",children:"."})]})]})}),s.jsx(e.span,{className:"vlist-s",children:"​"})]}),s.jsx(e.span,{className:"vlist-r",children:s.jsx(e.span,{className:"vlist",style:{height:"0.35em"},children:s.jsx(e.span,{})})})]})})})})]})})]})}),`
5`,s.jsxs(e.p,{children:["We will call these the ",s.jsx(e.em,{children:"shifted witness functions"}),"."]}),`
6`,s.jsxs(e.h2,{id:"as-polynomials",children:["As Polynomials",s.jsx(e.a,{"aria-hidden":"true",tabIndex:"-1",href:"#as-polynomials",children:s.jsx(e.div,{"data-autolink-icon":!0})})]}),`
7`,s.jsxs(e.p,{children:["What happens if we put both ",s.jsxs(e.span,{className:"katex",children:[s.jsx(e.span,{className:"katex-mathml",children:s.jsx(e.math,{xmlns:"http://www.w3.org/1998/Math/MathML",children:s.jsxs(e.semantics,{children:[s.jsx(e.mrow,{children:s.jsx(e.mi,{children:"w"})}),s.jsx(e.annotation,{encoding:"application/x-tex",children:"w"})]})})}),s.jsx(e.span,{className:"katex-html","aria-hidden":"true",children:s.jsxs(e.span,{className:"base",children:[s.jsx(e.span,{className:"strut",style:{height:"0.4306em"}}),s.jsx(e.span,{className:"mord mathnormal",style:{marginRight:"0.02691em"},children:"w"})]})})]})," and ",s.jsxs(e.span,{className:"katex",children:[s.jsx(e.span,{className:"katex-mathml",children:s.jsx(e.math,{xmlns:"http://www.w3.org/1998/Math/MathML",children:s.jsxs(e.semantics,{children:[s.jsxs(e.mrow,{children:[s.jsxs(e.msub,{children:[s.jsx(e.mtext,{mathvariant:"sans-serif",children:"shift"}),s.jsx(e.mtext,{children:"op"})]}),s.jsx(e.mo,{stretchy:"false",children:"("}),s.jsx(e.mi,{children:"w"}),s.jsx(e.mo,{stretchy:"false",children:")"}),s.jsx(e.mo,{stretchy:"false",children:"("}),s.jsx(e.mi,{children:"y"}),s.jsx(e.mo,{separator:"true",children:","}),s.jsx(e.mi,{children:"s"}),s.jsx(e.mo,{stretchy:"false",children:")"})]}),s.jsx(e.annotation,{encoding:"application/x-tex",children:"\\textsf{shift}_{\\text{op}}(w)(y, s)"})]})})}),s.jsx(e.span,{className:"katex-html","aria-hidden":"true",children:s.jsxs(e.span,{className:"base",children:[s.jsx(e.span,{className:"strut",style:{height:"1.0361em",verticalAlign:"-0.2861em"}}),s.jsxs(e.span,{className:"mord",children:[s.jsx(e.span,{className:"mord text",children:s.jsx(e.span,{className:"mord textsf",children:"shift"})}),s.jsx(e.span,{className:"msupsub",children:s.jsxs(e.span,{className:"vlist-t vlist-t2",children:[s.jsxs(e.span,{className:"vlist-r",children:[s.jsx(e.span,{className:"vlist",style:{height:"0.1514em"}

7,children:s.jsxs(e.span,{style:{top:"-2.55em",marginRight:"0.05em"},children:[s.jsx(e.span,{className:"pstrut",style:{height:"2.7em"}}),s.jsx(e.span,{className:"sizing reset-size6 size3 mtight",children:s.jsx(e.span,{className:"mord mtight",children:s.jsx(e.span,{className:"mord text mtight",children:s.jsx(e.span,{className:"mord mtight",children:"op"})})})})]})}),s.jsx(e.span,{className:"vlist-s",children:"​"})]}),s.jsx(e.span,{className:"vlist-r",children:s.jsx(e.span,{className:"vlist",style:{height:"0.2861em"},children:s.jsx(e.span,{})})})]})})]}),s.jsx(e.span,{className:"mopen",children:"("}),s.jsx(e.span,{className:"mord mathnormal",style:{marginRight:"0.02691em"},children:"w"}),s.jsx(e.span,{className:"mclose",children:")"}),s.jsx(e.span,{className:"mopen",children:"("}),s.jsx(e.span,{className:"mord mathnormal",style:{marginRight:"0.03588em"},children:"y"}),s.jsx(e.span,{className:"mpunct",children:","}),s.jsx(e.span,{className:"mspace",style:{marginRight:"0.1667em"}}),s.jsx(e.span,{className:"mord mathnormal",children:"s"}),s.jsx(e.span,{className:"mclose",children:")"})]})})]})," through ",s.jsx(e.a,{href:"/blueprint/math/multilinears",children:"multilinearization"}),"? How will the resulting multilinears relate to each other?"]}),`
8`,s.jsxs(e.p,{children:["At this point, we assume that ",s.jsxs(e.span,{className:"katex",children:[s.jsx(e.span,{className:"katex-mathml",children:s.jsx(e.math,{xmlns:"http://www.w3.org/1998/Math/MathML",children:s.jsxs(e.semantics,{children:[s.jsxs(e.mrow,{children:[s.jsxs(e.msub,{children:[s.jsx(e.mi,{children:"n"}),s.jsx(e.mtext,{children:"words"})]}),s.jsx(e.mo,{children:"="}),s.jsxs(e.msup,{children:[s.jsx(e.mn,{children:"2"}),s.jsxs(e.msub,{children:[s.jsx(e.mi,{mathvariant:"normal",children:"ℓ"}),s.jsx(e.mtext,{children:"words"})]})]})]}),s.jsx(e.annotation,{encoding:"application/x-tex",children:"n_\\text{words} = 2^{\\ell_\\text{words}}"})]})})}),s.jsxs(e.span,{className:"katex-html","aria-hidden":"true",children:[s.jsxs(e.span,{className:"base",children:[s.jsx(e.span,{className:"strut",style:{height:"0.5806em",verticalAlign:"-0.15em"}}),s.jsxs(e.span,{className:"mord",children:[s.jsx(e.span,{className:"mord mathnormal",children:"n"}),s.jsx(e.span,{className:"msupsub",children:s.jsxs(e.span,{className:"vlist-t vlist-t2",children:[s.jsxs(e.span,{className:"vlist-r",children:[s.jsx(e.span,{className:"vlist",style:{height:"0.3361em"},children:s.jsxs(e.span,{style:{top:"-2.55em",marginLeft:"0em",marginRight:"0.05em"},children:[s.jsx(e.span,{className:"pstrut",style:{height:"2.7em"}}),s.jsx(e.span,{className:"sizing reset-size6 size3 mtight",children:s.jsx(e.span,{className:"mord text mtight",children:s.jsx(e.span,{className:"mord mtight",children:"words"})})})]})}),s.jsx(e.span,{className:"vlist-s",children:"​"})]}),s.jsx(e.span,{className:"vlist-r",children:s.jsx(e.span,{className:"vlist",style:{height:"0.15em"},children:s.jsx(e.span,{})})})]})})]}),s.jsx(e.span,{className:"mspace",style:{marginRight:"0.2778em"}}),s.jsx(e.span,{className:"mrel",children:"="}),s.jsx(e.span,{className:"mspace",style:{marginRight:"0.2778em"}})]}),s.jsxs(e.span,{className:"base",children:[s.jsx(e.span,{className:"strut",style:{height:"0.8491em"}}),s.jsxs(e.span,{className:"mord",children:[s.jsx(e.span,{className:"mord",children:"2"}),s.jsx(e.span,{className:"msupsub",children:s.jsx(e.span,{className:"vlist-t",children:s.jsx(e.span,{className:"vlist-r",children:s.jsx(e.span,{className:"vlist",style:{height:"0.8491em"},children:s.jsxs(e.span,{style:{top:"-3.063em",marginRight:"0.05em"},children:[s.jsx(e.span,{className:"pstrut",style:{height:"2.7em"}}),s.jsx(e.span,{className:"sizing reset-size6 size3 mtight",children:s.jsx(e.span,{className:"mord mtight",children:s.jsxs(e.span,{className:"mord mtight",children:[s.jsx(e.span,{className:"mord mtight",children:"ℓ"}),s.jsx(e.span,{className:"msupsub",children:s.jsxs(e.span,{className:"vlist-t vlist-t2",children:[s.jsxs(e.span,{className:"vlist-r",children:[s.jsx(e.span,{className:"vlist",style:{height:"0.3448em"}

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41`,s.jsxs(e.p,{children:["For each such ",s.jsxs(e.span,{className:"katex",children:[s.jsx(e.span,{className:"katex-mathml",children:s.jsx(e.math,{xmlns:"http://www.w3.org/1998/Math/MathML",children:s.jsxs(e.semantics,{children:[s.jsx(e.mrow,{children:s.jsx(e.mtext,{children:"op"})}),s.jsx(e.annotation,{encoding:"application/x-tex",children:"\\text{op}"})]})})}),s.jsx(e.span,{className:"katex-html","aria-hidden":"true",children:s.jsxs(e.span,{className:"base",children:[s.jsx(e.span,{className:"strut",style:{height:"0.625em",verticalAlign:"-0.1944em"}}),s.jsx(e.span,{className:"mord text",children:s.jsx(e.span,{className:"mord",children:"op"})})]})})]}),", we define a corresponding ",s.jsx(e.em,{children:"function"})," from ",s.jsxs(e.span,{className:"katex",children:[s.jsx(e.span,{className:"katex-mathml",children:s.jsx(e.math,{xmlns:"http://www.w3.org/1998/Math/MathML",children:s.jsxs(e.semantics,{children:[s.jsxs(e.mrow,{children:[s.jsxs(e.msub,{children:[s.jsx(e.mi,{mathvariant:"script",children:"B"}),s.jsx(e.mn,{children:"6"})]}),s.jsx(e.mo,{children:"×"}),s.jsxs(e.msub,{children:[s.jsx(e.mi,{mathvariant:"script",children:"B"}),s.jsx(e.mn,{children:"6"})]}),s.jsx(e.mo,{children:"×"}),s.jsxs(e.msub,{children:[s.jsx(e.mi,{mathvariant:"script",children:"B"}),s.jsx(e.mn,{children:"6"})]}),s.jsx(e.mo,{children:"→"}),s.jsx(e.mo,{stretchy:"false",children:"{"}),s.jsx(e.mn,{children:"0"}),s.jsx(e.mo,{separator:"true",children:","}),s.jsx(e.mn,{children:"1"}),s.jsx(e.mo,{stretchy:"false",children:"}"})]}),s.jsx(e.annotation,{encoding:"application/x-tex",children:"\\mathcal{B}_6 \\times \\mathcal{B}_6 \\times \\mathcal{B}_6 \\to \\{0, 1\\}"})]})})}),s.jsxs(e.span,{className:"katex-html","aria-hidden":"true",children:[s.jsxs(e.span,{className:"base",children:[s.jsx(e.span,{className:"strut",style:{height:"0.8333em",verticalAlign:"-0.15em"}}),s.jsxs(e.span,{className:"mord",children:[s.jsx(e.span,{className:"mord mathcal",style:{marginRight:"0.03041em"},children:"B"}),s.jsx(e.span,{className:"msupsub",children:s.jsxs(e.span,{className:"vlist-t vlist-t2",children:[s.jsxs(e.span,{className:"vlist-r",children:[s.jsx(e.span,{className:"vlist",style:{height:"0.3011em"}

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42`,s.jsx(e.span,{className:"katex-display",children:s.jsxs(e.span,{className:"katex",children:[s.jsx(e.span,{className:"katex-mathml",children:s.jsx(e.math,{xmlns:"http://www.w3.org/1998/Math/MathML",display:"block",children:s.jsxs(e.semantics,{children:[s.jsx(e.mtable,{rowspacing:"0.16em",columnspacing:"1em",children:s.jsx(e.mtr,{children:s.jsx(e.mtd,{children:s.jsx(e.mstyle,{scriptlevel:"0",displaystyle:"true",children:s.jsxs(e.mrow,{children:[s.jsxs(e.msub,{children:[s.jsx(e.mtext,{mathvariant:"sans-serif",children:"shift-ind"}),s.jsx(e.mtext,{children:"op"})]}),s.jsx(e.mo,{stretchy:"false",children:"("}),s.jsx(e.mi,{children:"i"}),s.jsx(e.mo,{separator:"true",children:","}),s.jsx(e.mi,{children:"j"}),s.jsx(e.mo,{separator:"true",children:","}),s.jsx(e.mi,{children:"s"}),s.jsx(e.mo,{stretchy:"false",children:")"}),s.jsx(e.mo,{children:s.jsx(e.mi,{mathvariant:"normal",children:"≔"})}),s.jsxs(e.mrow,{children:[s.jsx(e.mo,{fence:"true",children:"{"}),s.jsxs(e.mtable,{rowspacing:"0.36em",columnalign:"left left",columnspacing:"1em",children:[s.jsxs(e.mtr,{children:[s.jsx(e.mtd,{children:s.jsx(e.mstyle,{scriptlevel:"0",displaystyle:"false",children:s.jsx(e.mn,{children:"1"})})}),s.jsx(e.mtd,{children:s.jsx(e.mstyle,{scriptlevel:"0",displaystyle:"false",children:s.jsxs(e.mrow,{children:[s.jsx(e.mtext,{children:" if the "}),s.jsx(e.mstyle,{scriptlevel:"0",displaystyle:"false",children:s.jsxs(e.msup,{children:[s.jsx(e.mi,{children:"i"}),s.jsx(e.mtext,{children:"th"})]})}),s.jsx(e.mtext,{children:" bit of "}),s.jsxs(e.mstyle,{scriptlevel:"0",displaystyle:"false",children:[s.jsx(e.mtext,{children:"op"}),s.jsx(e.mo,{stretchy:"false",children:"("}),s.jsx(e.mi,{children:"v"}),s.jsx(e.mo,{separator:"true",children:","}),s.jsx(e.mi,{children:"s"}),s.jsx(e.mo,{stretchy:"false",children:")"})]}),s.jsx(e.mtext,{children:" "}),s.jsxs(e.mrow,{children:[s.jsx(e.mtext,{children:"is the "}),s.jsx(e.mstyle,{scriptlevel:"0",displaystyle:"false",children:s.jsxs(e.msup,{children:[s.jsx(e.mi,{children:"j"}),s.jsx(e.mtext,{children:"th"})]})}),s.jsx(e.mtext,{children:" bit of "}),s.jsx(e.mstyle,{scriptlevel:"0",displaystyle:"false",children:s.jsx(e.mi,{children:"v"})})]})]})})})]}),s.jsxs(e.mtr,{children:[s.jsx(e.mtd,{children:s.jsx(e.mstyle,{scriptlevel:"0",displaystyle:"false",children:s.jsx(e.mn,{children:"0"})})}),s.jsx(e.mtd,{children:s.jsx(e.mstyle,{scriptlevel:"0",displaystyle:"false",children:s.jsx(e.mtext,{children:" otherwise"})})})]})]})]})]})})})})}),s.jsx(e.annotation,{encoding:"application/x-tex",children:`\\begin{equation*}\\textsf{shift-ind}_\\text{op}(i, j, s) \\coloneqq \\begin{cases}
431 & \\text{ if the $i^{\\text{th}}$ bit of $\\text{op}(v, s)$ \\text{is the $j^{\\text{th}}$ bit of $v$}} \\\\
440 & \\text{ otherwise}
45\\end{cases}\\end{equation*}`})]})})}),s.jsx(e.span,{className:"katex-html","aria-hidden":"true",children:s.jsxs(e.span,{className:"base",children:[s.jsx(e.span,{className:"strut",style:{height:"3em",verticalAlign:"-1.25em"}}),s.jsx(e.span,{className:"mord",children:s.jsx(e.span,{className:"mtable",children:s.jsx(e.span,{className:"col-align-c",children:s.jsxs(e.span,{className:"vlist-t vlist-t2",children:[s.jsxs(e.span,{className:"vlist-r",children:[s.jsx(e.span,{className:"vlist",style:{height:"1.75em"}

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46`,s.jsxs(e.p,{children:["For each triple of inputs ",s.jsxs(e.span,{className:"katex",children:[s.jsx(e.span,{className:"katex-mathml",children:s.jsx(e.math,{xmlns:"http://www.w3.org/1998/Math/MathML",children:s.jsxs(e.semantics,{children:[s.jsx(e.mrow,{children:s.jsx(e.mi,{children:"i"})}),s.jsx(e.annotation,{encoding:"application/x-tex",children:"i"})]})})}),s.jsx(e.span,{className:"katex-html","aria-hidden":"true",children:s.jsxs(e.span,{className:"base",children:[s.jsx(e.span,{className:"strut",style:{height:"0.6595em"}}),s.jsx(e.span,{className:"mord mathnormal",children:"i"})]})})]}),", 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The shift indicator picks up that fact."]}),`
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47,children:s.jsxs(e.span,{style:{top:"-2.55em",marginLeft:"0em",marginRight:"0.05em"},children:[s.jsx(e.span,{className:"pstrut",style:{height:"2.7em"}}),s.jsx(e.span,{className:"sizing reset-size6 size3 mtight",children:s.jsx(e.span,{className:"mord mtight",children:s.jsx(e.span,{className:"mord mtight",children:"30"})})})]})}),s.jsx(e.span,{className:"vlist-s",children:"​"})]}),s.jsx(e.span,{className:"vlist-r",children:s.jsx(e.span,{className:"vlist",style:{height:"0.15em"},children:s.jsx(e.span,{})})})]})})]}),s.jsx(e.span,{className:"mspace",style:{marginRight:"0.2778em"}}),s.jsx(e.span,{className:"mrel",children:"="}),s.jsx(e.span,{className:"mspace",style:{marginRight:"0.2778em"}})]}),s.jsxs(e.span,{className:"base",children:[s.jsx(e.span,{className:"strut",style:{height:"0.5806em",verticalAlign:"-0.15em"}}),s.jsxs(e.span,{className:"mord",children:[s.jsx(e.span,{className:"mord mathnormal",style:{marginRight:"0.03588em"},children:"v"}),s.jsx(e.span,{className:"msupsub",children:s.jsxs(e.span,{className:"vlist-t vlist-t2",children:[s.jsxs(e.span,{className:"vlist-r",children:[s.jsx(e.span,{className:"vlist",style:{height:"0.3011em"},children:s.jsxs(e.span,{style:{top:"-2.55em",marginLeft:"-0.0359em",marginRight:"0.05em"},children:[s.jsx(e.span,{className:"pstrut",style:{height:"2.7em"}}),s.jsx(e.span,{className:"sizing reset-size6 size3 mtight",children:s.jsx(e.span,{className:"mord mtight",children:s.jsx(e.span,{className:"mord mtight",children:"20"})})})]})}),s.jsx(e.span,{className:"vlist-s",children:"​"})]}),s.jsx(e.span,{className:"vlist-r",children:s.jsx(e.span,{className:"vlist",style:{height:"0.15em"},children:s.jsx(e.span,{})})})]})})]})]})]})]}),", regardless of ",s.jsxs(e.span,{className:"katex",children:[s.jsx(e.span,{className:"katex-mathml",children:s.jsx(e.math,{xmlns:"http://www.w3.org/1998/Math/MathML",children:s.jsxs(e.semantics,{children:[s.jsx(e.mrow,{children:s.jsx(e.mi,{children:"v"})}),s.jsx(e.annotation,{encoding:"application/x-tex",children:"v"})]})})}),s.jsx(e.span,{className:"katex-html","aria-hidden":"true",children:s.jsxs(e.span,{className:"base",children:[s.jsx(e.span,{className:"strut",style:{height:"0.4306em"}}),s.jsx(e.span,{className:"mord mathnormal",style:{marginRight:"0.03588em"},children:"v"})]})})]}),". This means that ",s.jsxs(e.span,{className:"katex",children:[s.jsx(e.span,{className:"katex-mathml",children:s.jsx(e.math,{xmlns:"http://www.w3.org/1998/Math/MathML",children:s.jsxs(e.semantics,{children:[s.jsxs(e.mrow,{children:[s.jsxs(e.msub,{children:[s.jsx(e.mtext,{mathvariant:"sans-serif",children:"shift-ind"}),s.jsxs(e.mrow,{children:[s.jsx(e.mi,{mathvariant:"sans-serif",children:"s"}),s.jsx(e.mi,{mathvariant:"sans-serif",children:"l"}),s.jsx(e.mi,{mathvariant:"sans-serif",children:"l"})]})]}),s.jsx(e.mo,{stretchy:"false",children:"("}),s.jsx(e.mn,{children:"30"}),s.jsx(e.mo,{separator:"true",children:","}),s.jsx(e.mn,{children:"20"}),s.jsx(e.mo,{separator:"true",children:","}),s.jsx(e.mn,{children:"10"}),s.jsx(e.mo,{stretchy:"false",children:")"}),s.jsx(e.mo,{children:"="}),s.jsx(e.mn,{children:"1"})]}),s.jsx(e.annotation,{encoding:"application/x-tex",children:"\\textsf{shift-ind}_\\mathsf{sll}(30, 20, 10) = 1"})]})})}),s.jsxs(e.span,{className:"katex-html","aria-hidden":"true",children:[s.jsxs(e.span,{className:"base",children:[s.jsx(e.span,{className:"strut",style:{height:"1em",verticalAlign:"-0.25em"}}),s.jsxs(e.span,{className:"mord",children:[s.jsx(e.span,{className:"mord text",children:s.jsx(e.span,{className:"mord textsf",children:"shift-ind"})}),s.jsx(e.span,{className:"msupsub",children:s.jsxs(e.span,{className:"vlist-t vlist-t2",children:[s.jsxs(e.span,{className:"vlist-r",children:[s.jsx(e.span,{className:"vlist",style:{height:"0.3361em"},children:s.jsxs(e.span,{style:{top:"-2.55em",marginRight:"0.05em"},children:[s.jsx(e.span,{className:"pstrut",style:{height:"2.7em"}}),s.jsx(e.span,{className:"sizing reset-size6 size3 mtight",children:s.jsx(e.span,{className:"mord mtight",children:s.jsx(e.span,{className:"mord mathsf mtight",children:"sll"})})})]})}),s.jsx(e.span,{className:"vlist-s",children:"​"})]}),s.jsx(e.span,{className:"vlist-r",children:s.jsx(e.span,{className:"vlist",style:{height:"0.15em"},children:s.jsx(e.span,{})})})]})})]}),s.jsx(e.span,{className:"mopen",children:"("}),s.jsx(e.span,{className:"mord",children:"30"}),s.jsx(e.span,{className:"mpunct",children:","}),s.jsx(e.span,{className:"mspace",style:{marginRight:"0.1667em"}}),s.jsx(e.span,{className:"mord",children:"20"}),s.jsx(e.span,{className:"mpunct",children:","}),s.jsx(e.span,{className:"mspace",style:{marginRight:"0.1667em"}}),s.jsx(e.span,{className:"mord",children:"10"}),s.jsx(e.span,{className:"mclose",children:")"}),s.jsx(e.span,{className:"mspace",style:{marginRight:"0.2778em"}}),s.jsx(e.span,{className:"mrel",children:"="}),s.jsx(e.span,{className:"mspace",style:{marginRight:"0.2778em"}})]}),s.jsxs(e.span,{className:"base",children:[s.jsx(e.span,{className:"strut",style:{height:"0.6444em"}}),s.jsx(e.span,{className:"mord",children:"1"})]})]})]}),". On the other hand, ",s.jsxs(e.span,{className:"katex",children:[s.jsx(e.span,{className:"katex-mathml",children:s.jsx(e.math,{xmlns:"http://www.w3.org/1998/Math/MathML",children:s.jsxs(e.semantics,{children:[s.jsxs(e.mrow,{children:[s.jsxs(e.mrow,{children:[s.jsx(e.mi,{mathvariant:"sans-serif",children:"s"}),s.jsx(e.mi,{mathvariant:"sans-serif",children:"l"}),s.jsx(e.mi,{mathvariant:"sans-serif",children:"l"})]}),s.jsx(e.mo,{stretchy:"false",children:"("}),s.jsx(e.mi,{children:"v"}),s.jsx(e.mo,{separator:"true",children:","}),s.jsx(e.mn,{children:"10"}),s.jsxs(e.msub,{children:[s.jsx(e.mo,{stretchy:"false",children:")"}),s.jsx(e.mn,{children:"30"})]})]}),s.jsx(e.annotation,{encoding:"application/x-tex",children:"\\mathsf{sll}(v, 10)_{30}"})]})})}),s.jsx(e.span,{className:"katex-html","aria-hidden":"true",children:s.jsxs(e.span,{className:"base",children:[s.jsx(e.span,{className:"strut",style:{height:"1em",verticalAlign:"-0.25em"}}),s.jsx(e.span,{className:"mord",children:s.jsx(e.span,{className:"mord mathsf",children:"sll"})}),s.jsx(e.span,{className:"mopen",children:"("}),s.jsx(e.span,{className:"mord mathnormal",style:{marginRight:"0.03588em"},children:"v"}),s.jsx(e.span,{className:"mpunct",children:","}),s.jsx(e.span,{className:"mspace",style:{marginRight:"0.1667em"}}),s.jsx(e.span,{className:"mord",children:"10"}),s.jsxs(e.span,{className:"mclose",children:[s.jsx(e.span,{className:"mclose",children:")"}),s.jsx(e.span,{className:"msupsub",children:s.jsxs(e.span,{className:"vlist-t vlist-t2",children:[s.jsxs(e.span,{className:"vlist-r",children:[s.jsx(e.span,{className:"vlist",style:{height:"0.3011em"}

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",s.jsxs(e.span,{className:"katex",children:[s.jsx(e.span,{className:"katex-mathml",children:s.jsx(e.math,{xmlns:"http://www.w3.org/1998/Math/MathML",children:s.jsxs(e.semantics,{children:[s.jsxs(e.mrow,{children:[s.jsx(e.mi,{children:"j"}),s.jsx(e.mo,{children:"∈"}),s.jsxs(e.msub,{children:[s.jsx(e.mi,{mathvariant:"script",children:"B"}),s.jsx(e.mn,{children:"6"})]})]}),s.jsx(e.annotation,{encoding:"application/x-tex",children:"j \\in \\mathcal{B}_6"})]})})}),s.jsxs(e.span,{className:"katex-html","aria-hidden":"true",children:[s.jsxs(e.span,{className:"base",children:[s.jsx(e.span,{className:"strut",style:{height:"0.854em",verticalAlign:"-0.1944em"}}),s.jsx(e.span,{className:"mord 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48`,s.jsxs(e.p,{children:["With a bit of work we can unwind exactly what these mean. For each ",s.jsxs(e.span,{className:"katex",children:[s.jsx(e.span,{className:"katex-mathml",children:s.jsx(e.math,{xmlns:"http://www.w3.org/1998/Math/MathML",children:s.jsxs(e.semantics,{children:[s.jsx(e.mrow,{children:s.jsx(e.mi,{children:"i"})}),s.jsx(e.annotation,{encoding:"application/x-tex",children:"i"})]})})}),s.jsx(e.span,{className:"katex-html","aria-hidden":"true",children:s.jsxs(e.span,{className:"base",children:[s.jsx(e.span,{className:"strut",style:{height:"0.6595em"}}),s.jsx(e.span,{className:"mord mathnormal",children:"i"})]})})]}),", ",s.jsxs(e.span,{className:"katex",children:[s.jsx(e.span,{className:"katex-mathml",children:s.jsx(e.math,{xmlns:"http://www.w3.org/1998/Math/MathML",children:s.jsxs(e.semantics,{children:[s.jsx(e.mrow,{children:s.jsx(e.mi,{children:"j"})}),s.jsx(e.annotation,{encoding:"application/x-tex",children:"j"})]})})}),s.jsx(e.span,{className:"katex-html","aria-hidden":"true",children:s.jsxs(e.span,{className:"base",children:[s.jsx(e.span,{className:"strut",style:{height:"0.854em",verticalAlign:"-0.1944em"}}),s.jsx(e.span,{className:"mord mathnormal",style:{marginRight:"0.05724em"},children:"j"})]})})]})," and ",s.jsxs(e.span,{className:"katex",children:[s.jsx(e.span,{className:"katex-mathml",children:s.jsx(e.math,{xmlns:"http://www.w3.org/1998/Math/MathML",children:s.jsxs(e.semantics,{children:[s.jsx(e.mrow,{children:s.jsx(e.mi,{children:"s"})}),s.jsx(e.annotation,{encoding:"application/x-tex",children:"s"})]})})}),s.jsx(e.span,{className:"katex-html","aria-hidden":"true",children:s.jsxs(e.span,{className:"base",children:[s.jsx(e.span,{className:"strut",style:{height:"0.4306em"}}),s.jsx(e.span,{className:"mord mathnormal",children:"s"})]})})]})," in ",s.jsxs(e.span,{className:"katex",children:[s.jsx(e.span,{className:"katex-mathml",children:s.jsx(e.math,{xmlns:"http://www.w3.org/1998/Math/MathML",children:s.jsxs(e.semantics,{children:[s.jsx(e.mrow,{children:s.jsxs(e.msub,{children:[s.jsx(e.mi,{mathvariant:"script",children:"B"}),s.jsx(e.mn,{children:"6"})]})}),s.jsx(e.annotation,{encoding:"application/x-tex",children:"\\mathcal{B}_6"})]})})}),s.jsx(e.span,{className:"katex-html","aria-hidden":"true",children:s.jsxs(e.span,{className:"base",children:[s.jsx(e.span,{className:"strut",style:{height:"0.8333em",verticalAlign:"-0.15em"}}),s.jsxs(e.span,{className:"mord",children:[s.jsx(e.span,{className:"mord mathcal",style:{marginRight:"0.03041em"},children:"B"}),s.jsx(e.span,{className:"msupsub",children:s.jsxs(e.span,{className:"vlist-t vlist-t2",children:[s.jsxs(e.span,{className:"vlist-r",children:[s.jsx(e.span,{className:"vlist",style:{height:"0.3011em"},children:s.jsxs(e.span,{style:{top:"-2.55em",marginLeft:"-0.0304em",marginRight:"0.05em"},children:[s.jsx(e.span,{className:"pstrut",style:{height:"2.7em"}}),s.jsx(e.span,{className:"sizing reset-size6 size3 mtight",children:s.jsx(e.span,{className:"mord mtight",children:"6"})})]})}),s.jsx(e.span,{className:"vlist-s",children:"​"})]}),s.jsx(e.span,{className:"vlist-r",children:s.jsx(e.span,{className:"vlist",style:{height:"0.15em"},children:s.jsx(e.span,{})})})]})})]})]})})]}),","]}),`
49`,s.jsxs(e.ul,{children:[`
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65`]}),`
66`,s.jsx(e.p,{children:"We will come back to these descriptions when we arithmetize the functions, in the next two pages."}),`
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68`,s.jsx(e.p,{children:"Now, we will write"}),`
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