From 488adc94d946fa21e2da1ac69f90da5e0e53e90c Mon Sep 17 00:00:00 2001 From: =?UTF-8?q?Gregor=20Poga=C4=8Dnik?= <1640719+fiksn@users.noreply.github.com> Date: Sat, 11 Sep 2021 14:44:47 +0200 Subject: [PATCH] Simple improvements --- README.md | 31 +++++++++++++++++++++++++++---- Secp256k1.png | Bin 0 -> 16012 bytes addition.gif | Bin 0 -> 396310 bytes ecc.png | Bin 0 -> 2834 bytes 4 files changed, 27 insertions(+), 4 deletions(-) create mode 100644 Secp256k1.png create mode 100644 addition.gif create mode 100644 ecc.png diff --git a/README.md b/README.md index 1cf82e6..f282136 100644 --- a/README.md +++ b/README.md @@ -1,14 +1,37 @@ # schnorr-intro -A gentle introduction to Schnorr signatures +A gentle introduction to Schnorr signature scheme using Eliptic Curve Cryptograph (ECC) + +It was invented by german mathematician Claus-Peter Schnorr. Unfortunately he patented the scheme in 1988 (patend expired in February 2008). So during the creation of Bitcoin it was "free", unfortunately the space lacked any kind of libraries. Therefore ECDSA scheme was used. ## Eliptic Curve Cryptography 101 -An elliptic curve is defined by formula +An elliptic curve is defined by formula: -![equation](http://www.sciweavers.org/tex2img.php?eq=y%5E2x%5E3&bc=White&fc=Black&im=jpg&fs=12&ff=arev&edit=) +![equation](http://www.sciweavers.org/tex2img.php?eq=y%5E2%3Dx%5E3%2Bax%2Bb&bc=Black&fc=White&im=jpg&fs=12&ff=arev&edit=) +![image](ecc.png) +a and b are parameters that define the curve and are carefully tuned. + +Secp256k1 curve used by Bitcoin (and others) has the formula + +![equation](http://www.sciweavers.org/tex2img.php?eq=y%5E2%3Dx%5E3%2B7&bc=Black&fc=White&im=jpg&fs=12&ff=arev&edit=) (a = 0, b = 7) and looks like this: + +![image](Secp256k1.png) + +### Operations + +We operate on points on the curve A, B, C, G, P (upper-case letters) all define points on the curve with an (x, y) coordinate. + +#### Addition + +We can define addition of two points A and B by drawing a line through those two points and where the line intersects the curve is the negative of the new point. After transposition over the x axis (y = 0) we get the actual result + +![image](addition.gif) + +The operation is commutative (A + B = B + A) + +#### Multiplications with a scalar -### Addition ## Schnorr signature scheme diff --git a/Secp256k1.png b/Secp256k1.png new file mode 100644 index 0000000000000000000000000000000000000000..de022553c70bcad1f94c06d56e227b1ae7124474 GIT binary patch literal 16012 zcmb7rd011|*0&c2Z~%%{sZyXK!gYeS6j5TZRxL&fwps-w3~ChxiJ=gp#6YPEDiupo zrOI@zv8aSW84^et6eC&`6eS2rKtP2M5>yD(VBlLP3FMr>eZTkRdH5%LpFOR4t=~HJ zt(Uv;j0H0$O`2p(*tB8mq)7%o$iLAv_@#W`@1jYQKAA+=uiE1D~p4VE+yIaZ#eUjWK$L=$|$n3B=V72!oMRc|25gB~iD)-IGSZSOu{O8tHOh~+W^>ADF=@LQ({WLEM|j|n-c zZ;a-co0~TtobAJPJX%zE`DAt{Ueq>?bI7H`%nnYH+hm(h4*VxXvfs=nc+G?(N6mZ^ z#*gHcZ!Il9nZ3+Pp0Br$civP>&Ehsg?7T>Ojs z?_BuX$E>{VPivUjvGhnul_{x}ba&A=MEENp1hp+CBqUb4cD4!TW9R3Pk34$yzxnYZ z*RS*Ka{;LgLk9LK9>u|wBokO*mi;f2r}GpJrUAkn?A4ujaavy93guC-H@E8%@YWm- zW#acoNU`_xe40Z$LXO}JV<%4$+xiFIX_#x;7)&0Da`SydDyULeVqf#uRMwu2LHMpVr3r?rp7%WKFK-Mg 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