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z$iFSryh%g_U`8_BV>}=bN|}*P&xFg6#FwO0kcqF%$Ur)Y{dBHgEyTYLHL?b^wYwC| zVAe%@FP8+cjB1BHR)@(C$UouSWZ-038D|VICI}cc0XTCJXx*57Wn>lfxsje zuW97feZS(eP2pw9QJo1qKLBEj%F6c{%K0jPG2FEpx1Z+pSMDf9LBQ#B!P y%lY9)j9rDp4=X!PJA=yicd=}S%XarzKSoug(uA#XXb6BHAP@v#W|r4f0Q?V{>I~@s diff --git a/weil-pairing.tex b/weil-pairing.tex index 8911373..6d660b6 100644 --- a/weil-pairing.tex +++ b/weil-pairing.tex @@ -34,7 +34,6 @@ \date{August 2022} \begin{document} - \maketitle \begin{abstract} @@ -45,15 +44,117 @@ \tableofcontents -\section{Divisors and rational functions} +\section{Rational functions} + +Let $E/\Bbbk$ be an elliptic curve defined by: $y^2 = x^3 + Ax + B$. + +\paragraph{set of polynomials over $E$:} +$\Bbbk[E] := \Bbbk[x,y] / (y^2 - x^3 - Ax - B =0)$ + +we can replace $y^2$ in the polynomial $f \in \Bbbk[E]$ with $x^3 + Ax + B$ + +\paragraph{canonical form:} $f(x,y) = v(x)+y w(x)$ for $v, w \in \Bbbk[x]$ +\paragraph{conjugate:} $\overline{f} = v(x) - y w(x)$ +\paragraph{norm:} $N_f = f \cdot \overline{f} = v(x)^2 - y^2 w(x)^2 = v(x)^2 - (x^3 + Ax + B) w(x)^2 \in \Bbbk[x] \subset \Bbbk[E]$ + +we can see that $N_{fg} = N_f \cdot N_g$ + +\paragraph{set of rational functions over $E$:} +$\Bbbk(E) := \Bbbk[E] \times \Bbbk[E]/ \thicksim$ + +For $r\in \Bbbk(E)$ and a finite point $P \in E(\Bbbk)$, $r$ is \emph{finite} at $P$ iff +$$\exists~ r=\frac{f}{g} ~\text{with}~ f,g \in \Bbbk[E],~ s.t.~ g(P) \neq 0$$ +We define $r(P)=\frac{f(P)}{g(P)}$. Otherwise, $r(P)=\infty$. + +Remark: $r=\frac{f}{g} \in \Bbbk(E)$, $r=\frac{f}{g}=\frac{f \cdot \overline{g}}{g \cdot \overline{g}} = \frac{f \overline{g}}{N_g}$, thus +$$r(x,y)=\frac{ (f \overline{g})(x,y)}{N_g(x,y)} = \underbrace{ \frac{v(x)}{N_g(x)} + y \frac{w(x)}{N_g(x)} }_\text{canonical form of $r(x,y)$}$$ + +\paragraph{degree of $f$:} Let $f\in \Bbbk[E]$, in canonical form: $f(x,y) = v(x) + y w(x)$, +$$deg(f) := max\{ 2 \cdot deg_x(v), 3+2 \cdot deg_x(w) \}$$ + +For $f,g \in \Bbbk[E]$: +\begin{enumerate}[i.] + \item $deg(f) = deg_x(N_f)$ + \item $deg(f \cdot g) = deg(f) + deg(g)$ +\end{enumerate} + +\begin{definition} + Let $r=\frac{f}{g} \in \Bbbk(E)$ + \begin{enumerate}[i.] + \item if $deg(f) < deg(g):~ r(0)=0$ + \item if $deg(f) > deg(g):~ r ~\text{is not finite at}~ 0$ + \item if $deg(f) = deg(g)$ with $deg(f)$ even:\\ + $f$'s canonical form leading terms $ax^d$\\ + $g$'s canonical form leading terms $bx^d$\\ + $a,b \in \Bbbk^\times,~ d=\frac{deg(f)}{2}$, set $r(0)=\frac{a}{b}$ + \item if $deg(f) = deg(g)$ with $deg(f)$ odd\\ + $f$'s canonical form leading terms $ax^d$\\ + $g$'s canonical form leading terms $bx^d$\\ + $a,b \in \Bbbk^\times,~ deg(f)=deg(g)=3+2d$, set $r(0)=\frac{a}{b}$ + \end{enumerate} +\end{definition} + +\subsection{Zeros, poles, uniformizers and multiplicities} + +$r \in \Bbbk(E)$ has a \emph{zero} in $P\in E$ if $r(P)=0$\\ +$r \in \Bbbk(E)$ has a \emph{pole} in $P\in E$ if $r(P)$ is not finite. + +\paragraph{uniformizer:} Let $P\in E$, +uniformizer: rational function $u \in \Bbbk(E)$ with $u(P)=0$ if +$\forall r\in \Bbbk(E) \setminus \{0\},~ \exists d \in \mathbb{Z},~ s\in \Bbbk(E)$ finite at $P$ with $s(P) \neq 0$ s.t. +$$r=u^d \cdot s$$ + +\paragraph{order:} Let $P \in E(\Bbbk)$, let $u \in \Bbbk(E)$ be a uniformizer at $P$. +For $r \in \Bbbk(E) \setminus \{0\}$ being a rational function with $r=u^d \cdot s$ with $s(P)\neq 0, \infty$, we say that $r$ has \emph{order} $d$ at $P$ ($ord_P(r)=d$). + +\paragraph{multiplicity:} \emph{multiplicity of a zero} of $r$ is the order of $r$ at that point, \emph{multiplicity of a pole} of $r$ is the order of $r$ at that point. + +if $P \in E(\Bbbk)$ is neither a zero or pole of $r$, then $ord_P(r)=0$ ($=d,~ r=u^0s$). + +\vspace{0.5cm} +\begin{minipage}{4.3 in} + \paragraph{Multiplicities, from the book "Elliptic Tales"} (p.69), to provide intuition + + Factorization into \emph{linear factors}: $p(x)=c\cdot (x-a_1) \cdots (x-a_d)$\\ + $d$: degree of $p(x)$, $a_i \in \Bbbk$\\ + Solutions to $p(x)=0$ are $x=a_1, \ldots, a_d$ (some $a_i$ can be repeated)\\ + eg.: $p(x)=(x-1)(x-1)(x-3)$, solutions to $p(x)=0:~ 1, 1, 3$\\ + $x=1$ is a solution to $p(x)=0$ of \emph{multiplicity} 2. + + The total number of solutions (counted with multiplicity) is $d$, the degree of the polynomial whose roots we are finding. +\end{minipage} + + +\section{Divisors} \begin{definition}{Divisor} - $$D= \sum_{P \in E(\mathbb{K})} n_p \cdot [P]$$ + $$D= \sum_{P \in E(\Bbbk)} n_p \cdot [P]$$ \end{definition} \begin{definition}{Degree \& Sum} - $$deg(D)= \sum_{P \in E(\mathbb{K})} n_p$$ - $$sum(D)= \sum_{P \in E(\mathbb{K})} n_p \cdot P$$ + $$deg(D)= \sum_{P \in E(\Bbbk)} n_p$$ + $$sum(D)= \sum_{P \in E(\Bbbk)} n_p \cdot P$$ +\end{definition} + + +The set of all divisors on $E$ forms a group: for $D = \sum_{P\in E(\Bbbk)} n_P[P]$ and $D' = \sum_{P\in E(\Bbbk)} m_P[P]$, +$$D+D' = \sum_{P\in E(\Bbbk)} (n_P + m_P)[P]$$ + +\begin{definition}{Associated divisor} + $$div(r) = \sum_{P \in E(\Bbbk)} ord_P(r)[P]$$ +\end{definition} + +Observe that +\begin{enumerate} + \item[] $div(rs) = div(r)+div(s)$ + \item[] $div(\frac{r}{s}) = div(r)-div(s)$ +\end{enumerate} + +Observe that +$$\sum{P \in E(\Bbbk)} ord_P(r) \cdot P = 0$$ + +\begin{definition}{Support} + $$\sum_P n_P[P], ~\forall P \in E(\Bbbk) \mid n_P \neq 0$$ \end{definition} \begin{definition}{Principal divisor} @@ -62,21 +163,21 @@ $D \sim D'$ iff $D - D'$ is principal. -\begin{definition}{Evaluation of a rational function} +\begin{definition}{Evaluation of a rational function} (function $r$ evaluated at $D$) $$r(D)= \prod r(P)^{n_p}$$ \end{definition} \section{Weil reciprocity} \begin{theorem}{(Weil reciprocity)} - Let $E/ \mathbb{K}$ be an e.c. over an alg. closed field. If $r,~s \in \mathbb{K}\setminus \{0\}$ are rational functions whose divisors have disjoint support, then + Let $E/ \Bbbk$ be an e.c. over an alg. closed field. If $r,~s \in \Bbbk\setminus \{0\}$ are rational functions whose divisors have disjoint support, then $$r(div(s)) = s(div(r))$$ \end{theorem} Proof. (todo) \section{Generic Weil Pairing} -Let $E(\mathbb{K})$, with $\mathbb{K}$ of char $p$, $n$ s.t. $p \nmid n$. +Let $E(\Bbbk)$, with $\Bbbk$ of char $p$, $n$ s.t. $p \nmid n$. -$\mathbb{K}$ large enough: $E(\mathbb{K})[n] = E(\mathbb{\overline{K}}) = \mathbb{Z}_n \oplus \mathbb{Z}_n$ (with $n^2$ elements). +$\Bbbk$ large enough: $E(\Bbbk)[n] = E(\overline{\Bbbk}) = \mathbb{Z}_n \oplus \mathbb{Z}_n$ (with $n^2$ elements). For $P, Q \in E[n]$, \begin{align*} @@ -143,7 +244,7 @@ with $S \neq \{O, P, -Q, P-Q \}$. \begin{solution}{6.29} $div(R(x) \cdot S(x)) = div( R(x)) + div( S(x))$, where $R(x), S(x)$ are rational functions. \\proof:\\ - \emph{Norm} of $f$: $N_f = f \cdot \overline{f}$, and we know that $N_{fg} = N_f \cdot N_g~\forall~\mathbb{K}[E]$,\\ + \emph{Norm} of $f$: $N_f = f \cdot \overline{f}$, and we know that $N_{fg} = N_f \cdot N_g~\forall~\Bbbk[E]$,\\ then $$deg(f) = deg_x(N_f)$$\\ and $$deg(f \cdot g) = deg(f) + deg(g)$$ @@ -151,8 +252,8 @@ with $S \neq \{O, P, -Q, P-Q \}$. $$deg(f \cdot g) = deg_x(N_{fg}) = deg_x(N_f \cdot N_g)$$ $$= deg_x(N_f) + deg_x(N_g) = deg(f) + deg(g)$$ - So, $\forall P \in E(\mathbb{K}),~ ord_P(rs) = ord_P(r) + ord_P(s)$.\\ - As $div(r) = \sum_{P\in E(\mathbb{K})} ord_P(r)[P]$, $div(s) = \sum ord_P(s)[P]$. + So, $\forall P \in E(\Bbbk),~ ord_P(rs) = ord_P(r) + ord_P(s)$.\\ + As $div(r) = \sum_{P\in E(\Bbbk)} ord_P(r)[P]$, $div(s) = \sum ord_P(s)[P]$. So, $$div(rs) = \sum ord_P(rs)[P]$$