From a2dcc8fc0162889b51704c309e74def165273b09 Mon Sep 17 00:00:00 2001 From: arnaucube Date: Sun, 12 Feb 2023 18:54:01 +0100 Subject: [PATCH] Add Caulk initial notes --- README.md | 21 +++ notes_caulk.pdf | Bin 0 -> 337934 bytes notes_caulk.tex | 347 +++++++++++++++++++++++++++++++++++++++++++++++ paper-notes.bib | 18 +++ weil-pairing.pdf | Bin 193407 -> 201677 bytes weil-pairing.tex | 75 ++++++++-- 6 files changed, 451 insertions(+), 10 deletions(-) create mode 100644 notes_caulk.pdf create mode 100644 notes_caulk.tex diff --git a/README.md b/README.md index 70e16fe..f3d1f0e 100644 --- a/README.md +++ b/README.md @@ -1,3 +1,24 @@ # math Notes, code and documents done while reading books and papers. + +### Sage code +- [blind signatures over elliptic curve](blind-sign-over-ec.sage) +- [BLS signatures](bls-sigs.sage) +- [FFT](fft.sage) +- [IPA](ipa.sage) +- [KZG commitments](kzg.sage) +- [Powers of Tau](powersoftau.sage) +- [Ring signatures](ring-signatures.sage) +- [Sigma protocol](sigma.sage) + +### Notes +- [Notes on "Abstract Algebra" book, by Charles C. Pinter](abstract-algebra-charles-pinter-notes.pdf) +- [Notes on Caulk paper](notes_caulk.pdf) +- [Notes on the DFT & FFT](fft-notes.pdf) +- [Notes on the BLS signatures](notes_bls-sig.pdf) +- [Notes on IPA from Halo paper](notes_halo.pdf) +- [Notes on Sonic paper](notes_sonic.pdf) +- [Notes on Weil pairing](weil-pairing.pdf) +- [Notes on Sigma protocol and OR proofs](sigma-or-notes.pdf) +- [Notes on Reed-Solomon codes](notes_reed-solomon.pdf) diff --git a/notes_caulk.pdf b/notes_caulk.pdf new file mode 100644 index 0000000000000000000000000000000000000000..ff31410ba2f311195871dd9232d8487540ab1389 GIT binary patch literal 337934 zcmb5VLzFJSwyayWZQHKDY}>YN+qS*RHdooU?W$F_`SyL|_V#I zt&yvlsF{g_sTmAEKa7j3vzd_{jOXTzmR8D^D3br40damJ?Skn|h$;b*7=U)9QL&QF zOVRdPdpI3K%N)8dKVRfQlHk_9BJqyQYuV*#?faQ|bVDt?xqkKze~xi>j)1S2f)qxH z#M9kBOQ|L?ktIcx$lTJscaq`ZpPj)q2KW%01MaV{_6?`Qm(51aXe0OpXH4+10|A|W zUcl!W_7FDlIF!)!VKfo3s=8ua3$g!olg;F}^RKs(Vw#4#x;j#9BpJ~d(!a$iJ{`+! 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zDkg-fk`n!yl`rY0;lx%An>9M=bpU>xkD?T z=Aj=;f@@4au8UdvaYuq`a4Q$hEJ+Mu2juPhhU@bIQhI+`Er}|Z%lK#wAC%j(z&{-P zh1{G)rDCawitH>Cg+{bLG`eQUoV!lfCb@}RLMCqSP`C9qJke)DgT@K8>jLo~GtP2a zuAbZZj^0A7?|4AL49_Jb`aOFW%uY|NxT0wTg91Z-?dI?4q90i{1#&j0`b diff --git a/weil-pairing.tex b/weil-pairing.tex index 4195bc4..8911373 100644 --- a/weil-pairing.tex +++ b/weil-pairing.tex @@ -4,6 +4,7 @@ \usepackage{amsthm} \usepackage{amsmath} \usepackage{amssymb} +\usepackage{mathtools} \usepackage{enumerate} \usepackage{hyperref} \hypersetup{ @@ -28,7 +29,7 @@ {\endinnersolution} -\title{Bilinear Pairings - study} +\title{Weil Pairing - study} \author{arnaucube} \date{August 2022} @@ -44,9 +45,7 @@ \tableofcontents -\section{Weil reciprocity} - -\section{Generic Weil Pairing} +\section{Divisors and rational functions} \begin{definition}{Divisor} $$D= \sum_{P \in E(\mathbb{K})} n_p \cdot [P]$$ @@ -67,21 +66,77 @@ $D \sim D'$ iff $D - D'$ is principal. $$r(D)= \prod r(P)^{n_p}$$ \end{definition} -\subsection{Generic Weil Pairing} +\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 +$$r(div(s)) = s(div(r))$$ +\end{theorem} +Proof. (todo) + +\section{Generic Weil Pairing} Let $E(\mathbb{K})$, with $\mathbb{K}$ of char $p$, $n$ s.t. $p \nmid n$. $\mathbb{K}$ large enough: $E(\mathbb{K})[n] = E(\mathbb{\overline{K}}) = \mathbb{Z}_n \oplus \mathbb{Z}_n$ (with $n^2$ elements). -$P, Q \in E[n]$: -$$D_P \sim [P] - [0]$$ -$$D_Q \sim [Q] - [0]$$ +For $P, Q \in E[n]$, +\begin{align*} + D_P &\sim [P] - [0]\\ + D_Q &\sim [Q] - [0] +\end{align*} + We need them to have disjoint support: -$$D_P \sim [P] - [0]$$ -$$D_Q \sim [Q+T] - [T]$$ +\begin{align*} + D_P &\sim [P] - [0]\\ + D_Q' &\sim [Q+T] - [T] +\end{align*} $$\Delta D = D_Q - D_Q' = [Q] - [0] - [Q+T] + [T]$$ +Note that $n D_P$ and $n D_Q$ are principal. Proof: +\begin{align*} + n D_P &= n [P] - n [O]\\ + deg(n D_P) &= n - n = 0\\ + sum(n D_P) &= nP - nO = 0 +\end{align*} +($nP = 0$ bcs. $P$ is n-torsion) + +Since $n D_P,~ n D_Q$ are principal, we know that $f_P,~ f_Q$ exist. + +Take +\begin{align*} + f_P &: div(f_P) = n D_P\\ + f_Q &: div(f_Q) = n D_Q +\end{align*} + +We define +$$ +e_n(P, Q) = \frac{f_P(D_Q)}{f_Q(D_P)} +$$ + +Remind: evaluation of a rational function over a divisor $D$: +\begin{align*} + D &= \sum n_P [P]\\ + r(D) &= \prod r(P)^{n_P} +\end{align*} + +If $D_P = [P+S] - [S],~~ D_Q=[Q-T]-[T]$ what is $e_n(P, Q)$? + +\begin{align*} + f_P(D_Q) &= f_P(Q+T)^1 \cdot f_P(T)^{-1}\\ + f_Q(D_P) &= f_Q(P+S)^1 \cdot f_Q(S)^{-1} +\end{align*} + +$$ +e_n(P, Q) = \frac{f_P(Q+T)}{f_P(T)} / \frac{f_Q(P+S)}{f_Q(S)} +$$ + +with $S \neq \{O, P, -Q, P-Q \}$. + + +\section{Properties} + + \section{Exercises} \emph{An Introduction to Mathematical Cryptography, 2nd Edition} - Section 6.8. Bilinear pairings on elliptic curves