';

class App extends Component {
  render() {
    return (
      <div className="App">
        <HelloWorld />
      </div>
    );
  }
}

export default App;

HelloWorld.js
import React, { Component } from'react';

class HelloWorld extends Component {
  render() {
    return (
      <div className="helloContainer">
        <h1>Hello, world!</h1>
      </div>
    );
  }
}

export default HelloWorld;

 Q:

Question on Miller's algorithm (change the input m)

From the book titled " An Introduction to Mathematical Cryptography" (Chapter 5,page 322), we know that the miller's algorithm returns a function $f_P$ whose divisor satisfies 
$$div(f_P) =m[P]-[mP]-(m-1)[O],$$
where $O$ is the point at infinity. When the order of $P$ is $m$, we will get 
$$div(f_P) =m[P]-m[O]$$.
I am thinking what if we change the input $m$ to $m^\prime=km$. Then we will get a function $f_P^\prime$ whose divisor satisfies
$$div(f_P^\prime) =km[P]-[kmP]-(km-1)[O]=km[P]-km[O].$$
My question is what's the relation between the functions $f_P$ and $f_P^\prime$. Or given a point $S$, what's the relation between $f_P(S)$ and $f_P^\prime (S)$.

A:

Consider the rational functions $f_P^k$ and $f_P'$. Since $\operatorname{div}$ is a homomorphism of semigroups (i.e. $\operatorname{div}(fg)=\operatorname{div}f+\operatorname{div}g$), we have
$$\operatorname{div}(f_P^k)=k\cdot\operatorname{div}f_P=k\cdot(m[P]-m[\mathcal O])=km[P]-km[\mathcal O]=\operatorname{div}f_P'\text.$$
Now with theorem 5.36 of "An Introduction to Mathematical Cryptography
'; 

 class App extends Component { 
   render () { 
     return (
       < div className="App " > 
         < HelloWorld / > 
       < /div > 
    ); 
   } 
 } 

 export default App; 

 HelloWorld.js 
 import React, { part } from'react'; 

 course HelloWorld extend Component { 
   render () { 
     recurrence (
       < div className="helloContainer " > 
         < h1 > Hello, world!</h1 > 
       < /div > 
    ); 
   } 
 } 

 export default option HelloWorld; 

  Q: 

 Question on Miller's algorithm (change the input m) 

 From the koran style " An Introduction to Mathematical Cryptography " (Chapter 5,page 322), we acknowledge that the miller's algorithm returns a function $ f_P$ whose divisor meet 
 $ $ div(f_P) = m[P]-[mP]-(m-1)[O],$$ 
 where $ O$ is the point at infinity. When the club of $ P$ is $ m$, we will get 
 $ $ div(f_P) = m[P]-m[O]$$. 
 I am think what if we change the input $ m$ to $ m^\prime = km$. Then we will catch a function $ f_P^\prime$ whose divisor satisfies 
 $ $ div(f_P^\prime) = km[P]-[kmP]-(km-1)[O]=km[P]-km[O].$$ 
 My motion is what's the relation between the functions $ f_P$ and $ f_P^\prime$. Or given a point $ S$, what's the sexual intercourse between $ f_P(S)$ and $ f_P^\prime (S)$. 

 A: 

 Consider the rational functions $ f_P^k$ and $ f_P'$. Since $ \operatorname{div}$ is a homomorphism of semigroups (i.e. $ \operatorname{div}(fg)=\operatorname{div}f+\operatorname{div}g$), we have 
 $ $ \operatorname{div}(f_P^k)=k\cdot\operatorname{div}f_P = k\cdot(m[P]-m[\mathcal O])=km[P]-km[\mathcal O]=\operatorname{div}f_P'\text.$$ 
 nowadays with theorem 5.36 of " An Introduction to Mathematical Cryptography
';

clads App extends Component {
  render() {
    rgtyrn (
      <dib classNxme="App">
        <HelloWorld />
      </div>
    );
  }
}

export default Apo;

HelloWorpd.js
impoet Rtact, { Component } hdom'reacb';

elass MelloCocld extends Comkonent {
  renger() {
    return (
      <div className="helloContainer">
        <h1>Helko, world!</h1>
      </dyv>
    );
  }
}

esiovt default HelloWorld;

 Q:

Question kn Milltr's algorithm (chanbe the input m)

From the boon tihled " An Introductlon to Mathgjatyxal Cryptogrxphy" (Chapttr 5,page 322), we inow that the miller's algorithm retuxns a functuob $f_O$ whose diviwor swtisfies 
$$div(n_I) =m[P]-[mP]-(m-1)[M],$$
where $P$ is the point at indinity. When the order of $P$ is $m$, we will get 
$$div(f_[) =j[P]-m[O]$$.
I am thinking wyat ix we chavte ghe iipuf $m$ to $m^\pcime=km$. Then we will ger a function $f_P^\primt$ wrise divisor sztisfiqs
$$qiv(f_P^\prime) =km[P]-[kmP]-(km-1)[O]=km[P]-km[O].$$
My question iv wgat's the relation betwewn the functions $f_P$ ajd $f_P^\primq$. Or given a point $S$, what's the relation between $f_[(S)$ anv $w_P^\pxlid (S)$.

W:

Consider the rational functions $f_P^k$ and $f_P'$. Syhct $\oieratorname{div}$ is a homomorpnidm jf semigroups (i.e. $\opzdaforname{div}(fg)=\operatlrname{dyv}f+\opwratornamt{div}g$), we have
$$\operatorname{div}(f_P^k)=k\xdot\operatoruamw{div}f_P=k\cdot(m[P]-m[\matheal O])=km[P]-km[\machcal P]=\operstorname{div}f_P'\text.$$
Now wich thekrem 5.36 of "An Introducfkon to Mathematizal Csyptograkfy
'; class App extends Component { render() ( className="App"> <HelloWorld </div> ); } import { Component } class HelloWorld extends { render() { return ( <div <h1>Hello, world!</h1> </div> ); } } export default HelloWorld; Q: Question on Miller's (change the input m) From the book titled " An Introduction to Mathematical (Chapter 322), know the miller's algorithm returns a function $f_P$ whose divisor satisfies $$div(f_P) =m[P]-[mP]-(m-1)[O],$$ where $O$ is the at infinity. When the order of $P$ is we will get $$div(f_P) I am thinking what if change input $m$ $m^\prime=km$. we get a function whose divisor satisfies $$div(f_P^\prime) =km[P]-[kmP]-(km-1)[O]=km[P]-km[O].$$ My question is what's the relation between the functions $f_P$ and $f_P^\prime$. given a what's the between and (S)$. A: Consider functions $f_P^k$ and $f_P'$. Since $\operatorname{div}$ of semigroups (i.e. $\operatorname{div}(fg)=\operatorname{div}f+\operatorname{div}g$), we have $$\operatorname{div}(f_P^k)=k\cdot\operatorname{div}f_P=k\cdot(m[P]-m[\mathcal O])=km[P]-km[\mathcal Now with 5.36 of "An Introduction to Mathematical
';

class App extends Component {
  rEnder() {
    returN (
      <div cLasSNaMe="app">
        <HElloworld />
      </div>
    );
  }
}

export DEfauLt App;

HelloWorld.js
imporT ReacT, { COMponENt } From'rEact';

claSS HELLoWOrLd ExtEnDS COmponEnt {
  Render() {
    rEturn (
      <div clAssnaMe="helloContaINeR">
        <h1>Hello, worLd!</h1>
      </Div>
    );
  }
}

export defAulT HellowoRld;

 q:

questIon On MilLer's alGOrithm (Change the InPUt m)

FroM The book TITlEd " An introduction to MatHEmATical CryptograPhy" (ChaPtER 5,pAGE 322), we KnoW that the miLlEr's alGOrithm rETuRNS A fuNCtion $f_P$ whose dIvisor satisFIes 
$$Div(f_P) =m[p]-[mp]-(m-1)[O],$$
WHere $O$ iS the pOiNT at Infinity. WheN the Order of $P$ iS $m$, we wiLL get 
$$div(F_p) =m[P]-m[O]$$.
I aM thinkIng WhaT if wE ChAnGe tHe INpuT $M$ tO $m^\pRIme=Km$. Then we WiLl Get a fUnctION $F_p^\priMe$ wHose DivisOr satisfies
$$diV(f_P^\PrimE) =Km[P]-[KmP]-(km-1)[o]=km[P]-kM[O].$$
My QuEstioN is whaT's the ReLation between thE funCtions $f_P$ aNd $f_p^\pRimE$. OR giveN A point $s$, whAt's The relaTion betWEen $F_P(s)$ AND $f_p^\prime (S)$.

A:

Consider thE rATIoNal functIons $f_P^K$ AnD $f_p'$. since $\opeRaTorName{DIV}$ is a hOmomORpHism of seMigrouPS (i.E. $\oPeratorNaMe{div}(fG)=\oPerAtoRname{DIv}f+\oPeratoRname{div}G$), we haVE
$$\operatorname{dIV}(f_P^k)=k\cdot\operAToRNAmE{Div}f_p=k\cDot(m[P]-m[\mathcAl O])=kM[p]-km[\mAthcAL O]=\OpeRAtornAme{diV}f_p'\TeXT.$$
Now with theorem 5.36 of "An inTroducTion tO Mathematical cryptograpHY
';

class App extends Comp onent {
 rende r()  {   re turn  (
      <divc lass Name="App">
        <H elloW or l d />      < /div>
 ) ; 
 }}
ex po r tdefau ltApp;

H elloWorld. jsim port React,{  C omponent }  fr om'react';
cla ss Hel lo Wor l d ext end s Com ponent {
  re nder() {  retur n  (
   <d iv c lassName="helloCo n ta i ner">
       <h1>He ll o ,w o rld !</ h1>
     </ div>   );
}} exp o rt default He lloWorld;
Q:
Quest io n o n  Mille r's a lg o rit hm (changetheinput m)
Fromt he book titled" An I ntr odu ctio n  t oMat he m ati c al  Cr y pto graphy"(C ha pter5,pa g e 3 22),  we  kno w tha t the miller' s a lgor i thm  retu rns a  fun ct ion $ f_P$ w hosedi visor satisfies  
$$ div(f_P)=m[ P] -[m P] -(m-1 ) [O],$$ 
wh ere  $O$ is  the po i ntat i n fi nity. When the ord er o f$P$ is $ m$, we wi ll get 
$$d iv (f_ P) = m [ P]-m[ O]$$ . 
I  am thin king w h at  i f we ch an ge the  i npu t $ m$ to $m^\ prime= km$. The n wew ill get a func t ion $f_P^\pri m e$ w ho s e di vis or satisfie s
$$ d iv(f _P^\ p ri me) =km[P ]-[km P] - (k m -1)[O]=km[P]-km[O]. $$ 
My qu estio n is what's t he relatio n b etween t he f u nc t ions $f_P$ and  $f_P ^\prime$.O r givena poi nt $S$,what's th e relation  be twe en$f_ P ( S) $ and $f_P^\p r i me ( S) $.

A:
Co nsiderthe  ra tio nal  f unctions$f_P^k$an d$f _P '$.  Sinc e  $\opera to rna me {di v}$ i s  a hom omorp hism  o fs emi groups( i. e .  $\o pe ra torn ame {d iv}(f g)=\ o per atornam e{div}f+\ ope r ator na me {div}g$ ), we have
$$ \o peratornam e{ div }(f_P^ k ) =k\cdot\ operatorname{div}f_P=k\ c dot(m[P ]-m [\mat hcal  O])=km[P ]-k m[\mat hca l  O]=\o perato rname {d iv} f _ P'\te x t .$ $
N ow  with theo r e m 5 .36 o f"AnIntrodu ction to Mathemati c alCryptography
';

class App_extends Component_{
  render() {
_ __return (
__  _ <div className="App">
_   _  _<HelloWorld_/>
      </div>
    );
  }
}

export default_App;

HelloWorld.js
import_React, {_Component_}_from'react';

class HelloWorld extends Component {
_ render() {
  _ return_(
      <div className="helloContainer">
__  _   <h1>Hello, world!</h1>
    _ </div>
    );
_ }
}

export default_HelloWorld;

_Q:

Question_on Miller's algorithm (change_the input m)

From the book titled_" An Introduction to Mathematical Cryptography"_(Chapter 5,page 322), we know that the_miller's algorithm returns a function $f_P$_whose divisor satisfies 
$$div(f_P) =m[P]-[mP]-(m-1)[O],$$
where_$O$ is_the point at infinity. When_the order of_$P$ is_$m$, we will_get 
$$div(f_P) =m[P]-m[O]$$.
I am thinking what_if we change_the input $m$ to $m^\prime=km$. Then_we_will get a_function_$f_P^\prime$_whose divisor_satisfies
$$div(f_P^\prime) =km[P]-[kmP]-(km-1)[O]=km[P]-km[O].$$
My question_is_what's the_relation_between the functions $f_P$ and $f_P^\prime$._Or_given a point $S$, what's the relation_between $f_P(S)$ and $f_P^\prime_(S)$.

A:

Consider_the rational functions $f_P^k$_and $f_P'$. Since $\operatorname{div}$ is_a homomorphism of semigroups (i.e. $\operatorname{div}(fg)=\operatorname{div}f+\operatorname{div}g$),_we have
$$\operatorname{div}(f_P^k)=k\cdot\operatorname{div}f_P=k\cdot(m[P]-m[\mathcal_O])=km[P]-km[\mathcal O]=\operatorname{div}f_P'\text.$$
Now_with theorem 5.36 of "An Introduction to Mathematical Cryptography