https://www.binius.xyz/assets/fields-iYf67usw.js

1import{u as l,j as s}from"./index-Bpb9-9J-.js";const m={title:"Binary Fields",description:"Basics on the fields we use"};function n(a){const e={a:"a",annotation:"annotation",div:"div",em:"em",h1:"h1",h2:"h2",header:"header",math:"math",mi:"mi",mn:"mn",mo:"mo",mrow:"mrow",msub:"msub",msubsup:"msubsup",msup:"msup",p:"p",semantics:"semantics",span:"span",...l(),...a.components};return s.jsxs(s.Fragment,{children:[s.jsxs(e.header,{children:[s.jsxs(e.h1,{id:"binary-fields",children:["Binary Fields",s.jsx(e.a,{"aria-hidden":"true",tabIndex:"-1",href:"#binary-fields",children:s.jsx(e.div,{"data-autolink-icon":!0})})]}),s.jsx(e.div,{role:"doc-subtitle",children:"Basics on the fields we use"})]}),`
2`,s.jsxs(e.p,{children:["A ",s.jsx(e.em,{children:"field"})," is a set equipped with addition and multiplication operations, which satisfy the commutative, associative and distributive laws, and where everything nonzero has an inverse. A ",s.jsx(e.em,{children:"finite field"})," is moreover finite."]}),`
3`,s.jsxs(e.p,{children:["The finite fields you're probably most used to look like ",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:"double-struck",children:"F"}),s.jsx(e.mi,{children:"p"})]}),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.mi,{children:"p"}),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:"\\mathbb{F}_p = \\{0, \\ldots , p - 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:"0.975em",verticalAlign:"-0.2861em"}}),s.jsxs(e.span,{className:"mord",children:[s.jsx(e.span,{className:"mord mathbb",children:"F"}),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",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 mathnormal mtight",children:"p"})})]})}),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:"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 mathnormal",children:"p"}),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:"}"})]})]})]}),", where ",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:"p"})}),s.jsx(e.annotation,{encoding:"application/x-tex",children:"p"})]})})}),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 mathnormal",children:"p"})]})})]})," is a large prime. Arithmetic in that set is carried out modulo ",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:"p"})}),s.jsx(e.annotation,{encoding:"application/x-tex",children:"p"})]})})}),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 mathnormal",children:"p"})]})})]}),". Elements of the field can be represented, concretely, as ",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.mo,{stretchy:"false",children:"⌈"}),s.jsxs(e.msub,{children:[s.jsxs(e.mrow,{children:[s.jsx(e.mi,{children:"log"}),s.jsx(e.mo,{children:"⁡"})]}),s.jsx(e.mn,{children:"2"})]}),s.jsx(e.mi,{children:"p"}),s.jsx(e.mo,{stretchy:"false",children:"⌉"})]}),s.jsx(e.annotation,{encoding:"application/x-tex",children:"\\lceil \\log_2 p \\rceil"})]})})}),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:"mopen",children:"⌈"}),s.jsxs(e.span,{className:"mop",children:[s.jsxs(e.span,{className:"mop",children:["lo",s.jsx(e.span,{style:{marginRight:"0.01389em"},children:"g"})]}),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:"0.207em"}

3,children:s.jsxs(e.span,{style:{top:"-2.4559em",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:"2"})})]})}),s.jsx(e.span,{className:"vlist-s",children:"​"})]}),s.jsx(e.span,{className:"vlist-r",children:s.jsx(e.span,{className:"vlist",style:{height:"0.2441em"},children:s.jsx(e.span,{})})})]})})]}),s.jsx(e.span,{className:"mspace",style:{marginRight:"0.1667em"}}),s.jsx(e.span,{className:"mord mathnormal",children:"p"}),s.jsx(e.span,{className:"mclose",children:"⌉"})]})})]}),"-bit strings. Some of those strings—i.e., the ones representing integers in the high range ",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.mo,{stretchy:"false",children:"{"}),s.jsx(e.mi,{children:"p"}),s.jsx(e.mo,{separator:"true",children:","}),s.jsx(e.mo,{children:"…"}),s.jsx(e.mo,{separator:"true",children:","}),s.jsx(e.mo,{stretchy:"false",children:"⌈"}),s.jsxs(e.msub,{children:[s.jsxs(e.mrow,{children:[s.jsx(e.mi,{children:"log"}),s.jsx(e.mo,{children:"⁡"})]}),s.jsx(e.mn,{children:"2"})]}),s.jsx(e.mi,{children:"p"}),s.jsx(e.mo,{stretchy:"false",children:"⌉"}),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:"\\{p, \\ldots , \\lceil \\log_2 p \\rceil - 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.jsx(e.span,{className:"mopen",children:"{"}),s.jsx(e.span,{className:"mord mathnormal",children:"p"}),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:"mopen",children:"⌈"}),s.jsxs(e.span,{className:"mop",children:[s.jsxs(e.span,{className:"mop",children:["lo",s.jsx(e.span,{style:{marginRight:"0.01389em"},children:"g"})]}),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:"0.207em"},children:s.jsxs(e.span,{style:{top:"-2.4559em",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:"2"})})]})}),s.jsx(e.span,{className:"vlist-s",children:"​"})]}),s.jsx(e.span,{className:"vlist-r",children:s.jsx(e.span,{className:"vlist",style:{height:"0.2441em"},children:s.jsx(e.span,{})})})]})})]}),s.jsx(e.span,{className:"mspace",style:{marginRight:"0.1667em"}}),s.jsx(e.span,{className:"mord mathnormal",children:"p"}),s.jsx(e.span,{className:"mclose",children:"⌉"}),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:"}"})]})]})]}),'—will be "off-limits".']}),`
4`,s.jsx(e.p,{children:"Here, we work in characteristic 2. This means that we construct our fields differently."}),`
5`,s.jsxs(e.h2,{id:"construction",children:["Construction",s.jsx(e.a,{"aria-hidden":"true",tabIndex:"-1",href:"#construction",children:s.jsx(e.div,{"data-autolink-icon":!0})})]}),`
6`,s.jsxs(e.p,{children:["The simplest binary field is ",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:"double-struck",children:"F"}),s.jsx(e.mn,{children:"2"})]}),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:"\\mathbb{F}_2 = \\{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:"0.8389em",verticalAlign:"-0.15em"}}),s.jsxs(e.span,{className:"mord",children:[s.jsx(e.span,{className:"mord mathbb",children:"F"}),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"}

6,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:"2"})})]})}),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:"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:"mord",children:"1"}),s.jsx(e.span,{className:"mclose",children:"}"})]})]})]}),", the field with two elements."]}),`
7`,s.jsxs(e.p,{children:["By taking algebraic extensions of this field, we can get further binary fields. We pick a bit-length. For now, we'll stick with ",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.mn,{children:"128"})}),s.jsx(e.annotation,{encoding:"application/x-tex",children:"128"})]})})}),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:"0.6444em"}}),s.jsx(e.span,{className:"mord",children:"128"})]})})]}),". We fix once and for all a global irreducible polynomial ",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:"f"}),s.jsx(e.mo,{stretchy:"false",children:"("}),s.jsx(e.mi,{children:"X"}),s.jsx(e.mo,{stretchy:"false",children:")"}),s.jsx(e.mo,{children:"∈"}),s.jsxs(e.msub,{children:[s.jsx(e.mi,{mathvariant:"double-struck",children:"F"}),s.jsx(e.mn,{children:"2"})]}),s.jsx(e.mo,{stretchy:"false",children:"["}),s.jsx(e.mi,{children:"X"}),s.jsx(e.mo,{stretchy:"false",children:"]"})]}),s.jsx(e.annotation,{encoding:"application/x-tex",children:"f(X) \\in \\mathbb{F}_2[X]"})]})})}),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.jsx(e.span,{className:"mord mathnormal",style:{marginRight:"0.10764em"},children:"f"}),s.jsx(e.span,{className:"mopen",children:"("}),s.jsx(e.span,{className:"mord mathnormal",style:{marginRight:"0.07847em"},children:"X"}),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:"1em",verticalAlign:"-0.25em"}}),s.jsxs(e.span,{className:"mord",children:[s.jsx(e.span,{className:"mord mathbb",children:"F"}),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:"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:"2"})})]})}),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 mathnormal",style:{marginRight:"0.07847em"},children:"X"}),s.jsx(e.span,{className:"mclose",children:"]"})]})]})]})," of degree ",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.mn,{children:"128"})}),s.jsx(e.annotation,{encoding:"application/x-tex",children:"128"})]})})}),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:"0.6444em"}}),s.jsx(e.span,{className:"mord",children:"128"})]})})]}),". The corresponding quotient ring ",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:"double-struck",children:"F"}),s.jsx(e.mn,{children:"2"})]}),s.jsx(e.mo,{stretchy:"false",children:"["}),s.jsx(e.mi,{children:"X"}),s.jsx(e.mo,{stretchy:"false",children:"]"}),s.jsx(e.mi,{mathvariant:"normal",children:"/"}),s.jsx(e.mo,{stretchy:"false",children:"("}),s.jsx(e.mi,{children:"f"}),s.jsx(e.mo,{stretchy:"false",children:"("}),s.jsx(e.mi,{children:"X"}),s.jsx(e.mo,{stretchy:"false",children:")"}),s.jsx(e.mo,{stretchy:"false",children:")"})]}),s.jsx(e.annotation,{encoding:"application/x-tex",children:"\\mathbb{F}_2[X] / (f(X))"})]})})}),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.jsxs(e.span,{className:"mord",children:[s.jsx(e.span,{className:"mord mathbb",children:"F"}),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"}

7,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:"2"})})]})}),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 mathnormal",style:{marginRight:"0.07847em"},children:"X"}),s.jsx(e.span,{className:"mclose",children:"]"}),s.jsx(e.span,{className:"mord",children:"/"}),s.jsx(e.span,{className:"mopen",children:"("}),s.jsx(e.span,{className:"mord mathnormal",style:{marginRight:"0.10764em"},children:"f"}),s.jsx(e.span,{className:"mopen",children:"("}),s.jsx(e.span,{className:"mord mathnormal",style:{marginRight:"0.07847em"},children:"X"}),s.jsx(e.span,{className:"mclose",children:"))"})]})})]})," is a field, namely ",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:"double-struck",children:"F"}),s.jsxs(e.msup,{children:[s.jsx(e.mn,{children:"2"}),s.jsx(e.mn,{children:"128"})]})]})}),s.jsx(e.annotation,{encoding:"application/x-tex",children:"\\mathbb{F}_{2^{128}}"})]})})}),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:"0.8665em",verticalAlign:"-0.1776em"}}),s.jsxs(e.span,{className:"mord",children:[s.jsx(e.span,{className:"mord mathbb",children:"F"}),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"},children:s.jsxs(e.span,{style:{top:"-2.5224em",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.jsxs(e.span,{className:"mord mtight",children:[s.jsx(e.span,{className:"mord mtight",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:"0.7463em"},children:s.jsxs(e.span,{style:{top:"-2.786em",marginRight:"0.0714em"},children:[s.jsx(e.span,{className:"pstrut",style:{height:"2.5em"}}),s.jsx(e.span,{className:"sizing reset-size3 size1 mtight",children:s.jsx(e.span,{className:"mord mtight",children:s.jsx(e.span,{className:"mord mtight",children:"128"})})})]})})})})})]})})})]})}),s.jsx(e.span,{className:"vlist-s",children:"​"})]}),s.jsx(e.span,{className:"vlist-r",children:s.jsx(e.span,{className:"vlist",style:{height:"0.1776em"},children:s.jsx(e.span,{})})})]})})]})]})})]}),"."]}),`
8`,s.jsxs(e.p,{children:["Concretely, we can represent elements ",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:"α"}),s.jsx(e.mo,{children:"∈"}),s.jsxs(e.msub,{children:[s.jsx(e.mi,{mathvariant:"double-struck",children:"F"}),s.jsxs(e.msup,{children:[s.jsx(e.mn,{children:"2"}),s.jsx(e.mn,{children:"128"})]})]})]}),s.jsx(e.annotation,{encoding:"application/x-tex",children:"\\alpha \\in \\mathbb{F}_{2^{128}}"})]})})}),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",style:{marginRight:"0.0037em"},children:"α"}),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:"0.8665em",verticalAlign:"-0.1776em"}}

8),s.jsxs(e.span,{className:"mord",children:[s.jsx(e.span,{className:"mord mathbb",children:"F"}),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"},children:s.jsxs(e.span,{style:{top:"-2.5224em",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.jsxs(e.span,{className:"mord mtight",children:[s.jsx(e.span,{className:"mord mtight",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:"0.7463em"},children:s.jsxs(e.span,{style:{top:"-2.786em",marginRight:"0.0714em"},children:[s.jsx(e.span,{className:"pstrut",style:{height:"2.5em"}}),s.jsx(e.span,{className:"sizing reset-size3 size1 mtight",children:s.jsx(e.span,{className:"mord mtight",children:s.jsx(e.span,{className:"mord mtight",children:"128"})})})]})})})})})]})})})]})}),s.jsx(e.span,{className:"vlist-s",children:"​"})]}),s.jsx(e.span,{className:"vlist-r",children:s.jsx(e.span,{className:"vlist",style:{height:"0.1776em"},children:s.jsx(e.span,{})})})]})})]})]})]})]})," as polynomials of degree less than ",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.mn,{children:"128"})}),s.jsx(e.annotation,{encoding:"application/x-tex",children:"128"})]})})}),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:"0.6444em"}}),s.jsx(e.span,{className:"mord",children:"128"})]})})]}),"; say, ",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:"α"}),s.jsx(e.mo,{children:"="}),s.jsxs(e.msubsup,{children:[s.jsx(e.mo,{children:"∑"}),s.jsxs(e.mrow,{children:[s.jsx(e.mi,{children:"i"}),s.jsx(e.mo,{children:"="}),s.jsx(e.mn,{children:"0"})]}),s.jsx(e.mn,{children:"127"})]}),s.jsxs(e.msub,{children:[s.jsx(e.mi,{children:"α"}),s.jsx(e.mi,{children:"i"})]}),s.jsx(e.mo,{children:"⋅"}),s.jsxs(e.msup,{children:[s.jsx(e.mi,{children:"X"}),s.jsx(e.mi,{children:"i"})]})]}),s.jsx(e.annotation,{encoding:"application/x-tex",children:"\\alpha = \\sum_{i = 0}^{127} \\alpha_i \\cdot X^i"})]})})}),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:"0.4306em"}}),s.jsx(e.span,{className:"mord mathnormal",style:{marginRight:"0.0037em"},children:"α"}),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:"1.2537em",verticalAlign:"-0.2997em"}}),s.jsxs(e.span,{className:"mop",children:[s.jsx(e.span,{className:"mop op-symbol small-op",style:{position:"relative",top:"0em"},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.jsxs(e.span,{className:"vlist",style:{height:"0.954em"},children:[s.jsxs(e.span,{style:{top:"-2.4003em",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.jsxs(e.span,{className:"mord mtight",children:[s.jsx(e.span,{className:"mord mathnormal mtight",children:"i"}),s.jsx(e.span,{className:"mrel mtight",children:"="}),s.jsx(e.span,{className:"mord mtight",children:"0"})]})})]}

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8,children:s.jsxs(e.span,{style:{top:"-2.786em",marginRight:"0.0714em"},children:[s.jsx(e.span,{className:"pstrut",style:{height:"2.5em"}}),s.jsx(e.span,{className:"sizing reset-size3 size1 mtight",children:s.jsx(e.span,{className:"mord mtight",children:s.jsx(e.span,{className:"mord mtight",children:"128"})})})]})})})})})]})})})]})}),s.jsx(e.span,{className:"vlist-s",children:"​"})]}),s.jsx(e.span,{className:"vlist-r",children:s.jsx(e.span,{className:"vlist",style:{height:"0.1776em"},children:s.jsx(e.span,{})})})]})})]})]})})]})," get represented in practice as ",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.mn,{children:"128"})}),s.jsx(e.annotation,{encoding:"application/x-tex",children:"128"})]})})}),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:"0.6444em"}}),s.jsx(e.span,{className:"mord",children:"128"})]})})]}),"-bit words (or say as pairs of 64-bit words)."]}),`
9`,s.jsxs(e.h2,{id:"algebraic-properties",children:["Algebraic Properties",s.jsx(e.a,{"aria-hidden":"true",tabIndex:"-1",href:"#algebraic-properties",children:s.jsx(e.div,{"data-autolink-icon":!0})})]}),`
10`,s.jsxs(e.p,{children:["We fix a binary field, say ",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:"double-struck",children:"F"}),s.jsxs(e.msup,{children:[s.jsx(e.mn,{children:"2"}),s.jsx(e.mn,{children:"128"})]})]})}),s.jsx(e.annotation,{encoding:"application/x-tex",children:"\\mathbb{F}_{2^{128}}"})]})})}),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:"0.8665em",verticalAlign:"-0.1776em"}}),s.jsxs(e.span,{className:"mord",children:[s.jsx(e.span,{className:"mord mathbb",children:"F"}),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"},children:s.jsxs(e.span,{style:{top:"-2.5224em",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.jsxs(e.span,{className:"mord mtight",children:[s.jsx(e.span,{className:"mord mtight",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:"0.7463em"},children:s.jsxs(e.span,{style:{top:"-2.786em",marginRight:"0.0714em"},children:[s.jsx(e.span,{className:"pstrut",style:{height:"2.5em"}}),s.jsx(e.span,{className:"sizing reset-size3 size1 mtight",children:s.jsx(e.span,{className:"mord mtight",children:s.jsx(e.span,{className:"mord mtight",children:"128"})})})]})})})})})]})})})]})}),s.jsx(e.span,{className:"vlist-s",children:"​"})]}),s.jsx(e.span,{className:"vlist-r",children:s.jsx(e.span,{className:"vlist",style:{height:"0.1776em"},children:s.jsx(e.span,{})})})]})})]})]})})]}),". The ",s.jsx(e.em,{children:"multiplicative group of units"})," 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.jsxs(e.msub,{children:[s.jsx(e.mi,{mathvariant:"double-struck",children:"F"}),s.jsxs(e.msup,{children:[s.jsx(e.mn,{children:"2"}),s.jsx(e.mn,{children:"128"})]})]})}),s.jsx(e.annotation,{encoding:"application/x-tex",children:"\\mathbb{F}_{2^{128}}"})]})})}),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:"0.8665em",verticalAlign:"-0.1776em"}}

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This means that we can fix an element ",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:"g"}),s.jsx(e.mo,{children:"∈"}),s.jsxs(e.msubsup,{children:[s.jsx(e.mi,{mathvariant:"double-struck",children:"F"}),s.jsxs(e.msup,{children:[s.jsx(e.mn,{children:"2"}),s.jsx(e.mn,{children:"128"})]}),s.jsx(e.mo,{children:"∗"})]})]}),s.jsx(e.annotation,{encoding:"application/x-tex",children:"g \\in \\mathbb{F}_{2^{128}}^*"})]})})}),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:"g"}),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:"1.0083em",verticalAlign:"-0.3194em"}}),s.jsxs(e.span,{className:"mord",children:[s.jsx(e.span,{className:"mord mathbb",children:"F"}),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.jsxs(e.span,{className:"vlist",style:{height:"0.6887em"}

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11`,s.jsxs(e.p,{children:["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:"double-struck",children:"F"}),s.jsxs(e.msup,{children:[s.jsx(e.mn,{children:"2"}),s.jsx(e.mn,{children:"128"})]})]})}),s.jsx(e.annotation,{encoding:"application/x-tex",children:"\\mathbb{F}_{2^{128}}"})]})})}),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:"0.8665em",verticalAlign:"-0.1776em"}}),s.jsxs(e.span,{className:"mord",children:[s.jsx(e.span,{className:"mord mathbb",children:"F"}),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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