From 11e7cc56134b81cf6d9a2730a4b419258442ea17 Mon Sep 17 00:00:00 2001 From: arnaucube Date: Fri, 26 Aug 2022 10:24:32 +0200 Subject: [PATCH] Add pairings initial notes --- abstract-algebra-charles-pinter-notes.pdf | Bin 209652 -> 224786 bytes abstract-algebra-charles-pinter-notes.tex | 15 ++- blind-sign-over-ec.sage | 8 +- fft.sage | 16 +-- ipa.sage | 15 ++- notes_bls-sig.pdf | Bin 155189 -> 140121 bytes notes_bls-sig.tex | 2 +- pairings.pdf | Bin 0 -> 193407 bytes pairings.tex | 123 ++++++++++++++++++++++ sigma-or-notes.pdf | Bin 194068 -> 209492 bytes 10 files changed, 158 insertions(+), 21 deletions(-) create mode 100644 pairings.pdf create mode 100644 pairings.tex diff --git a/abstract-algebra-charles-pinter-notes.pdf b/abstract-algebra-charles-pinter-notes.pdf index a4f2f4c5bf2859e7b69c0736b1641c54a69feffc..f1f3462db15e2396a82d741198f6b5d9b0679f34 100644 GIT binary patch delta 66965 zcmV(@K-RzXFg?Hf`zq7-b%4yjk>rLsu?`INPqPMOVK2 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z23#A3g8#d^u9&5F*&ZvBP|3L`pc*ezoFpI^mvs0|E(F|+N!3p(FXvZkQ#t^@gzST! zC>=4qlbb6UDTp6w1f~e}KCH*i(G!d0OYAL8&E34Nj5;B*jy;JlT;d~^o8qKa1cBd* zM`uTb=}y%pC3Fo`2NK5>GE2EL(I?J94xlh7dZQM2Q-y8BOJz`AG2%lCE=~m zO+rXXPSe%uwh_Jl>#JFTu#LCbn1#aiLa58s12ebmmvSzp%tUFf_^icpdHpvz4N zNf0p9cv3P*?ZrajqMPi+7*#5dUEvXr#?w^rx*MoB_$lK}x_70b|M7CK5o^IeKL>xA z_tq6WiEO}`6)~&oZuxmhT)}rKB6~_}w2D04z#W&qK9d!>KQ#`weZaio?2tT92$ZQP zKb48)j~6-eS)f>gN1fkA|1^bJF8UH)tU0Nj@e@xwJ zqmMp4efMZ>?WKyRqeE5|=C#B{Tf60KkBZ#LXh-jn{Q;Q@DfTCz)MFd~4&>Gz(E1$ s.t. $a^n=1~mod~p$). + + By Lagrange's theorem, $h~|~|G| = p - 1$, so $p-1 = h \cdot m$, thus + $$ + a^{p-1} = (a^h)^m \equiv 1^m \equiv 1~mod~p + $$ + + ~\\\emph{Another perspective:}\\ + We have $a^p \equiv a \pmod{p}$, by dividing by $a$ on both sides, we obtain $a^{p-1} \equiv 1 \pmod{p}$. + \end{theorem} \begin{theorem}[Euler's $\phi$ function] diff --git a/blind-sign-over-ec.sage b/blind-sign-over-ec.sage index 6b7d82e..6ef25fc 100644 --- a/blind-sign-over-ec.sage +++ b/blind-sign-over-ec.sage @@ -52,19 +52,19 @@ def verify(G, Q, sig, m): # ethereum elliptic curve -p = 0xFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFEFFFFFC2F +p = 0xFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFEFFFFFC2F # base field a = 0 b = 7 -F = GF(p) +F = GF(p) # base field E = EllipticCurve(F, [a,b]) GX = 0x79BE667EF9DCBBAC55A06295CE870B07029BFCDB2DCE28D959F2815B16F81798 GY = 0x483ADA7726A3C4655DA4FBFC0E1108A8FD17B448A68554199C47D08FFB10D4B8 g = E(GX,GY) n = 0xFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFEBAAEDCE6AF48A03BBFD25E8CD0364141 -q = g.order() +q = g.order() # order of Fp assert is_prime(p) assert is_prime(q) -Fq = GF(q) +Fq = GF(q) # scalar field diff --git a/fft.sage b/fft.sage index c01d17e..c5494fd 100644 --- a/fft.sage +++ b/fft.sage @@ -105,21 +105,25 @@ print("nth roots of unity:", w) print("Vandermonde matrix:") print(ft) -a = vector([3,4,5,9]) -print("a:", a) +fa_eval = vector([3,4,5,9]) +print("fa_eval:", fa_eval) # interpolate f_a(x) -fa_coef = ft_inv * a +fa_coef = ft_inv * fa_eval print("fa_coef:", fa_coef) P. = PolynomialRing(F) fa = P(list(fa_coef)) print("f_a(x):", fa) -# check that evaluating fa(x) at the roots of unity returns the expected values of a -for i in range(len(a)): - assert fa(w[i]) == a[i] +# check that evaluating fa(x) at the roots of unity returns the expected values of fa_eval +for i in range(len(fa_eval)): + assert fa(w[i]) == fa_eval[i] +# go from coefficient form to evaluation form +fa_eval2 = ft * fa_coef +print("fa_eval'", fa_eval) +assert fa_eval2 == fa_eval # Fast polynomial multiplicaton using FFT diff --git a/ipa.sage b/ipa.sage index 55dbada..f5e2158 100644 --- a/ipa.sage +++ b/ipa.sage @@ -181,7 +181,7 @@ class IPA_halo: # a, b, G have length=1 # l, r are random blinding factors # L, R are the "cross-terms" of the inner product - return a[0], b[0], G[0], l, r, L, R + return a[0], l, r, L, R def verify(self, P, a, v, x_powers, r, u, U, lj, rj, L, R): print("methid verify()") @@ -358,7 +358,7 @@ class TestUtils(unittest.TestCase): class TestIPA_bulletproofs(unittest.TestCase): - def test_inner_product(self): + def test_inner_product_argument(self): d = 8 ipa = IPA_bulletproofs(Fq, E, g, d) @@ -374,9 +374,6 @@ class TestIPA_bulletproofs(unittest.TestCase): v = ipa.evaluate(a, b) print("v", v) - # verifier - # r = int(ipa.F.random_element()) - # verifier generate random challenges {uᵢ} ∈ 𝕀 and U ∈ 𝔾 U = ipa.E.random_element() k = int(math.log(d, 2)) @@ -418,7 +415,7 @@ class TestIPA_halo(unittest.TestCase): vc_c = vc_a + vc_b assert vc_c == expected_vc_c - def test_inner_product(self): + def test_inner_product_argument(self): d = 8 ipa = IPA_halo(Fq, E, g, d) @@ -428,7 +425,7 @@ class TestIPA_halo(unittest.TestCase): x = ipa.F(3) x_powers = powers_of(x, ipa.d) # = b - # verifier + # blinding factor r = int(ipa.F.random_element()) # prover @@ -438,6 +435,8 @@ class TestIPA_halo(unittest.TestCase): print("v", v) # verifier generate random challenges {uᵢ} ∈ 𝕀 and U ∈ 𝔾 + # This might be obtained from the hash of the transcript + # (Fiat-Shamir heuristic for non-interactive version) U = ipa.E.random_element() k = int(math.log(ipa.d, 2)) u = [None] * k @@ -449,7 +448,7 @@ class TestIPA_halo(unittest.TestCase): P = P + int(v) * U # prover - a_ipa, b_ipa, G_ipa, lj, rj, L, R = ipa.ipa(a, x_powers, u, U) + a_ipa, lj, rj, L, R = ipa.ipa(a, x_powers, u, U) # verifier print("P", P) diff --git a/notes_bls-sig.pdf b/notes_bls-sig.pdf index 537b48feb8224e9fb22573f42a4a880b8b1303a5..524afbbdcff15f0cef2a54a05b138ebc8dbe5aa5 100644 GIT binary patch delta 21313 zcmV(&K;gf&yb0OH2(X_912i-_lhLLqf9+aJbK*u2zVj<|6qTr%_rob)QnkCu9#WN! zlZ)4t5)Wg_SaLvQEB`({i~z|RKp60{ReM1oHT{~eA2TfyjUy6$=#cXI^G&DsUaH7~ zRZK-U_YvhBv@(&B8W@++%`m!+|LR&5k7kq4go;60O!BOoa3U#$6vZk$0gNbUVbSaMxK72 zK9#oy`3wzvcGBZK8&1}Is!qUve>^X`H11C)1OM}JY0iIF-XoM&T|yX8rK5zQ-qd)# zM@`oWP~5IO_nAl%;KjjcxuO|+0v#!^nj6nUONt%>VTC860WJcBjw&=ibrVz&WpZ9p zlL}8tfU+9T)cHik{&N-pfd)cq63`rqF4ZQ!z78nF1X%n7Ioda3Mf3c)f5j4O_nJ~) 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+ linkcolor=black, + urlcolor=blue +} +% \usepackage{xcolor} + +% prevent warnings of underfull \hbox: +% \usepackage{etoolbox} +% \apptocmd{\sloppy}{\hbadness 4000\relax}{}{} + +\theoremstyle{definition} +\newtheorem{definition}{Def}[section] +\newtheorem{theorem}[definition]{Thm} +\newtheorem{innersolution}{} +\newenvironment{solution}[1] +{\renewcommand\theinnersolution{#1}\innersolution} +{\endinnersolution} + + +\title{Bilinear Pairings - study} +\author{arnaucube} +\date{August 2022} + +\begin{document} + +\maketitle + +\begin{abstract} + Notes taken from \href{https://sites.google.com/site/matanprasma/artifact}{Matan Prsma} math seminars and also while reading about Bilinear Pairings. Usually while reading papers and books I take handwritten notes, this document contains some of them re-written to $LaTeX$. + + The notes are not complete, don't include all the steps neither all the proofs. I use these notes to revisit the concepts after some time of reading the topic. +\end{abstract} + +\tableofcontents + +\section{Weil reciprocity} + +\section{Generic Weil Pairing} + +\begin{definition}{Divisor} + $$D= \sum_{P \in E(\mathbb{K})} n_p \cdot [P]$$ +\end{definition} + +\begin{definition}{Degree \& Sum} + $$deg(D)= \sum_{P \in E(\mathbb{K})} n_p$$ + $$sum(D)= \sum_{P \in E(\mathbb{K})} n_p \cdot P$$ +\end{definition} + +\begin{definition}{Principal divisor} + iff $deg(D)=0$ and $sum(D)=0$ +\end{definition} +$D \sim D'$ iff $D - D'$ is principal. + + +\begin{definition}{Evaluation of a rational function} + $$r(D)= \prod r(P)^{n_p}$$ +\end{definition} + +\subsection{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). + +$P, Q \in E[n]$: +$$D_P \sim [P] - [0]$$ +$$D_Q \sim [Q] - [0]$$ +We need them to have disjoint support: +$$D_P \sim [P] - [0]$$ +$$D_Q \sim [Q+T] - [T]$$ + +$$\Delta D = D_Q - D_Q' = [Q] - [0] - [Q+T] + [T]$$ + + +\section{Exercises} +\emph{An Introduction to Mathematical Cryptography, 2nd Edition} - Section 6.8. Bilinear pairings on elliptic curves + +\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 $N_{fg} = N_f \cdot N_g~\forall~\mathbb{K}[E]$,\\ + then $$deg(f) = deg_x(N_f)$$\\ + and $$deg(f \cdot g) = deg(f) + deg(g)$$ + + Proof: + $$deg(f \cdot g) = deg_x(N_{fg}) = deg_x(N_f \cdot N_g)$$ + $$= deg_x(N_f) + deg_x(N_g) = deg(f) + deg(g)$$ + + 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, + $$div(rs) = \sum ord_P(rs)[P]$$ + $$= \sum ord_P(r)[P] + \sum ord_P(s)[P] = div(r) + div(s)$$ +\end{solution} + +\vspace{0.5cm} + +\begin{solution}{6.31} + $$e_m(P, Q) = e_m(Q, P)^{-1} \forall P, Q \in E[m]$$ + Proof: + We know that $e_m(P, P) = 1$, so: + $$1 = e_m(P+Q, P+Q) = e_m(P, P) \cdot e_m(P, Q) \cdot e_m(Q, P) \cdot e_m(Q, Q)$$ + + and we know that $e_m(P, P) = 1$, then we have: + $$1 = e_m(P, Q) \cdot e_m(Q, P)$$ + $$\Longrightarrow e_m(P, Q) = e_m(Q, P)^{-1}$$ +\end{solution} + + + + +\end{document} diff --git a/sigma-or-notes.pdf b/sigma-or-notes.pdf index e3a87053fadde1aed5c53418204574979b16aa03..6481642edb916ad0b9e683a2a3e7ce200fa6930f 100644 GIT binary patch delta 31089 zcmeFZV~{7^yCqt-ZFkwWZQHi3U)k!iZQFKLm#ezVwrx$n_nw(Jcg}wz?w5%;6LY>~ z?97bF$cS7k*V^lO_D;TlT{?iPmxKUeWlREwrvhlo#ILg>^&V@Ss6)qM^%8~?unTbr z&NmA-3Pzdx%&lqbM9|s&;EKMiyG}^9oG4w+f)GP}5Y51eGw*#p652OS**SP*@iq7J zdb}9-Jq4ZOd1c`;ULz|p%!Q;2Jjw#m=YmZG{)IX~vCOSYKv&&A);%4ssQVg_Db`4XF59;ZB zLBP}TL;rSOdO)hqW?B8wn&Aov&C?hj{DW~A(T7F91QJ@ zWfC4F93jY_p!D}l--dr*u81GpDYbv3>vBo43ZNu8(_mU zS&sDQTl5=(VqGtbUC(Q>c1C;&^TNUp$A#w`!cQzIG?O&I2H6ofH_}qrXM*}D_MOnqiACMU_ zmPa$exkEu*Nrt54g7SQFN@S9wIZbrK;nWi25pvMw+w!r#Pi<;$+;`Sn!jaB5e51sm z1mf43d?FVG+#~*68G46qTmI=F7*tWDDX#`48U(F29Fh?P3{sR3VzUPq_>7H)FJY5} zq1WbuMuA07XU`n8DF$fjS$`BWU~_9&<$YOhOZ~ixG1&MNUVRrMOM+Le*|`+MV;~k5 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