arnaucube
6 years ago
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+97 -0README.md
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+674 -0bn128/LICENSE
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+186 -0bn128/README.md
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+407 -0bn128/bn128.go
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+66 -0bn128/bn128_test.go
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+129 -0bn128/fq.go
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+161 -0bn128/fq12.go
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+154 -0bn128/fq2.go
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+192 -0bn128/fq6.go
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+160 -0bn128/fqn_test.go
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+191 -0bn128/g1.go
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+31 -0bn128/g1_test.go
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+221 -0bn128/g2.go
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+24 -0bn128/g2_test.go
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+6 -0go.mod
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+8 -0go.sum
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+145 -0r1csqap.go
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+112 -0r1csqap_test.go
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# go-snark [](https://goreportcard.com/report/github.com/arnaucube/go-snark) |
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Not finished, work in progress (doing this in my free time, so I don't have much time). |
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#### Test |
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``` |
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go test ./... -v |
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``` |
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## R1CS to Quadratic Arithmetic Program |
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- `Succinct Non-Interactive Zero Knowledge for a von Neumann Architecture`, Eli Ben-Sasson, Alessandro Chiesa, Eran Tromer, Madars Virza https://eprint.iacr.org/2013/879.pdf |
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- Vitalik Buterin blog post about QAP https://medium.com/@VitalikButerin/quadratic-arithmetic-programs-from-zero-to-hero-f6d558cea649 |
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- Ariel Gabizon in Zcash blog https://z.cash/blog/snark-explain5 |
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- Lagrange polynomial Wikipedia article https://en.wikipedia.org/wiki/Lagrange_polynomial |
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#### Usage |
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- R1CS to QAP |
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```go |
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b0 := big.NewFloat(float64(0)) |
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b1 := big.NewFloat(float64(1)) |
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b5 := big.NewFloat(float64(5)) |
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a := [][]*big.Float{ |
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[]*big.Float{b0, b1, b0, b0, b0, b0}, |
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[]*big.Float{b0, b0, b0, b1, b0, b0}, |
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[]*big.Float{b0, b1, b0, b0, b1, b0}, |
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[]*big.Float{b5, b0, b0, b0, b0, b1}, |
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} |
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b := [][]*big.Float{ |
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[]*big.Float{b0, b1, b0, b0, b0, b0}, |
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[]*big.Float{b0, b1, b0, b0, b0, b0}, |
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[]*big.Float{b1, b0, b0, b0, b0, b0}, |
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[]*big.Float{b1, b0, b0, b0, b0, b0}, |
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} |
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c := [][]*big.Float{ |
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[]*big.Float{b0, b0, b0, b1, b0, b0}, |
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[]*big.Float{b0, b0, b0, b0, b1, b0}, |
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[]*big.Float{b0, b0, b0, b0, b0, b1}, |
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[]*big.Float{b0, b0, b1, b0, b0, b0}, |
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} |
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alpha, beta, gamma, z := R1CSToQAP(a, b, c) |
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fmt.Println(alpha) |
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fmt.Println(beta) |
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fmt.Println(gamma) |
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fmt.Println(z) |
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/* |
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out: |
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alpha: [[-5 9.166666666666666 -5 0.8333333333333334] [8 -11.333333333333332 5 -0.6666666666666666] [0 0 0 0] [-6 9.5 -4 0.5] [4 -7 3.5 -0.5] [-1 1.8333333333333333 -1 0.16666666666666666]] |
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beta: [[3 -5.166666666666667 2.5 -0.33333333333333337] [-2 5.166666666666667 -2.5 0.33333333333333337] [0 0 0 0] [0 0 0 0] [0 0 0 0] [0 0 0 0]] |
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gamma: [[0 0 0 0] [0 0 0 0] [-1 1.8333333333333333 -1 0.16666666666666666] [4 -4.333333333333333 1.5 -0.16666666666666666] [-6 9.5 -4 0.5] [4 -7 3.5 -0.5]] |
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z: [24 -50 35 -10 1] |
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*/ |
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``` |
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## Bn128 |
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Implementation of the bn128 pairing. |
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Implementation followng the information and the implementations from: |
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- `Multiplication and Squaring on Pairing-Friendly |
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Fields`, Augusto Jun Devegili, Colm Ó hÉigeartaigh, Michael Scott, and Ricardo Dahab https://pdfs.semanticscholar.org/3e01/de88d7428076b2547b60072088507d881bf1.pdf |
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- `Optimal Pairings`, Frederik Vercauteren https://www.cosic.esat.kuleuven.be/bcrypt/optimal.pdf , https://eprint.iacr.org/2008/096.pdf |
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- `Double-and-Add with Relative Jacobian |
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Coordinates`, Björn Fay https://eprint.iacr.org/2014/1014.pdf |
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- `Fast and Regular Algorithms for Scalar Multiplication |
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over Elliptic Curves`, Matthieu Rivain https://eprint.iacr.org/2011/338.pdf |
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- `High-Speed Software Implementation of the Optimal Ate Pairing over Barreto–Naehrig Curves`, Jean-Luc Beuchat, Jorge E. González-Díaz, Shigeo Mitsunari, Eiji Okamoto, Francisco Rodríguez-Henríquez, and Tadanori Teruya https://eprint.iacr.org/2010/354.pdf |
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- `New software speed records for cryptographic pairings`, Michael Naehrig, Ruben Niederhagen, Peter Schwabe https://cryptojedi.org/papers/dclxvi-20100714.pdf |
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- `Implementing Cryptographic Pairings over Barreto-Naehrig Curves`, Augusto Jun Devegili, Michael Scott, Ricardo Dahab https://eprint.iacr.org/2007/390.pdf |
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- https://github.com/zcash/zcash/tree/master/src/snark |
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- https://github.com/iden3/snarkjs |
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- https://github.com/ethereum/py_ecc/tree/master/py_ecc/bn128 |
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#### Usage |
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- Pairing |
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```go |
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bn128, err := NewBn128() |
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assert.Nil(t, err) |
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big25 := big.NewInt(int64(25)) |
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big30 := big.NewInt(int64(30)) |
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g1a := bn128.G1.MulScalar(bn128.G1.G, big25) |
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g2a := bn128.G2.MulScalar(bn128.G2.G, big30) |
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g1b := bn128.G1.MulScalar(bn128.G1.G, big30) |
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g2b := bn128.G2.MulScalar(bn128.G2.G, big25) |
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pA, err := bn128.Pairing(g1a, g2a) |
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assert.Nil(t, err) |
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pB, err := bn128.Pairing(g1b, g2b) |
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assert.Nil(t, err) |
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assert.True(t, bn128.Fq12.Equal(pA, pB)) |
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``` |
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@ -0,0 +1,674 @@ |
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GNU GENERAL PUBLIC LICENSE |
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Version 3, 29 June 2007 |
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Copyright (C) 2007 Free Software Foundation, Inc. <https://fsf.org/> |
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Everyone is permitted to copy and distribute verbatim copies |
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of this license document, but changing it is not allowed. |
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Preamble |
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The GNU General Public License is a free, copyleft license for |
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The licenses for most software and other practical works are designed |
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GNU General Public License for most of our software; it applies also to |
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any other work released this way by its authors. You can apply it to |
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When we speak of free software, we are referring to freedom, not |
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||||
|
more than your reasonable cost of physically performing this |
||||
|
conveying of source, or (2) access to copy the |
||||
|
Corresponding Source from a network server at no charge. |
||||
|
|
||||
|
c) Convey individual copies of the object code with a copy of the |
||||
|
written offer to provide the Corresponding Source. This |
||||
|
alternative is allowed only occasionally and noncommercially, and |
||||
|
only if you received the object code with such an offer, in accord |
||||
|
with subsection 6b. |
||||
|
|
||||
|
d) Convey the object code by offering access from a designated |
||||
|
place (gratis or for a charge), and offer equivalent access to the |
||||
|
Corresponding Source in the same way through the same place at no |
||||
|
further charge. You need not require recipients to copy the |
||||
|
Corresponding Source along with the object code. If the place to |
||||
|
copy the object code is a network server, the Corresponding Source |
||||
|
may be on a different server (operated by you or a third party) |
||||
|
that supports equivalent copying facilities, provided you maintain |
||||
|
clear directions next to the object code saying where to find the |
||||
|
Corresponding Source. Regardless of what server hosts the |
||||
|
Corresponding Source, you remain obligated to ensure that it is |
||||
|
available for as long as needed to satisfy these requirements. |
||||
|
|
||||
|
e) Convey the object code using peer-to-peer transmission, provided |
||||
|
you inform other peers where the object code and Corresponding |
||||
|
Source of the work are being offered to the general public at no |
||||
|
charge under subsection 6d. |
||||
|
|
||||
|
A separable portion of the object code, whose source code is excluded |
||||
|
from the Corresponding Source as a System Library, need not be |
||||
|
included in conveying the object code work. |
||||
|
|
||||
|
A "User Product" is either (1) a "consumer product", which means any |
||||
|
tangible personal property which is normally used for personal, family, |
||||
|
or household purposes, or (2) anything designed or sold for incorporation |
||||
|
into a dwelling. In determining whether a product is a consumer product, |
||||
|
doubtful cases shall be resolved in favor of coverage. For a particular |
||||
|
product received by a particular user, "normally used" refers to a |
||||
|
typical or common use of that class of product, regardless of the status |
||||
|
of the particular user or of the way in which the particular user |
||||
|
actually uses, or expects or is expected to use, the product. A product |
||||
|
is a consumer product regardless of whether the product has substantial |
||||
|
commercial, industrial or non-consumer uses, unless such uses represent |
||||
|
the only significant mode of use of the product. |
||||
|
|
||||
|
"Installation Information" for a User Product means any methods, |
||||
|
procedures, authorization keys, or other information required to install |
||||
|
and execute modified versions of a covered work in that User Product from |
||||
|
a modified version of its Corresponding Source. The information must |
||||
|
suffice to ensure that the continued functioning of the modified object |
||||
|
code is in no case prevented or interfered with solely because |
||||
|
modification has been made. |
||||
|
|
||||
|
If you convey an object code work under this section in, or with, or |
||||
|
specifically for use in, a User Product, and the conveying occurs as |
||||
|
part of a transaction in which the right of possession and use of the |
||||
|
User Product is transferred to the recipient in perpetuity or for a |
||||
|
fixed term (regardless of how the transaction is characterized), the |
||||
|
Corresponding Source conveyed under this section must be accompanied |
||||
|
by the Installation Information. But this requirement does not apply |
||||
|
if neither you nor any third party retains the ability to install |
||||
|
modified object code on the User Product (for example, the work has |
||||
|
been installed in ROM). |
||||
|
|
||||
|
The requirement to provide Installation Information does not include a |
||||
|
requirement to continue to provide support service, warranty, or updates |
||||
|
for a work that has been modified or installed by the recipient, or for |
||||
|
the User Product in which it has been modified or installed. Access to a |
||||
|
network may be denied when the modification itself materially and |
||||
|
adversely affects the operation of the network or violates the rules and |
||||
|
protocols for communication across the network. |
||||
|
|
||||
|
Corresponding Source conveyed, and Installation Information provided, |
||||
|
in accord with this section must be in a format that is publicly |
||||
|
documented (and with an implementation available to the public in |
||||
|
source code form), and must require no special password or key for |
||||
|
unpacking, reading or copying. |
||||
|
|
||||
|
7. Additional Terms. |
||||
|
|
||||
|
"Additional permissions" are terms that supplement the terms of this |
||||
|
License by making exceptions from one or more of its conditions. |
||||
|
Additional permissions that are applicable to the entire Program shall |
||||
|
be treated as though they were included in this License, to the extent |
||||
|
that they are valid under applicable law. If additional permissions |
||||
|
apply only to part of the Program, that part may be used separately |
||||
|
under those permissions, but the entire Program remains governed by |
||||
|
this License without regard to the additional permissions. |
||||
|
|
||||
|
When you convey a copy of a covered work, you may at your option |
||||
|
remove any additional permissions from that copy, or from any part of |
||||
|
it. (Additional permissions may be written to require their own |
||||
|
removal in certain cases when you modify the work.) You may place |
||||
|
additional permissions on material, added by you to a covered work, |
||||
|
for which you have or can give appropriate copyright permission. |
||||
|
|
||||
|
Notwithstanding any other provision of this License, for material you |
||||
|
add to a covered work, you may (if authorized by the copyright holders of |
||||
|
that material) supplement the terms of this License with terms: |
||||
|
|
||||
|
a) Disclaiming warranty or limiting liability differently from the |
||||
|
terms of sections 15 and 16 of this License; or |
||||
|
|
||||
|
b) Requiring preservation of specified reasonable legal notices or |
||||
|
author attributions in that material or in the Appropriate Legal |
||||
|
Notices displayed by works containing it; or |
||||
|
|
||||
|
c) Prohibiting misrepresentation of the origin of that material, or |
||||
|
requiring that modified versions of such material be marked in |
||||
|
reasonable ways as different from the original version; or |
||||
|
|
||||
|
d) Limiting the use for publicity purposes of names of licensors or |
||||
|
authors of the material; or |
||||
|
|
||||
|
e) Declining to grant rights under trademark law for use of some |
||||
|
trade names, trademarks, or service marks; or |
||||
|
|
||||
|
f) Requiring indemnification of licensors and authors of that |
||||
|
material by anyone who conveys the material (or modified versions of |
||||
|
it) with contractual assumptions of liability to the recipient, for |
||||
|
any liability that these contractual assumptions directly impose on |
||||
|
those licensors and authors. |
||||
|
|
||||
|
All other non-permissive additional terms are considered "further |
||||
|
restrictions" within the meaning of section 10. If the Program as you |
||||
|
received it, or any part of it, contains a notice stating that it is |
||||
|
governed by this License along with a term that is a further |
||||
|
restriction, you may remove that term. If a license document contains |
||||
|
a further restriction but permits relicensing or conveying under this |
||||
|
License, you may add to a covered work material governed by the terms |
||||
|
of that license document, provided that the further restriction does |
||||
|
not survive such relicensing or conveying. |
||||
|
|
||||
|
If you add terms to a covered work in accord with this section, you |
||||
|
must place, in the relevant source files, a statement of the |
||||
|
additional terms that apply to those files, or a notice indicating |
||||
|
where to find the applicable terms. |
||||
|
|
||||
|
Additional terms, permissive or non-permissive, may be stated in the |
||||
|
form of a separately written license, or stated as exceptions; |
||||
|
the above requirements apply either way. |
||||
|
|
||||
|
8. Termination. |
||||
|
|
||||
|
You may not propagate or modify a covered work except as expressly |
||||
|
provided under this License. Any attempt otherwise to propagate or |
||||
|
modify it is void, and will automatically terminate your rights under |
||||
|
this License (including any patent licenses granted under the third |
||||
|
paragraph of section 11). |
||||
|
|
||||
|
However, if you cease all violation of this License, then your |
||||
|
license from a particular copyright holder is reinstated (a) |
||||
|
provisionally, unless and until the copyright holder explicitly and |
||||
|
finally terminates your license, and (b) permanently, if the copyright |
||||
|
holder fails to notify you of the violation by some reasonable means |
||||
|
prior to 60 days after the cessation. |
||||
|
|
||||
|
Moreover, your license from a particular copyright holder is |
||||
|
reinstated permanently if the copyright holder notifies you of the |
||||
|
violation by some reasonable means, this is the first time you have |
||||
|
received notice of violation of this License (for any work) from that |
||||
|
copyright holder, and you cure the violation prior to 30 days after |
||||
|
your receipt of the notice. |
||||
|
|
||||
|
Termination of your rights under this section does not terminate the |
||||
|
licenses of parties who have received copies or rights from you under |
||||
|
this License. If your rights have been terminated and not permanently |
||||
|
reinstated, you do not qualify to receive new licenses for the same |
||||
|
material under section 10. |
||||
|
|
||||
|
9. Acceptance Not Required for Having Copies. |
||||
|
|
||||
|
You are not required to accept this License in order to receive or |
||||
|
run a copy of the Program. Ancillary propagation of a covered work |
||||
|
occurring solely as a consequence of using peer-to-peer transmission |
||||
|
to receive a copy likewise does not require acceptance. However, |
||||
|
nothing other than this License grants you permission to propagate or |
||||
|
modify any covered work. These actions infringe copyright if you do |
||||
|
not accept this License. Therefore, by modifying or propagating a |
||||
|
covered work, you indicate your acceptance of this License to do so. |
||||
|
|
||||
|
10. Automatic Licensing of Downstream Recipients. |
||||
|
|
||||
|
Each time you convey a covered work, the recipient automatically |
||||
|
receives a license from the original licensors, to run, modify and |
||||
|
propagate that work, subject to this License. You are not responsible |
||||
|
for enforcing compliance by third parties with this License. |
||||
|
|
||||
|
An "entity transaction" is a transaction transferring control of an |
||||
|
organization, or substantially all assets of one, or subdividing an |
||||
|
organization, or merging organizations. If propagation of a covered |
||||
|
work results from an entity transaction, each party to that |
||||
|
transaction who receives a copy of the work also receives whatever |
||||
|
licenses to the work the party's predecessor in interest had or could |
||||
|
give under the previous paragraph, plus a right to possession of the |
||||
|
Corresponding Source of the work from the predecessor in interest, if |
||||
|
the predecessor has it or can get it with reasonable efforts. |
||||
|
|
||||
|
You may not impose any further restrictions on the exercise of the |
||||
|
rights granted or affirmed under this License. For example, you may |
||||
|
not impose a license fee, royalty, or other charge for exercise of |
||||
|
rights granted under this License, and you may not initiate litigation |
||||
|
(including a cross-claim or counterclaim in a lawsuit) alleging that |
||||
|
any patent claim is infringed by making, using, selling, offering for |
||||
|
sale, or importing the Program or any portion of it. |
||||
|
|
||||
|
11. Patents. |
||||
|
|
||||
|
A "contributor" is a copyright holder who authorizes use under this |
||||
|
License of the Program or a work on which the Program is based. The |
||||
|
work thus licensed is called the contributor's "contributor version". |
||||
|
|
||||
|
A contributor's "essential patent claims" are all patent claims |
||||
|
owned or controlled by the contributor, whether already acquired or |
||||
|
hereafter acquired, that would be infringed by some manner, permitted |
||||
|
by this License, of making, using, or selling its contributor version, |
||||
|
but do not include claims that would be infringed only as a |
||||
|
consequence of further modification of the contributor version. For |
||||
|
purposes of this definition, "control" includes the right to grant |
||||
|
patent sublicenses in a manner consistent with the requirements of |
||||
|
this License. |
||||
|
|
||||
|
Each contributor grants you a non-exclusive, worldwide, royalty-free |
||||
|
patent license under the contributor's essential patent claims, to |
||||
|
make, use, sell, offer for sale, import and otherwise run, modify and |
||||
|
propagate the contents of its contributor version. |
||||
|
|
||||
|
In the following three paragraphs, a "patent license" is any express |
||||
|
agreement or commitment, however denominated, not to enforce a patent |
||||
|
(such as an express permission to practice a patent or covenant not to |
||||
|
sue for patent infringement). To "grant" such a patent license to a |
||||
|
party means to make such an agreement or commitment not to enforce a |
||||
|
patent against the party. |
||||
|
|
||||
|
If you convey a covered work, knowingly relying on a patent license, |
||||
|
and the Corresponding Source of the work is not available for anyone |
||||
|
to copy, free of charge and under the terms of this License, through a |
||||
|
publicly available network server or other readily accessible means, |
||||
|
then you must either (1) cause the Corresponding Source to be so |
||||
|
available, or (2) arrange to deprive yourself of the benefit of the |
||||
|
patent license for this particular work, or (3) arrange, in a manner |
||||
|
consistent with the requirements of this License, to extend the patent |
||||
|
license to downstream recipients. "Knowingly relying" means you have |
||||
|
actual knowledge that, but for the patent license, your conveying the |
||||
|
covered work in a country, or your recipient's use of the covered work |
||||
|
in a country, would infringe one or more identifiable patents in that |
||||
|
country that you have reason to believe are valid. |
||||
|
|
||||
|
If, pursuant to or in connection with a single transaction or |
||||
|
arrangement, you convey, or propagate by procuring conveyance of, a |
||||
|
covered work, and grant a patent license to some of the parties |
||||
|
receiving the covered work authorizing them to use, propagate, modify |
||||
|
or convey a specific copy of the covered work, then the patent license |
||||
|
you grant is automatically extended to all recipients of the covered |
||||
|
work and works based on it. |
||||
|
|
||||
|
A patent license is "discriminatory" if it does not include within |
||||
|
the scope of its coverage, prohibits the exercise of, or is |
||||
|
conditioned on the non-exercise of one or more of the rights that are |
||||
|
specifically granted under this License. You may not convey a covered |
||||
|
work if you are a party to an arrangement with a third party that is |
||||
|
in the business of distributing software, under which you make payment |
||||
|
to the third party based on the extent of your activity of conveying |
||||
|
the work, and under which the third party grants, to any of the |
||||
|
parties who would receive the covered work from you, a discriminatory |
||||
|
patent license (a) in connection with copies of the covered work |
||||
|
conveyed by you (or copies made from those copies), or (b) primarily |
||||
|
for and in connection with specific products or compilations that |
||||
|
contain the covered work, unless you entered into that arrangement, |
||||
|
or that patent license was granted, prior to 28 March 2007. |
||||
|
|
||||
|
Nothing in this License shall be construed as excluding or limiting |
||||
|
any implied license or other defenses to infringement that may |
||||
|
otherwise be available to you under applicable patent law. |
||||
|
|
||||
|
12. No Surrender of Others' Freedom. |
||||
|
|
||||
|
If conditions are imposed on you (whether by court order, agreement or |
||||
|
otherwise) that contradict the conditions of this License, they do not |
||||
|
excuse you from the conditions of this License. If you cannot convey a |
||||
|
covered work so as to satisfy simultaneously your obligations under this |
||||
|
License and any other pertinent obligations, then as a consequence you may |
||||
|
not convey it at all. For example, if you agree to terms that obligate you |
||||
|
to collect a royalty for further conveying from those to whom you convey |
||||
|
the Program, the only way you could satisfy both those terms and this |
||||
|
License would be to refrain entirely from conveying the Program. |
||||
|
|
||||
|
13. Use with the GNU Affero General Public License. |
||||
|
|
||||
|
Notwithstanding any other provision of this License, you have |
||||
|
permission to link or combine any covered work with a work licensed |
||||
|
under version 3 of the GNU Affero General Public License into a single |
||||
|
combined work, and to convey the resulting work. The terms of this |
||||
|
License will continue to apply to the part which is the covered work, |
||||
|
but the special requirements of the GNU Affero General Public License, |
||||
|
section 13, concerning interaction through a network will apply to the |
||||
|
combination as such. |
||||
|
|
||||
|
14. Revised Versions of this License. |
||||
|
|
||||
|
The Free Software Foundation may publish revised and/or new versions of |
||||
|
the GNU General Public License from time to time. Such new versions will |
||||
|
be similar in spirit to the present version, but may differ in detail to |
||||
|
address new problems or concerns. |
||||
|
|
||||
|
Each version is given a distinguishing version number. If the |
||||
|
Program specifies that a certain numbered version of the GNU General |
||||
|
Public License "or any later version" applies to it, you have the |
||||
|
option of following the terms and conditions either of that numbered |
||||
|
version or of any later version published by the Free Software |
||||
|
Foundation. If the Program does not specify a version number of the |
||||
|
GNU General Public License, you may choose any version ever published |
||||
|
by the Free Software Foundation. |
||||
|
|
||||
|
If the Program specifies that a proxy can decide which future |
||||
|
versions of the GNU General Public License can be used, that proxy's |
||||
|
public statement of acceptance of a version permanently authorizes you |
||||
|
to choose that version for the Program. |
||||
|
|
||||
|
Later license versions may give you additional or different |
||||
|
permissions. However, no additional obligations are imposed on any |
||||
|
author or copyright holder as a result of your choosing to follow a |
||||
|
later version. |
||||
|
|
||||
|
15. Disclaimer of Warranty. |
||||
|
|
||||
|
THERE IS NO WARRANTY FOR THE PROGRAM, TO THE EXTENT PERMITTED BY |
||||
|
APPLICABLE LAW. EXCEPT WHEN OTHERWISE STATED IN WRITING THE COPYRIGHT |
||||
|
HOLDERS AND/OR OTHER PARTIES PROVIDE THE PROGRAM "AS IS" WITHOUT WARRANTY |
||||
|
OF ANY KIND, EITHER EXPRESSED OR IMPLIED, INCLUDING, BUT NOT LIMITED TO, |
||||
|
THE IMPLIED WARRANTIES OF MERCHANTABILITY AND FITNESS FOR A PARTICULAR |
||||
|
PURPOSE. THE ENTIRE RISK AS TO THE QUALITY AND PERFORMANCE OF THE PROGRAM |
||||
|
IS WITH YOU. SHOULD THE PROGRAM PROVE DEFECTIVE, YOU ASSUME THE COST OF |
||||
|
ALL NECESSARY SERVICING, REPAIR OR CORRECTION. |
||||
|
|
||||
|
16. Limitation of Liability. |
||||
|
|
||||
|
IN NO EVENT UNLESS REQUIRED BY APPLICABLE LAW OR AGREED TO IN WRITING |
||||
|
WILL ANY COPYRIGHT HOLDER, OR ANY OTHER PARTY WHO MODIFIES AND/OR CONVEYS |
||||
|
THE PROGRAM AS PERMITTED ABOVE, BE LIABLE TO YOU FOR DAMAGES, INCLUDING ANY |
||||
|
GENERAL, SPECIAL, INCIDENTAL OR CONSEQUENTIAL DAMAGES ARISING OUT OF THE |
||||
|
USE OR INABILITY TO USE THE PROGRAM (INCLUDING BUT NOT LIMITED TO LOSS OF |
||||
|
DATA OR DATA BEING RENDERED INACCURATE OR LOSSES SUSTAINED BY YOU OR THIRD |
||||
|
PARTIES OR A FAILURE OF THE PROGRAM TO OPERATE WITH ANY OTHER PROGRAMS), |
||||
|
EVEN IF SUCH HOLDER OR OTHER PARTY HAS BEEN ADVISED OF THE POSSIBILITY OF |
||||
|
SUCH DAMAGES. |
||||
|
|
||||
|
17. Interpretation of Sections 15 and 16. |
||||
|
|
||||
|
If the disclaimer of warranty and limitation of liability provided |
||||
|
above cannot be given local legal effect according to their terms, |
||||
|
reviewing courts shall apply local law that most closely approximates |
||||
|
an absolute waiver of all civil liability in connection with the |
||||
|
Program, unless a warranty or assumption of liability accompanies a |
||||
|
copy of the Program in return for a fee. |
||||
|
|
||||
|
END OF TERMS AND CONDITIONS |
||||
|
|
||||
|
How to Apply These Terms to Your New Programs |
||||
|
|
||||
|
If you develop a new program, and you want it to be of the greatest |
||||
|
possible use to the public, the best way to achieve this is to make it |
||||
|
free software which everyone can redistribute and change under these terms. |
||||
|
|
||||
|
To do so, attach the following notices to the program. It is safest |
||||
|
to attach them to the start of each source file to most effectively |
||||
|
state the exclusion of warranty; and each file should have at least |
||||
|
the "copyright" line and a pointer to where the full notice is found. |
||||
|
|
||||
|
<one line to give the program's name and a brief idea of what it does.> |
||||
|
Copyright (C) <year> <name of author> |
||||
|
|
||||
|
This program is free software: you can redistribute it and/or modify |
||||
|
it under the terms of the GNU General Public License as published by |
||||
|
the Free Software Foundation, either version 3 of the License, or |
||||
|
(at your option) any later version. |
||||
|
|
||||
|
This program is distributed in the hope that it will be useful, |
||||
|
but WITHOUT ANY WARRANTY; without even the implied warranty of |
||||
|
MERCHANTABILITY or FITNESS FOR A PARTICULAR PURPOSE. See the |
||||
|
GNU General Public License for more details. |
||||
|
|
||||
|
You should have received a copy of the GNU General Public License |
||||
|
along with this program. If not, see <https://www.gnu.org/licenses/>. |
||||
|
|
||||
|
Also add information on how to contact you by electronic and paper mail. |
||||
|
|
||||
|
If the program does terminal interaction, make it output a short |
||||
|
notice like this when it starts in an interactive mode: |
||||
|
|
||||
|
<program> Copyright (C) <year> <name of author> |
||||
|
This program comes with ABSOLUTELY NO WARRANTY; for details type `show w'. |
||||
|
This is free software, and you are welcome to redistribute it |
||||
|
under certain conditions; type `show c' for details. |
||||
|
|
||||
|
The hypothetical commands `show w' and `show c' should show the appropriate |
||||
|
parts of the General Public License. Of course, your program's commands |
||||
|
might be different; for a GUI interface, you would use an "about box". |
||||
|
|
||||
|
You should also get your employer (if you work as a programmer) or school, |
||||
|
if any, to sign a "copyright disclaimer" for the program, if necessary. |
||||
|
For more information on this, and how to apply and follow the GNU GPL, see |
||||
|
<https://www.gnu.org/licenses/>. |
||||
|
|
||||
|
The GNU General Public License does not permit incorporating your program |
||||
|
into proprietary programs. If your program is a subroutine library, you |
||||
|
may consider it more useful to permit linking proprietary applications with |
||||
|
the library. If this is what you want to do, use the GNU Lesser General |
||||
|
Public License instead of this License. But first, please read |
||||
|
<https://www.gnu.org/licenses/why-not-lgpl.html>. |
||||
@ -0,0 +1,186 @@ |
|||||
|
## Bn128 |
||||
|
Implementation of the bn128 pairing. |
||||
|
|
||||
|
|
||||
|
Implementation followng the information and the implementations from: |
||||
|
- `Multiplication and Squaring on Pairing-Friendly |
||||
|
Fields`, Augusto Jun Devegili, Colm Ó hÉigeartaigh, Michael Scott, and Ricardo Dahab https://pdfs.semanticscholar.org/3e01/de88d7428076b2547b60072088507d881bf1.pdf |
||||
|
- `Optimal Pairings`, Frederik Vercauteren https://www.cosic.esat.kuleuven.be/bcrypt/optimal.pdf , https://eprint.iacr.org/2008/096.pdf |
||||
|
- `Double-and-Add with Relative Jacobian |
||||
|
Coordinates`, Björn Fay https://eprint.iacr.org/2014/1014.pdf |
||||
|
- `Fast and Regular Algorithms for Scalar Multiplication |
||||
|
over Elliptic Curves`, Matthieu Rivain https://eprint.iacr.org/2011/338.pdf |
||||
|
- `High-Speed Software Implementation of the Optimal Ate Pairing over Barreto–Naehrig Curves`, Jean-Luc Beuchat, Jorge E. González-Díaz, Shigeo Mitsunari, Eiji Okamoto, Francisco Rodríguez-Henríquez, and Tadanori Teruya https://eprint.iacr.org/2010/354.pdf |
||||
|
- `New software speed records for cryptographic pairings`, Michael Naehrig, Ruben Niederhagen, Peter Schwabe https://cryptojedi.org/papers/dclxvi-20100714.pdf |
||||
|
- `Implementing Cryptographic Pairings over Barreto-Naehrig Curves`, Augusto Jun Devegili, Michael Scott, Ricardo Dahab https://eprint.iacr.org/2007/390.pdf |
||||
|
- https://github.com/zcash/zcash/tree/master/src/snark |
||||
|
- https://github.com/iden3/snarkjs |
||||
|
- https://github.com/ethereum/py_ecc/tree/master/py_ecc/bn128 |
||||
|
|
||||
|
- [x] Fq, Fq2, Fq6, Fq12 operations |
||||
|
- [x] G1, G2 operations |
||||
|
- [x] preparePairing |
||||
|
- [x] PreComupteG1, PreComupteG2 |
||||
|
- [x] DoubleStep, AddStep |
||||
|
- [x] MillerLoop |
||||
|
- [x] Pairing |
||||
|
|
||||
|
### Installation |
||||
|
``` |
||||
|
go get github.com/arnaucube/bn128 |
||||
|
``` |
||||
|
|
||||
|
#### Usage |
||||
|
|
||||
|
- Pairing |
||||
|
```go |
||||
|
bn128, err := NewBn128() |
||||
|
assert.Nil(t, err) |
||||
|
|
||||
|
big25 := big.NewInt(int64(25)) |
||||
|
big30 := big.NewInt(int64(30)) |
||||
|
|
||||
|
g1a := bn128.G1.MulScalar(bn128.G1.G, big25) |
||||
|
g2a := bn128.G2.MulScalar(bn128.G2.G, big30) |
||||
|
|
||||
|
g1b := bn128.G1.MulScalar(bn128.G1.G, big30) |
||||
|
g2b := bn128.G2.MulScalar(bn128.G2.G, big25) |
||||
|
|
||||
|
pA, err := bn128.Pairing(g1a, g2a) |
||||
|
assert.Nil(t, err) |
||||
|
pB, err := bn128.Pairing(g1b, g2b) |
||||
|
assert.Nil(t, err) |
||||
|
assert.True(t, bn128.Fq12.Equal(pA, pB)) |
||||
|
``` |
||||
|
|
||||
|
#### Test |
||||
|
``` |
||||
|
go test -v |
||||
|
``` |
||||
|
|
||||
|
##### Internal operations more deeply |
||||
|
|
||||
|
First let's assume that we have these three basic functions to convert integer compositions to big integer compositions: |
||||
|
```go |
||||
|
func iToBig(a int) *big.Int { |
||||
|
return big.NewInt(int64(a)) |
||||
|
} |
||||
|
|
||||
|
func iiToBig(a, b int) [2]*big.Int { |
||||
|
return [2]*big.Int{iToBig(a), iToBig(b)} |
||||
|
} |
||||
|
|
||||
|
func iiiToBig(a, b int) [2]*big.Int { |
||||
|
return [2]*big.Int{iToBig(a), iToBig(b)} |
||||
|
} |
||||
|
``` |
||||
|
- Finite Fields (1, 2, 6, 12) operations |
||||
|
```go |
||||
|
// new finite field of order 1 |
||||
|
fq1 := NewFq(iToBig(7)) |
||||
|
|
||||
|
// basic operations of finite field 1 |
||||
|
res := fq1.Add(iToBig(4), iToBig(4)) |
||||
|
res = fq1.Double(iToBig(5)) |
||||
|
res = fq1.Sub(iToBig(5), iToBig(7)) |
||||
|
res = fq1.Neg(iToBig(5)) |
||||
|
res = fq1.Mul(iToBig(5), iToBig(11)) |
||||
|
res = fq1.Inverse(iToBig(4)) |
||||
|
res = fq1.Square(iToBig(5)) |
||||
|
|
||||
|
// new finite field of order 2 |
||||
|
nonResidueFq2str := "-1" // i/j |
||||
|
nonResidueFq2, ok := new(big.Int).SetString(nonResidueFq2str, 10) |
||||
|
fq2 := Fq2{fq1, nonResidueFq2} |
||||
|
nonResidueFq6 := iiToBig(9, 1) |
||||
|
|
||||
|
// basic operations of finite field of order 2 |
||||
|
res := fq2.Add(iiToBig(4, 4), iiToBig(3, 4)) |
||||
|
res = fq2.Double(iiToBig(5, 3)) |
||||
|
res = fq2.Sub(iiToBig(5, 3), iiToBig(7, 2)) |
||||
|
res = fq2.Neg(iiToBig(4, 4)) |
||||
|
res = fq2.Mul(iiToBig(4, 4), iiToBig(3, 4)) |
||||
|
res = fq2.Inverse(iiToBig(4, 4)) |
||||
|
res = fq2.Div(iiToBig(4, 4), iiToBig(3, 4)) |
||||
|
res = fq2.Square(iiToBig(4, 4)) |
||||
|
|
||||
|
|
||||
|
// new finite field of order 6 |
||||
|
nonResidueFq6 := iiToBig(9, 1) // TODO |
||||
|
fq6 := Fq6{fq2, nonResidueFq6} |
||||
|
|
||||
|
// define two new values of Finite Field 6, in order to be able to perform the operations |
||||
|
a := [3][2]*big.Int{ |
||||
|
iiToBig(1, 2), |
||||
|
iiToBig(3, 4), |
||||
|
iiToBig(5, 6)} |
||||
|
b := [3][2]*big.Int{ |
||||
|
iiToBig(12, 11), |
||||
|
iiToBig(10, 9), |
||||
|
iiToBig(8, 7)} |
||||
|
|
||||
|
// basic operations of finite field order 6 |
||||
|
res := fq6.Add(a, b) |
||||
|
res = fq6.Sub(a, b) |
||||
|
res = fq6.Mul(a, b) |
||||
|
divRes := fq6.Div(mulRes, b) |
||||
|
|
||||
|
|
||||
|
// new finite field of order 12 |
||||
|
q, ok := new(big.Int).SetString("21888242871839275222246405745257275088696311157297823662689037894645226208583", 10) // i |
||||
|
if !ok { |
||||
|
fmt.Println("error parsing string to big integer") |
||||
|
} |
||||
|
|
||||
|
fq1 := NewFq(q) |
||||
|
nonResidueFq2, ok := new(big.Int).SetString("21888242871839275222246405745257275088696311157297823662689037894645226208582", 10) // i |
||||
|
assert.True(t, ok) |
||||
|
nonResidueFq6 := iiToBig(9, 1) |
||||
|
|
||||
|
fq2 := Fq2{fq1, nonResidueFq2} |
||||
|
fq6 := Fq6{fq2, nonResidueFq6} |
||||
|
fq12 := Fq12{fq6, fq2, nonResidueFq6} |
||||
|
|
||||
|
``` |
||||
|
|
||||
|
- G1 operations |
||||
|
```go |
||||
|
bn128, err := NewBn128() |
||||
|
assert.Nil(t, err) |
||||
|
|
||||
|
r1 := big.NewInt(int64(33)) |
||||
|
r2 := big.NewInt(int64(44)) |
||||
|
|
||||
|
gr1 := bn128.G1.MulScalar(bn128.G1.G, bn128.Fq1.Copy(r1)) |
||||
|
gr2 := bn128.G1.MulScalar(bn128.G1.G, bn128.Fq1.Copy(r2)) |
||||
|
|
||||
|
grsum1 := bn128.G1.Add(gr1, gr2) |
||||
|
r1r2 := bn128.Fq1.Add(r1, r2) |
||||
|
grsum2 := bn128.G1.MulScalar(bn128.G1.G, r1r2) |
||||
|
|
||||
|
a := bn128.G1.Affine(grsum1) |
||||
|
b := bn128.G1.Affine(grsum2) |
||||
|
assert.Equal(t, a, b) |
||||
|
assert.Equal(t, "0x2f978c0ab89ebaa576866706b14787f360c4d6c3869efe5a72f7c3651a72ff00", utils.BytesToHex(a[0].Bytes())) |
||||
|
assert.Equal(t, "0x12e4ba7f0edca8b4fa668fe153aebd908d322dc26ad964d4cd314795844b62b2", utils.BytesToHex(a[1].Bytes())) |
||||
|
``` |
||||
|
|
||||
|
- G2 operations |
||||
|
```go |
||||
|
bn128, err := NewBn128() |
||||
|
assert.Nil(t, err) |
||||
|
|
||||
|
r1 := big.NewInt(int64(33)) |
||||
|
r2 := big.NewInt(int64(44)) |
||||
|
|
||||
|
gr1 := bn128.G2.MulScalar(bn128.G2.G, bn128.Fq1.Copy(r1)) |
||||
|
gr2 := bn128.G2.MulScalar(bn128.G2.G, bn128.Fq1.Copy(r2)) |
||||
|
|
||||
|
grsum1 := bn128.G2.Add(gr1, gr2) |
||||
|
r1r2 := bn128.Fq1.Add(r1, r2) |
||||
|
grsum2 := bn128.G2.MulScalar(bn128.G2.G, r1r2) |
||||
|
|
||||
|
a := bn128.G2.Affine(grsum1) |
||||
|
b := bn128.G2.Affine(grsum2) |
||||
|
assert.Equal(t, a, b) |
||||
|
``` |
||||
@ -0,0 +1,407 @@ |
|||||
|
package bn128 |
||||
|
|
||||
|
import ( |
||||
|
"bytes" |
||||
|
"errors" |
||||
|
"math/big" |
||||
|
) |
||||
|
|
||||
|
type Bn128 struct { |
||||
|
Q *big.Int |
||||
|
Gg1 [2]*big.Int |
||||
|
Gg2 [2][2]*big.Int |
||||
|
NonResidueFq2 *big.Int |
||||
|
NonResidueFq6 [2]*big.Int |
||||
|
Fq1 Fq |
||||
|
Fq2 Fq2 |
||||
|
Fq6 Fq6 |
||||
|
Fq12 Fq12 |
||||
|
G1 G1 |
||||
|
G2 G2 |
||||
|
LoopCount *big.Int |
||||
|
LoopCountNeg bool |
||||
|
|
||||
|
TwoInv *big.Int |
||||
|
CoefB *big.Int |
||||
|
TwistCoefB [2]*big.Int |
||||
|
Twist [2]*big.Int |
||||
|
FrobeniusCoeffsC11 *big.Int |
||||
|
TwistMulByQX [2]*big.Int |
||||
|
TwistMulByQY [2]*big.Int |
||||
|
FinalExp *big.Int |
||||
|
} |
||||
|
|
||||
|
func NewBn128() (Bn128, error) { |
||||
|
var b Bn128 |
||||
|
q, ok := new(big.Int).SetString("21888242871839275222246405745257275088696311157297823662689037894645226208583", 10) // i
|
||||
|
if !ok { |
||||
|
return b, errors.New("err with q") |
||||
|
} |
||||
|
b.Q = q |
||||
|
|
||||
|
b.Gg1 = [2]*big.Int{ |
||||
|
big.NewInt(int64(1)), |
||||
|
big.NewInt(int64(2)), |
||||
|
} |
||||
|
|
||||
|
g2_00, ok := new(big.Int).SetString("10857046999023057135944570762232829481370756359578518086990519993285655852781", 10) |
||||
|
if !ok { |
||||
|
return b, errors.New("err with g2_00") |
||||
|
} |
||||
|
g2_01, ok := new(big.Int).SetString("11559732032986387107991004021392285783925812861821192530917403151452391805634", 10) |
||||
|
if !ok { |
||||
|
return b, errors.New("err with g2_00") |
||||
|
} |
||||
|
g2_10, ok := new(big.Int).SetString("8495653923123431417604973247489272438418190587263600148770280649306958101930", 10) |
||||
|
if !ok { |
||||
|
return b, errors.New("err with g2_00") |
||||
|
} |
||||
|
g2_11, ok := new(big.Int).SetString("4082367875863433681332203403145435568316851327593401208105741076214120093531", 10) |
||||
|
if !ok { |
||||
|
return b, errors.New("err with g2_00") |
||||
|
} |
||||
|
|
||||
|
b.Gg2 = [2][2]*big.Int{ |
||||
|
[2]*big.Int{ |
||||
|
g2_00, |
||||
|
g2_01, |
||||
|
}, |
||||
|
[2]*big.Int{ |
||||
|
g2_10, |
||||
|
g2_11, |
||||
|
}, |
||||
|
} |
||||
|
|
||||
|
b.Fq1 = NewFq(q) |
||||
|
b.NonResidueFq2, ok = new(big.Int).SetString("21888242871839275222246405745257275088696311157297823662689037894645226208582", 10) // i
|
||||
|
if !ok { |
||||
|
return b, errors.New("err with nonResidueFq2") |
||||
|
} |
||||
|
b.NonResidueFq6 = [2]*big.Int{ |
||||
|
big.NewInt(int64(9)), |
||||
|
big.NewInt(int64(1)), |
||||
|
} |
||||
|
|
||||
|
b.Fq2 = NewFq2(b.Fq1, b.NonResidueFq2) |
||||
|
b.Fq6 = NewFq6(b.Fq2, b.NonResidueFq6) |
||||
|
b.Fq12 = NewFq12(b.Fq6, b.Fq2, b.NonResidueFq6) |
||||
|
|
||||
|
b.G1 = NewG1(b.Fq1, b.Gg1) |
||||
|
b.G2 = NewG2(b.Fq2, b.Gg2) |
||||
|
|
||||
|
err := b.preparePairing() |
||||
|
if err != nil { |
||||
|
return b, err |
||||
|
} |
||||
|
|
||||
|
return b, nil |
||||
|
} |
||||
|
|
||||
|
func BigIsOdd(n *big.Int) bool { |
||||
|
one := big.NewInt(int64(1)) |
||||
|
and := new(big.Int).And(n, one) |
||||
|
return bytes.Equal(and.Bytes(), big.NewInt(int64(1)).Bytes()) |
||||
|
} |
||||
|
|
||||
|
func (bn128 *Bn128) preparePairing() error { |
||||
|
var ok bool |
||||
|
bn128.LoopCount, ok = new(big.Int).SetString("29793968203157093288", 10) |
||||
|
if !ok { |
||||
|
return errors.New("err with LoopCount from string") |
||||
|
} |
||||
|
|
||||
|
bn128.LoopCountNeg = false |
||||
|
|
||||
|
bn128.TwoInv = bn128.Fq1.Inverse(big.NewInt(int64(2))) |
||||
|
|
||||
|
bn128.CoefB = big.NewInt(int64(3)) |
||||
|
bn128.Twist = [2]*big.Int{ |
||||
|
big.NewInt(int64(9)), |
||||
|
big.NewInt(int64(1)), |
||||
|
} |
||||
|
bn128.TwistCoefB = bn128.Fq2.MulScalar(bn128.Fq2.Inverse(bn128.Twist), bn128.CoefB) |
||||
|
|
||||
|
bn128.FrobeniusCoeffsC11, ok = new(big.Int).SetString("21888242871839275222246405745257275088696311157297823662689037894645226208582", 10) |
||||
|
if !ok { |
||||
|
return errors.New("error parsing frobeniusCoeffsC11") |
||||
|
} |
||||
|
|
||||
|
a, ok := new(big.Int).SetString("21575463638280843010398324269430826099269044274347216827212613867836435027261", 10) |
||||
|
if !ok { |
||||
|
return errors.New("error parsing a") |
||||
|
} |
||||
|
b, ok := new(big.Int).SetString("10307601595873709700152284273816112264069230130616436755625194854815875713954", 10) |
||||
|
if !ok { |
||||
|
return errors.New("error parsing b") |
||||
|
} |
||||
|
bn128.TwistMulByQX = [2]*big.Int{ |
||||
|
a, |
||||
|
b, |
||||
|
} |
||||
|
|
||||
|
a, ok = new(big.Int).SetString("2821565182194536844548159561693502659359617185244120367078079554186484126554", 10) |
||||
|
if !ok { |
||||
|
return errors.New("error parsing a") |
||||
|
} |
||||
|
b, ok = new(big.Int).SetString("3505843767911556378687030309984248845540243509899259641013678093033130930403", 10) |
||||
|
if !ok { |
||||
|
return errors.New("error parsing b") |
||||
|
} |
||||
|
bn128.TwistMulByQY = [2]*big.Int{ |
||||
|
a, |
||||
|
b, |
||||
|
} |
||||
|
|
||||
|
bn128.FinalExp, ok = new(big.Int).SetString("552484233613224096312617126783173147097382103762957654188882734314196910839907541213974502761540629817009608548654680343627701153829446747810907373256841551006201639677726139946029199968412598804882391702273019083653272047566316584365559776493027495458238373902875937659943504873220554161550525926302303331747463515644711876653177129578303191095900909191624817826566688241804408081892785725967931714097716709526092261278071952560171111444072049229123565057483750161460024353346284167282452756217662335528813519139808291170539072125381230815729071544861602750936964829313608137325426383735122175229541155376346436093930287402089517426973178917569713384748081827255472576937471496195752727188261435633271238710131736096299798168852925540549342330775279877006784354801422249722573783561685179618816480037695005515426162362431072245638324744480", 10) |
||||
|
if !ok { |
||||
|
return errors.New("error parsing finalExp") |
||||
|
} |
||||
|
|
||||
|
return nil |
||||
|
|
||||
|
} |
||||
|
|
||||
|
func (bn128 Bn128) Pairing(p1 [3]*big.Int, p2 [3][2]*big.Int) ([2][3][2]*big.Int, error) { |
||||
|
pre1 := bn128.PreComputeG1(p1) |
||||
|
pre2, err := bn128.PreComputeG2(p2) |
||||
|
if err != nil { |
||||
|
return [2][3][2]*big.Int{}, err |
||||
|
} |
||||
|
|
||||
|
r1 := bn128.MillerLoop(pre1, pre2) |
||||
|
res := bn128.FinalExponentiation(r1) |
||||
|
return res, nil |
||||
|
} |
||||
|
|
||||
|
type AteG1Precomp struct { |
||||
|
Px *big.Int |
||||
|
Py *big.Int |
||||
|
} |
||||
|
|
||||
|
func (bn128 Bn128) PreComputeG1(p [3]*big.Int) AteG1Precomp { |
||||
|
pCopy := bn128.G1.Affine(p) |
||||
|
res := AteG1Precomp{ |
||||
|
Px: pCopy[0], |
||||
|
Py: pCopy[1], |
||||
|
} |
||||
|
return res |
||||
|
} |
||||
|
|
||||
|
type EllCoeffs struct { |
||||
|
Ell0 [2]*big.Int |
||||
|
EllVW [2]*big.Int |
||||
|
EllVV [2]*big.Int |
||||
|
} |
||||
|
type AteG2Precomp struct { |
||||
|
Qx [2]*big.Int |
||||
|
Qy [2]*big.Int |
||||
|
Coeffs []EllCoeffs |
||||
|
} |
||||
|
|
||||
|
func (bn128 Bn128) PreComputeG2(p [3][2]*big.Int) (AteG2Precomp, error) { |
||||
|
qCopy := bn128.G2.Affine(p) |
||||
|
res := AteG2Precomp{ |
||||
|
qCopy[0], |
||||
|
qCopy[1], |
||||
|
[]EllCoeffs{}, |
||||
|
} |
||||
|
r := [3][2]*big.Int{ |
||||
|
bn128.Fq2.Copy(qCopy[0]), |
||||
|
bn128.Fq2.Copy(qCopy[1]), |
||||
|
bn128.Fq2.One(), |
||||
|
} |
||||
|
var c EllCoeffs |
||||
|
for i := bn128.LoopCount.BitLen() - 2; i >= 0; i-- { |
||||
|
bit := bn128.LoopCount.Bit(i) |
||||
|
|
||||
|
c, r = bn128.DoublingStep(r) |
||||
|
res.Coeffs = append(res.Coeffs, c) |
||||
|
if bit == 1 { |
||||
|
c, r = bn128.MixedAdditionStep(qCopy, r) |
||||
|
res.Coeffs = append(res.Coeffs, c) |
||||
|
} |
||||
|
} |
||||
|
|
||||
|
q1 := bn128.G2.Affine(bn128.G2MulByQ(qCopy)) |
||||
|
if !bn128.Fq2.Equal(q1[2], bn128.Fq2.One()) { |
||||
|
return res, errors.New("q1[2] != Fq2.One") |
||||
|
} |
||||
|
q2 := bn128.G2.Affine(bn128.G2MulByQ(q1)) |
||||
|
if !bn128.Fq2.Equal(q2[2], bn128.Fq2.One()) { |
||||
|
return res, errors.New("q2[2] != Fq2.One") |
||||
|
} |
||||
|
|
||||
|
if bn128.LoopCountNeg { |
||||
|
r[1] = bn128.Fq2.Neg(r[1]) |
||||
|
} |
||||
|
q2[1] = bn128.Fq2.Neg(q2[1]) |
||||
|
|
||||
|
c, r = bn128.MixedAdditionStep(q1, r) |
||||
|
res.Coeffs = append(res.Coeffs, c) |
||||
|
|
||||
|
c, r = bn128.MixedAdditionStep(q2, r) |
||||
|
res.Coeffs = append(res.Coeffs, c) |
||||
|
|
||||
|
return res, nil |
||||
|
} |
||||
|
|
||||
|
func (bn128 Bn128) DoublingStep(current [3][2]*big.Int) (EllCoeffs, [3][2]*big.Int) { |
||||
|
x := current[0] |
||||
|
y := current[1] |
||||
|
z := current[2] |
||||
|
|
||||
|
a := bn128.Fq2.MulScalar(bn128.Fq2.Mul(x, y), bn128.TwoInv) |
||||
|
b := bn128.Fq2.Square(y) |
||||
|
c := bn128.Fq2.Square(z) |
||||
|
d := bn128.Fq2.Add(c, bn128.Fq2.Add(c, c)) |
||||
|
e := bn128.Fq2.Mul(bn128.TwistCoefB, d) |
||||
|
f := bn128.Fq2.Add(e, bn128.Fq2.Add(e, e)) |
||||
|
g := bn128.Fq2.MulScalar(bn128.Fq2.Add(b, f), bn128.TwoInv) |
||||
|
h := bn128.Fq2.Sub( |
||||
|
bn128.Fq2.Square(bn128.Fq2.Add(y, z)), |
||||
|
bn128.Fq2.Add(b, c)) |
||||
|
i := bn128.Fq2.Sub(e, b) |
||||
|
j := bn128.Fq2.Square(x) |
||||
|
eSqr := bn128.Fq2.Square(e) |
||||
|
current[0] = bn128.Fq2.Mul(a, bn128.Fq2.Sub(b, f)) |
||||
|
current[1] = bn128.Fq2.Sub(bn128.Fq2.Sub(bn128.Fq2.Square(g), eSqr), |
||||
|
bn128.Fq2.Add(eSqr, eSqr)) |
||||
|
current[2] = bn128.Fq2.Mul(b, h) |
||||
|
res := EllCoeffs{ |
||||
|
Ell0: bn128.Fq2.Mul(i, bn128.Twist), |
||||
|
EllVW: bn128.Fq2.Neg(h), |
||||
|
EllVV: bn128.Fq2.Add(j, bn128.Fq2.Add(j, j)), |
||||
|
} |
||||
|
|
||||
|
return res, current |
||||
|
} |
||||
|
|
||||
|
func (bn128 Bn128) MixedAdditionStep(base, current [3][2]*big.Int) (EllCoeffs, [3][2]*big.Int) { |
||||
|
x1 := current[0] |
||||
|
y1 := current[1] |
||||
|
z1 := current[2] |
||||
|
x2 := base[0] |
||||
|
y2 := base[1] |
||||
|
|
||||
|
d := bn128.Fq2.Sub(x1, bn128.Fq2.Mul(x2, z1)) |
||||
|
e := bn128.Fq2.Sub(y1, bn128.Fq2.Mul(y2, z1)) |
||||
|
f := bn128.Fq2.Square(d) |
||||
|
g := bn128.Fq2.Square(e) |
||||
|
h := bn128.Fq2.Mul(d, f) |
||||
|
i := bn128.Fq2.Mul(x1, f) |
||||
|
j := bn128.Fq2.Sub( |
||||
|
bn128.Fq2.Add(h, bn128.Fq2.Mul(z1, g)), |
||||
|
bn128.Fq2.Add(i, i)) |
||||
|
|
||||
|
current[0] = bn128.Fq2.Mul(d, j) |
||||
|
current[1] = bn128.Fq2.Sub( |
||||
|
bn128.Fq2.Mul(e, bn128.Fq2.Sub(i, j)), |
||||
|
bn128.Fq2.Mul(h, y1)) |
||||
|
current[2] = bn128.Fq2.Mul(z1, h) |
||||
|
|
||||
|
coef := EllCoeffs{ |
||||
|
Ell0: bn128.Fq2.Mul( |
||||
|
bn128.Twist, |
||||
|
bn128.Fq2.Sub( |
||||
|
bn128.Fq2.Mul(e, x2), |
||||
|
bn128.Fq2.Mul(d, y2))), |
||||
|
EllVW: d, |
||||
|
EllVV: bn128.Fq2.Neg(e), |
||||
|
} |
||||
|
return coef, current |
||||
|
} |
||||
|
func (bn128 Bn128) G2MulByQ(p [3][2]*big.Int) [3][2]*big.Int { |
||||
|
fmx := [2]*big.Int{ |
||||
|
p[0][0], |
||||
|
bn128.Fq1.Mul(p[0][1], bn128.Fq1.Copy(bn128.FrobeniusCoeffsC11)), |
||||
|
} |
||||
|
fmy := [2]*big.Int{ |
||||
|
p[1][0], |
||||
|
bn128.Fq1.Mul(p[1][1], bn128.Fq1.Copy(bn128.FrobeniusCoeffsC11)), |
||||
|
} |
||||
|
fmz := [2]*big.Int{ |
||||
|
p[2][0], |
||||
|
bn128.Fq1.Mul(p[2][1], bn128.Fq1.Copy(bn128.FrobeniusCoeffsC11)), |
||||
|
} |
||||
|
|
||||
|
return [3][2]*big.Int{ |
||||
|
bn128.Fq2.Mul(bn128.TwistMulByQX, fmx), |
||||
|
bn128.Fq2.Mul(bn128.TwistMulByQY, fmy), |
||||
|
fmz, |
||||
|
} |
||||
|
} |
||||
|
|
||||
|
func (bn128 Bn128) MillerLoop(pre1 AteG1Precomp, pre2 AteG2Precomp) [2][3][2]*big.Int { |
||||
|
// https://cryptojedi.org/papers/dclxvi-20100714.pdf
|
||||
|
// https://eprint.iacr.org/2008/096.pdf
|
||||
|
|
||||
|
idx := 0 |
||||
|
var c EllCoeffs |
||||
|
f := bn128.Fq12.One() |
||||
|
|
||||
|
for i := bn128.LoopCount.BitLen() - 2; i >= 0; i-- { |
||||
|
bit := bn128.LoopCount.Bit(i) |
||||
|
|
||||
|
c = pre2.Coeffs[idx] |
||||
|
idx++ |
||||
|
f = bn128.Fq12.Square(f) |
||||
|
|
||||
|
f = bn128.MulBy024(f, |
||||
|
c.Ell0, |
||||
|
bn128.Fq2.MulScalar(c.EllVW, pre1.Py), |
||||
|
bn128.Fq2.MulScalar(c.EllVV, pre1.Px)) |
||||
|
|
||||
|
if bit == 1 { |
||||
|
c = pre2.Coeffs[idx] |
||||
|
idx++ |
||||
|
f = bn128.MulBy024( |
||||
|
f, |
||||
|
c.Ell0, |
||||
|
bn128.Fq2.MulScalar(c.EllVW, pre1.Py), |
||||
|
bn128.Fq2.MulScalar(c.EllVV, pre1.Px)) |
||||
|
} |
||||
|
} |
||||
|
if bn128.LoopCountNeg { |
||||
|
f = bn128.Fq12.Inverse(f) |
||||
|
} |
||||
|
|
||||
|
c = pre2.Coeffs[idx] |
||||
|
idx++ |
||||
|
f = bn128.MulBy024( |
||||
|
f, |
||||
|
c.Ell0, |
||||
|
bn128.Fq2.MulScalar(c.EllVW, pre1.Py), |
||||
|
bn128.Fq2.MulScalar(c.EllVV, pre1.Px)) |
||||
|
|
||||
|
c = pre2.Coeffs[idx] |
||||
|
idx++ |
||||
|
|
||||
|
f = bn128.MulBy024( |
||||
|
f, |
||||
|
c.Ell0, |
||||
|
bn128.Fq2.MulScalar(c.EllVW, pre1.Py), |
||||
|
bn128.Fq2.MulScalar(c.EllVV, pre1.Px)) |
||||
|
|
||||
|
return f |
||||
|
} |
||||
|
|
||||
|
func (bn128 Bn128) MulBy024(a [2][3][2]*big.Int, ell0, ellVW, ellVV [2]*big.Int) [2][3][2]*big.Int { |
||||
|
b := [2][3][2]*big.Int{ |
||||
|
[3][2]*big.Int{ |
||||
|
ell0, |
||||
|
bn128.Fq2.Zero(), |
||||
|
ellVV, |
||||
|
}, |
||||
|
[3][2]*big.Int{ |
||||
|
bn128.Fq2.Zero(), |
||||
|
ellVW, |
||||
|
bn128.Fq2.Zero(), |
||||
|
}, |
||||
|
} |
||||
|
return bn128.Fq12.Mul(a, b) |
||||
|
} |
||||
|
|
||||
|
func (bn128 Bn128) FinalExponentiation(r [2][3][2]*big.Int) [2][3][2]*big.Int { |
||||
|
res := bn128.Fq12.Exp(r, bn128.FinalExp) |
||||
|
return res |
||||
|
} |
||||
@ -0,0 +1,66 @@ |
|||||
|
package bn128 |
||||
|
|
||||
|
import ( |
||||
|
"math/big" |
||||
|
"testing" |
||||
|
|
||||
|
"github.com/stretchr/testify/assert" |
||||
|
) |
||||
|
|
||||
|
func TestBN128(t *testing.T) { |
||||
|
bn128, err := NewBn128() |
||||
|
assert.Nil(t, err) |
||||
|
|
||||
|
big40 := big.NewInt(int64(40)) |
||||
|
big75 := big.NewInt(int64(75)) |
||||
|
|
||||
|
g1a := bn128.G1.MulScalar(bn128.G1.G, bn128.Fq1.Copy(big40)) |
||||
|
g2a := bn128.G2.MulScalar(bn128.G2.G, bn128.Fq1.Copy(big75)) |
||||
|
|
||||
|
g1b := bn128.G1.MulScalar(bn128.G1.G, bn128.Fq1.Copy(big75)) |
||||
|
g2b := bn128.G2.MulScalar(bn128.G2.G, bn128.Fq1.Copy(big40)) |
||||
|
|
||||
|
pre1a := bn128.PreComputeG1(g1a) |
||||
|
pre2a, err := bn128.PreComputeG2(g2a) |
||||
|
assert.Nil(t, err) |
||||
|
pre1b := bn128.PreComputeG1(g1b) |
||||
|
pre2b, err := bn128.PreComputeG2(g2b) |
||||
|
assert.Nil(t, err) |
||||
|
|
||||
|
r1 := bn128.MillerLoop(pre1a, pre2a) |
||||
|
r2 := bn128.MillerLoop(pre1b, pre2b) |
||||
|
|
||||
|
rbe := bn128.Fq12.Mul(r1, bn128.Fq12.Inverse(r2)) |
||||
|
|
||||
|
res := bn128.FinalExponentiation(rbe) |
||||
|
|
||||
|
a := bn128.Fq12.Affine(res) |
||||
|
b := bn128.Fq12.Affine(bn128.Fq12.One()) |
||||
|
|
||||
|
assert.True(t, bn128.Fq12.Equal(a, b)) |
||||
|
assert.True(t, bn128.Fq12.Equal(res, bn128.Fq12.One())) |
||||
|
} |
||||
|
|
||||
|
func TestBN128Pairing(t *testing.T) { |
||||
|
bn128, err := NewBn128() |
||||
|
assert.Nil(t, err) |
||||
|
|
||||
|
big25 := big.NewInt(int64(25)) |
||||
|
big30 := big.NewInt(int64(30)) |
||||
|
|
||||
|
g1a := bn128.G1.MulScalar(bn128.G1.G, big25) |
||||
|
g2a := bn128.G2.MulScalar(bn128.G2.G, big30) |
||||
|
|
||||
|
g1b := bn128.G1.MulScalar(bn128.G1.G, big30) |
||||
|
g2b := bn128.G2.MulScalar(bn128.G2.G, big25) |
||||
|
|
||||
|
pA, err := bn128.Pairing(g1a, g2a) |
||||
|
assert.Nil(t, err) |
||||
|
pB, err := bn128.Pairing(g1b, g2b) |
||||
|
assert.Nil(t, err) |
||||
|
|
||||
|
assert.True(t, bn128.Fq12.Equal(pA, pB)) |
||||
|
|
||||
|
assert.Equal(t, pA[0][0][0].String(), "73680848340331011700282047627232219336104151861349893575958589557226556635706") |
||||
|
assert.Equal(t, bn128.Fq12.Affine(pA)[0][0][0].String(), "8016119724813186033542830391460394070015218389456422587891475873290878009957") |
||||
|
} |
||||
@ -0,0 +1,129 @@ |
|||||
|
package bn128 |
||||
|
|
||||
|
import ( |
||||
|
"bytes" |
||||
|
"math/big" |
||||
|
) |
||||
|
|
||||
|
// Fq is the Z field over modulus Q
|
||||
|
type Fq struct { |
||||
|
Q *big.Int // Q
|
||||
|
} |
||||
|
|
||||
|
// NewFq generates a new Fq
|
||||
|
func NewFq(q *big.Int) Fq { |
||||
|
return Fq{ |
||||
|
q, |
||||
|
} |
||||
|
} |
||||
|
|
||||
|
// Zero returns a Zero value on the Fq
|
||||
|
func (fq Fq) Zero() *big.Int { |
||||
|
return big.NewInt(int64(0)) |
||||
|
} |
||||
|
|
||||
|
// One returns a One value on the Fq
|
||||
|
func (fq Fq) One() *big.Int { |
||||
|
return big.NewInt(int64(1)) |
||||
|
} |
||||
|
|
||||
|
// Add performs an addition on the Fq
|
||||
|
func (fq Fq) Add(a, b *big.Int) *big.Int { |
||||
|
r := new(big.Int).Add(a, b) |
||||
|
// return new(big.Int).Mod(r, fq.Q)
|
||||
|
return r |
||||
|
} |
||||
|
|
||||
|
// Double performs a doubling on the Fq
|
||||
|
func (fq Fq) Double(a *big.Int) *big.Int { |
||||
|
r := new(big.Int).Add(a, a) |
||||
|
// return new(big.Int).Mod(r, fq.Q)
|
||||
|
return r |
||||
|
} |
||||
|
|
||||
|
// Sub performs a subtraction on the Fq
|
||||
|
func (fq Fq) Sub(a, b *big.Int) *big.Int { |
||||
|
r := new(big.Int).Sub(a, b) |
||||
|
// return new(big.Int).Mod(r, fq.Q)
|
||||
|
return r |
||||
|
} |
||||
|
|
||||
|
// Neg performs a negation on the Fq
|
||||
|
func (fq Fq) Neg(a *big.Int) *big.Int { |
||||
|
m := new(big.Int).Neg(a) |
||||
|
// return new(big.Int).Mod(m, fq.Q)
|
||||
|
return m |
||||
|
} |
||||
|
|
||||
|
// Mul performs a multiplication on the Fq
|
||||
|
func (fq Fq) Mul(a, b *big.Int) *big.Int { |
||||
|
m := new(big.Int).Mul(a, b) |
||||
|
return new(big.Int).Mod(m, fq.Q) |
||||
|
// return m
|
||||
|
} |
||||
|
|
||||
|
func (fq Fq) MulScalar(base, e *big.Int) *big.Int { |
||||
|
return fq.Mul(base, e) |
||||
|
} |
||||
|
|
||||
|
// Inverse returns the inverse on the Fq
|
||||
|
func (fq Fq) Inverse(a *big.Int) *big.Int { |
||||
|
return new(big.Int).ModInverse(a, fq.Q) |
||||
|
// q := bigCopy(fq.Q)
|
||||
|
// t := big.NewInt(int64(0))
|
||||
|
// r := fq.Q
|
||||
|
// newt := big.NewInt(int64(0))
|
||||
|
// newr := fq.Affine(a)
|
||||
|
// for !bytes.Equal(newr.Bytes(), big.NewInt(int64(0)).Bytes()) {
|
||||
|
// q := new(big.Int).Div(bigCopy(r), bigCopy(newr))
|
||||
|
//
|
||||
|
// t = bigCopy(newt)
|
||||
|
// newt = fq.Sub(t, fq.Mul(q, newt))
|
||||
|
//
|
||||
|
// r = bigCopy(newr)
|
||||
|
// newr = fq.Sub(r, fq.Mul(q, newr))
|
||||
|
// }
|
||||
|
// if t.Cmp(big.NewInt(0)) == -1 { // t< 0
|
||||
|
// t = fq.Add(t, q)
|
||||
|
// }
|
||||
|
// return t
|
||||
|
} |
||||
|
|
||||
|
// Square performs a square operation on the Fq
|
||||
|
func (fq Fq) Square(a *big.Int) *big.Int { |
||||
|
m := new(big.Int).Mul(a, a) |
||||
|
return new(big.Int).Mod(m, fq.Q) |
||||
|
} |
||||
|
|
||||
|
func (fq Fq) IsZero(a *big.Int) bool { |
||||
|
return bytes.Equal(a.Bytes(), fq.Zero().Bytes()) |
||||
|
} |
||||
|
|
||||
|
func (fq Fq) Copy(a *big.Int) *big.Int { |
||||
|
return new(big.Int).SetBytes(a.Bytes()) |
||||
|
} |
||||
|
|
||||
|
func (fq Fq) Affine(a *big.Int) *big.Int { |
||||
|
nq := fq.Neg(fq.Q) |
||||
|
|
||||
|
aux := a |
||||
|
if aux.Cmp(big.NewInt(int64(0))) == -1 { // negative value
|
||||
|
if aux.Cmp(nq) != 1 { // aux less or equal nq
|
||||
|
aux = new(big.Int).Mod(aux, fq.Q) |
||||
|
} |
||||
|
if aux.Cmp(big.NewInt(int64(0))) == -1 { // negative value
|
||||
|
aux = new(big.Int).Add(aux, fq.Q) |
||||
|
} |
||||
|
} else { |
||||
|
if aux.Cmp(fq.Q) != -1 { // aux greater or equal nq
|
||||
|
aux = new(big.Int).Mod(aux, fq.Q) |
||||
|
} |
||||
|
} |
||||
|
return aux |
||||
|
} |
||||
|
|
||||
|
func (fq Fq) Equal(a, b *big.Int) bool { |
||||
|
aAff := fq.Affine(a) |
||||
|
bAff := fq.Affine(b) |
||||
|
return bytes.Equal(aAff.Bytes(), bAff.Bytes()) |
||||
|
} |
||||
@ -0,0 +1,161 @@ |
|||||
|
package bn128 |
||||
|
|
||||
|
import ( |
||||
|
"bytes" |
||||
|
"math/big" |
||||
|
) |
||||
|
|
||||
|
// Fq12 uses the same algorithms than Fq2, but with [2][3][2]*big.Int data structure
|
||||
|
|
||||
|
// Fq12 is Field 12
|
||||
|
type Fq12 struct { |
||||
|
F Fq6 |
||||
|
Fq2 Fq2 |
||||
|
NonResidue [2]*big.Int |
||||
|
} |
||||
|
|
||||
|
// NewFq12 generates a new Fq12
|
||||
|
func NewFq12(f Fq6, fq2 Fq2, nonResidue [2]*big.Int) Fq12 { |
||||
|
fq12 := Fq12{ |
||||
|
f, |
||||
|
fq2, |
||||
|
nonResidue, |
||||
|
} |
||||
|
return fq12 |
||||
|
} |
||||
|
|
||||
|
// Zero returns a Zero value on the Fq12
|
||||
|
func (fq12 Fq12) Zero() [2][3][2]*big.Int { |
||||
|
return [2][3][2]*big.Int{fq12.F.Zero(), fq12.F.Zero()} |
||||
|
} |
||||
|
|
||||
|
// One returns a One value on the Fq12
|
||||
|
func (fq12 Fq12) One() [2][3][2]*big.Int { |
||||
|
return [2][3][2]*big.Int{fq12.F.One(), fq12.F.Zero()} |
||||
|
} |
||||
|
|
||||
|
func (fq12 Fq12) mulByNonResidue(a [3][2]*big.Int) [3][2]*big.Int { |
||||
|
return [3][2]*big.Int{ |
||||
|
fq12.Fq2.Mul(fq12.NonResidue, a[2]), |
||||
|
a[0], |
||||
|
a[1], |
||||
|
} |
||||
|
} |
||||
|
|
||||
|
// Add performs an addition on the Fq12
|
||||
|
func (fq12 Fq12) Add(a, b [2][3][2]*big.Int) [2][3][2]*big.Int { |
||||
|
return [2][3][2]*big.Int{ |
||||
|
fq12.F.Add(a[0], b[0]), |
||||
|
fq12.F.Add(a[1], b[1]), |
||||
|
} |
||||
|
} |
||||
|
|
||||
|
// Double performs a doubling on the Fq12
|
||||
|
func (fq12 Fq12) Double(a [2][3][2]*big.Int) [2][3][2]*big.Int { |
||||
|
return fq12.Add(a, a) |
||||
|
} |
||||
|
|
||||
|
// Sub performs a subtraction on the Fq12
|
||||
|
func (fq12 Fq12) Sub(a, b [2][3][2]*big.Int) [2][3][2]*big.Int { |
||||
|
return [2][3][2]*big.Int{ |
||||
|
fq12.F.Sub(a[0], b[0]), |
||||
|
fq12.F.Sub(a[1], b[1]), |
||||
|
} |
||||
|
} |
||||
|
|
||||
|
// Neg performs a negation on the Fq12
|
||||
|
func (fq12 Fq12) Neg(a [2][3][2]*big.Int) [2][3][2]*big.Int { |
||||
|
return fq12.Sub(fq12.Zero(), a) |
||||
|
} |
||||
|
|
||||
|
// Mul performs a multiplication on the Fq12
|
||||
|
func (fq12 Fq12) Mul(a, b [2][3][2]*big.Int) [2][3][2]*big.Int { |
||||
|
// Multiplication and Squaring on Pairing-Friendly .pdf; Section 3 (Karatsuba)
|
||||
|
v0 := fq12.F.Mul(a[0], b[0]) |
||||
|
v1 := fq12.F.Mul(a[1], b[1]) |
||||
|
return [2][3][2]*big.Int{ |
||||
|
fq12.F.Add(v0, fq12.mulByNonResidue(v1)), |
||||
|
fq12.F.Sub( |
||||
|
fq12.F.Mul( |
||||
|
fq12.F.Add(a[0], a[1]), |
||||
|
fq12.F.Add(b[0], b[1])), |
||||
|
fq12.F.Add(v0, v1)), |
||||
|
} |
||||
|
} |
||||
|
|
||||
|
func (fq12 Fq12) MulScalar(base [2][3][2]*big.Int, e *big.Int) [2][3][2]*big.Int { |
||||
|
// for more possible implementations see g2.go file, at the function g2.MulScalar()
|
||||
|
|
||||
|
res := fq12.Zero() |
||||
|
rem := e |
||||
|
exp := base |
||||
|
|
||||
|
for !bytes.Equal(rem.Bytes(), big.NewInt(int64(0)).Bytes()) { |
||||
|
// if rem % 2 == 1
|
||||
|
if bytes.Equal(new(big.Int).Rem(rem, big.NewInt(int64(2))).Bytes(), big.NewInt(int64(1)).Bytes()) { |
||||
|
res = fq12.Add(res, exp) |
||||
|
} |
||||
|
exp = fq12.Double(exp) |
||||
|
rem = rem.Rsh(rem, 1) // rem = rem >> 1
|
||||
|
} |
||||
|
return res |
||||
|
} |
||||
|
|
||||
|
// Inverse returns the inverse on the Fq12
|
||||
|
func (fq12 Fq12) Inverse(a [2][3][2]*big.Int) [2][3][2]*big.Int { |
||||
|
t0 := fq12.F.Square(a[0]) |
||||
|
t1 := fq12.F.Square(a[1]) |
||||
|
t2 := fq12.F.Sub(t0, fq12.mulByNonResidue(t1)) |
||||
|
t3 := fq12.F.Inverse(t2) |
||||
|
return [2][3][2]*big.Int{ |
||||
|
fq12.F.Mul(a[0], t3), |
||||
|
fq12.F.Neg(fq12.F.Mul(a[1], t3)), |
||||
|
} |
||||
|
} |
||||
|
|
||||
|
// Div performs a division on the Fq12
|
||||
|
func (fq12 Fq12) Div(a, b [2][3][2]*big.Int) [2][3][2]*big.Int { |
||||
|
return fq12.Mul(a, fq12.Inverse(b)) |
||||
|
} |
||||
|
|
||||
|
// Square performs a square operation on the Fq12
|
||||
|
func (fq12 Fq12) Square(a [2][3][2]*big.Int) [2][3][2]*big.Int { |
||||
|
ab := fq12.F.Mul(a[0], a[1]) |
||||
|
|
||||
|
return [2][3][2]*big.Int{ |
||||
|
fq12.F.Sub( |
||||
|
fq12.F.Mul( |
||||
|
fq12.F.Add(a[0], a[1]), |
||||
|
fq12.F.Add( |
||||
|
a[0], |
||||
|
fq12.mulByNonResidue(a[1]))), |
||||
|
fq12.F.Add( |
||||
|
ab, |
||||
|
fq12.mulByNonResidue(ab))), |
||||
|
fq12.F.Add(ab, ab), |
||||
|
} |
||||
|
} |
||||
|
|
||||
|
func (fq12 Fq12) Exp(base [2][3][2]*big.Int, e *big.Int) [2][3][2]*big.Int { |
||||
|
res := fq12.One() |
||||
|
rem := fq12.Fq2.F.Copy(e) |
||||
|
exp := base |
||||
|
|
||||
|
for !bytes.Equal(rem.Bytes(), big.NewInt(int64(0)).Bytes()) { |
||||
|
if BigIsOdd(rem) { |
||||
|
res = fq12.Mul(res, exp) |
||||
|
} |
||||
|
exp = fq12.Square(exp) |
||||
|
rem = new(big.Int).Rsh(rem, 1) |
||||
|
} |
||||
|
return res |
||||
|
} |
||||
|
func (fq12 Fq12) Affine(a [2][3][2]*big.Int) [2][3][2]*big.Int { |
||||
|
return [2][3][2]*big.Int{ |
||||
|
fq12.F.Affine(a[0]), |
||||
|
fq12.F.Affine(a[1]), |
||||
|
} |
||||
|
} |
||||
|
func (fq12 Fq12) Equal(a, b [2][3][2]*big.Int) bool { |
||||
|
return fq12.F.Equal(a[0], b[0]) && fq12.F.Equal(a[1], b[1]) |
||||
|
} |
||||
@ -0,0 +1,154 @@ |
|||||
|
package bn128 |
||||
|
|
||||
|
import ( |
||||
|
"math/big" |
||||
|
) |
||||
|
|
||||
|
// Fq2 is Field 2
|
||||
|
type Fq2 struct { |
||||
|
F Fq |
||||
|
NonResidue *big.Int |
||||
|
} |
||||
|
|
||||
|
// NewFq2 generates a new Fq2
|
||||
|
func NewFq2(f Fq, nonResidue *big.Int) Fq2 { |
||||
|
fq2 := Fq2{ |
||||
|
f, |
||||
|
nonResidue, |
||||
|
} |
||||
|
return fq2 |
||||
|
} |
||||
|
|
||||
|
// Zero returns a Zero value on the Fq2
|
||||
|
func (fq2 Fq2) Zero() [2]*big.Int { |
||||
|
return [2]*big.Int{fq2.F.Zero(), fq2.F.Zero()} |
||||
|
} |
||||
|
|
||||
|
// One returns a One value on the Fq2
|
||||
|
func (fq2 Fq2) One() [2]*big.Int { |
||||
|
return [2]*big.Int{fq2.F.One(), fq2.F.Zero()} |
||||
|
} |
||||
|
|
||||
|
func (fq2 Fq2) mulByNonResidue(a *big.Int) *big.Int { |
||||
|
return fq2.F.Mul(fq2.NonResidue, a) |
||||
|
} |
||||
|
|
||||
|
// Add performs an addition on the Fq2
|
||||
|
func (fq2 Fq2) Add(a, b [2]*big.Int) [2]*big.Int { |
||||
|
return [2]*big.Int{ |
||||
|
fq2.F.Add(a[0], b[0]), |
||||
|
fq2.F.Add(a[1], b[1]), |
||||
|
} |
||||
|
} |
||||
|
|
||||
|
// Double performs a doubling on the Fq2
|
||||
|
func (fq2 Fq2) Double(a [2]*big.Int) [2]*big.Int { |
||||
|
return fq2.Add(a, a) |
||||
|
} |
||||
|
|
||||
|
// Sub performs a subtraction on the Fq2
|
||||
|
func (fq2 Fq2) Sub(a, b [2]*big.Int) [2]*big.Int { |
||||
|
return [2]*big.Int{ |
||||
|
fq2.F.Sub(a[0], b[0]), |
||||
|
fq2.F.Sub(a[1], b[1]), |
||||
|
} |
||||
|
} |
||||
|
|
||||
|
// Neg performs a negation on the Fq2
|
||||
|
func (fq2 Fq2) Neg(a [2]*big.Int) [2]*big.Int { |
||||
|
return fq2.Sub(fq2.Zero(), a) |
||||
|
} |
||||
|
|
||||
|
// Mul performs a multiplication on the Fq2
|
||||
|
func (fq2 Fq2) Mul(a, b [2]*big.Int) [2]*big.Int { |
||||
|
// Multiplication and Squaring on Pairing-Friendly.pdf; Section 3 (Karatsuba)
|
||||
|
// https://pdfs.semanticscholar.org/3e01/de88d7428076b2547b60072088507d881bf1.pdf
|
||||
|
v0 := fq2.F.Mul(a[0], b[0]) |
||||
|
v1 := fq2.F.Mul(a[1], b[1]) |
||||
|
return [2]*big.Int{ |
||||
|
fq2.F.Add(v0, fq2.mulByNonResidue(v1)), |
||||
|
fq2.F.Sub( |
||||
|
fq2.F.Mul( |
||||
|
fq2.F.Add(a[0], a[1]), |
||||
|
fq2.F.Add(b[0], b[1])), |
||||
|
fq2.F.Add(v0, v1)), |
||||
|
} |
||||
|
} |
||||
|
|
||||
|
func (fq2 Fq2) MulScalar(p [2]*big.Int, e *big.Int) [2]*big.Int { |
||||
|
// for more possible implementations see g2.go file, at the function g2.MulScalar()
|
||||
|
|
||||
|
q := fq2.Zero() |
||||
|
d := fq2.F.Copy(e) |
||||
|
r := p |
||||
|
|
||||
|
foundone := false |
||||
|
for i := d.BitLen(); i >= 0; i-- { |
||||
|
if foundone { |
||||
|
q = fq2.Double(q) |
||||
|
} |
||||
|
if d.Bit(i) == 1 { |
||||
|
foundone = true |
||||
|
q = fq2.Add(q, r) |
||||
|
} |
||||
|
} |
||||
|
return q |
||||
|
} |
||||
|
|
||||
|
// Inverse returns the inverse on the Fq2
|
||||
|
func (fq2 Fq2) Inverse(a [2]*big.Int) [2]*big.Int { |
||||
|
// High-Speed Software Implementation of the Optimal Ate Pairing over Barreto–Naehrig Curves .pdf
|
||||
|
// https://eprint.iacr.org/2010/354.pdf , algorithm 8
|
||||
|
t0 := fq2.F.Square(a[0]) |
||||
|
t1 := fq2.F.Square(a[1]) |
||||
|
t2 := fq2.F.Sub(t0, fq2.mulByNonResidue(t1)) |
||||
|
t3 := fq2.F.Inverse(t2) |
||||
|
return [2]*big.Int{ |
||||
|
fq2.F.Mul(a[0], t3), |
||||
|
fq2.F.Neg(fq2.F.Mul(a[1], t3)), |
||||
|
} |
||||
|
} |
||||
|
|
||||
|
// Div performs a division on the Fq2
|
||||
|
func (fq2 Fq2) Div(a, b [2]*big.Int) [2]*big.Int { |
||||
|
return fq2.Mul(a, fq2.Inverse(b)) |
||||
|
} |
||||
|
|
||||
|
// Square performs a square operation on the Fq2
|
||||
|
func (fq2 Fq2) Square(a [2]*big.Int) [2]*big.Int { |
||||
|
// https://pdfs.semanticscholar.org/3e01/de88d7428076b2547b60072088507d881bf1.pdf , complex squaring
|
||||
|
ab := fq2.F.Mul(a[0], a[1]) |
||||
|
return [2]*big.Int{ |
||||
|
fq2.F.Sub( |
||||
|
fq2.F.Mul( |
||||
|
fq2.F.Add(a[0], a[1]), |
||||
|
fq2.F.Add( |
||||
|
a[0], |
||||
|
fq2.mulByNonResidue(a[1]))), |
||||
|
fq2.F.Add( |
||||
|
ab, |
||||
|
fq2.mulByNonResidue(ab))), |
||||
|
fq2.F.Add(ab, ab), |
||||
|
} |
||||
|
} |
||||
|
|
||||
|
func (fq2 Fq2) IsZero(a [2]*big.Int) bool { |
||||
|
return fq2.F.IsZero(a[0]) && fq2.F.IsZero(a[1]) |
||||
|
} |
||||
|
|
||||
|
func (fq2 Fq2) Affine(a [2]*big.Int) [2]*big.Int { |
||||
|
return [2]*big.Int{ |
||||
|
fq2.F.Affine(a[0]), |
||||
|
fq2.F.Affine(a[1]), |
||||
|
} |
||||
|
} |
||||
|
func (fq2 Fq2) Equal(a, b [2]*big.Int) bool { |
||||
|
return fq2.F.Equal(a[0], b[0]) && fq2.F.Equal(a[1], b[1]) |
||||
|
} |
||||
|
|
||||
|
func (fq2 Fq2) Copy(a [2]*big.Int) [2]*big.Int { |
||||
|
return [2]*big.Int{ |
||||
|
fq2.F.Copy(a[0]), |
||||
|
fq2.F.Copy(a[1]), |
||||
|
} |
||||
|
} |
||||
@ -0,0 +1,192 @@ |
|||||
|
package bn128 |
||||
|
|
||||
|
import ( |
||||
|
"bytes" |
||||
|
"math/big" |
||||
|
) |
||||
|
|
||||
|
// Fq6 is Field 6
|
||||
|
type Fq6 struct { |
||||
|
F Fq2 |
||||
|
NonResidue [2]*big.Int |
||||
|
} |
||||
|
|
||||
|
// NewFq6 generates a new Fq6
|
||||
|
func NewFq6(f Fq2, nonResidue [2]*big.Int) Fq6 { |
||||
|
fq6 := Fq6{ |
||||
|
f, |
||||
|
nonResidue, |
||||
|
} |
||||
|
return fq6 |
||||
|
} |
||||
|
|
||||
|
// Zero returns a Zero value on the Fq6
|
||||
|
func (fq6 Fq6) Zero() [3][2]*big.Int { |
||||
|
return [3][2]*big.Int{fq6.F.Zero(), fq6.F.Zero(), fq6.F.Zero()} |
||||
|
} |
||||
|
|
||||
|
// One returns a One value on the Fq6
|
||||
|
func (fq6 Fq6) One() [3][2]*big.Int { |
||||
|
return [3][2]*big.Int{fq6.F.One(), fq6.F.Zero(), fq6.F.Zero()} |
||||
|
} |
||||
|
|
||||
|
func (fq6 Fq6) mulByNonResidue(a [2]*big.Int) [2]*big.Int { |
||||
|
return fq6.F.Mul(fq6.NonResidue, a) |
||||
|
} |
||||
|
|
||||
|
// Add performs an addition on the Fq6
|
||||
|
func (fq6 Fq6) Add(a, b [3][2]*big.Int) [3][2]*big.Int { |
||||
|
return [3][2]*big.Int{ |
||||
|
fq6.F.Add(a[0], b[0]), |
||||
|
fq6.F.Add(a[1], b[1]), |
||||
|
fq6.F.Add(a[2], b[2]), |
||||
|
} |
||||
|
} |
||||
|
|
||||
|
func (fq6 Fq6) Double(a [3][2]*big.Int) [3][2]*big.Int { |
||||
|
return fq6.Add(a, a) |
||||
|
} |
||||
|
|
||||
|
// Sub performs a subtraction on the Fq6
|
||||
|
func (fq6 Fq6) Sub(a, b [3][2]*big.Int) [3][2]*big.Int { |
||||
|
return [3][2]*big.Int{ |
||||
|
fq6.F.Sub(a[0], b[0]), |
||||
|
fq6.F.Sub(a[1], b[1]), |
||||
|
fq6.F.Sub(a[2], b[2]), |
||||
|
} |
||||
|
} |
||||
|
|
||||
|
// Neg performs a negation on the Fq6
|
||||
|
func (fq6 Fq6) Neg(a [3][2]*big.Int) [3][2]*big.Int { |
||||
|
return fq6.Sub(fq6.Zero(), a) |
||||
|
} |
||||
|
|
||||
|
// Mul performs a multiplication on the Fq6
|
||||
|
func (fq6 Fq6) Mul(a, b [3][2]*big.Int) [3][2]*big.Int { |
||||
|
v0 := fq6.F.Mul(a[0], b[0]) |
||||
|
v1 := fq6.F.Mul(a[1], b[1]) |
||||
|
v2 := fq6.F.Mul(a[2], b[2]) |
||||
|
return [3][2]*big.Int{ |
||||
|
fq6.F.Add( |
||||
|
v0, |
||||
|
fq6.mulByNonResidue( |
||||
|
fq6.F.Sub( |
||||
|
fq6.F.Mul( |
||||
|
fq6.F.Add(a[1], a[2]), |
||||
|
fq6.F.Add(b[1], b[2])), |
||||
|
fq6.F.Add(v1, v2)))), |
||||
|
|
||||
|
fq6.F.Add( |
||||
|
fq6.F.Sub( |
||||
|
fq6.F.Mul( |
||||
|
fq6.F.Add(a[0], a[1]), |
||||
|
fq6.F.Add(b[0], b[1])), |
||||
|
fq6.F.Add(v0, v1)), |
||||
|
fq6.mulByNonResidue(v2)), |
||||
|
|
||||
|
fq6.F.Add( |
||||
|
fq6.F.Sub( |
||||
|
fq6.F.Mul( |
||||
|
fq6.F.Add(a[0], a[2]), |
||||
|
fq6.F.Add(b[0], b[2])), |
||||
|
fq6.F.Add(v0, v2)), |
||||
|
v1), |
||||
|
} |
||||
|
} |
||||
|
|
||||
|
func (fq6 Fq6) MulScalar(base [3][2]*big.Int, e *big.Int) [3][2]*big.Int { |
||||
|
// for more possible implementations see g2.go file, at the function g2.MulScalar()
|
||||
|
|
||||
|
res := fq6.Zero() |
||||
|
rem := e |
||||
|
exp := base |
||||
|
|
||||
|
for !bytes.Equal(rem.Bytes(), big.NewInt(int64(0)).Bytes()) { |
||||
|
// if rem % 2 == 1
|
||||
|
if bytes.Equal(new(big.Int).Rem(rem, big.NewInt(int64(2))).Bytes(), big.NewInt(int64(1)).Bytes()) { |
||||
|
res = fq6.Add(res, exp) |
||||
|
} |
||||
|
exp = fq6.Double(exp) |
||||
|
rem = rem.Rsh(rem, 1) // rem = rem >> 1
|
||||
|
} |
||||
|
return res |
||||
|
} |
||||
|
|
||||
|
// Inverse returns the inverse on the Fq6
|
||||
|
func (fq6 Fq6) Inverse(a [3][2]*big.Int) [3][2]*big.Int { |
||||
|
t0 := fq6.F.Square(a[0]) |
||||
|
t1 := fq6.F.Square(a[1]) |
||||
|
t2 := fq6.F.Square(a[2]) |
||||
|
t3 := fq6.F.Mul(a[0], a[1]) |
||||
|
t4 := fq6.F.Mul(a[0], a[2]) |
||||
|
t5 := fq6.F.Mul(a[1], a[2]) |
||||
|
|
||||
|
c0 := fq6.F.Sub(t0, fq6.mulByNonResidue(t5)) |
||||
|
c1 := fq6.F.Sub(fq6.mulByNonResidue(t2), t3) |
||||
|
c2 := fq6.F.Sub(t1, t4) |
||||
|
|
||||
|
t6 := fq6.F.Inverse( |
||||
|
fq6.F.Add( |
||||
|
fq6.F.Mul(a[0], c0), |
||||
|
fq6.mulByNonResidue( |
||||
|
fq6.F.Add( |
||||
|
fq6.F.Mul(a[2], c1), |
||||
|
fq6.F.Mul(a[1], c2))))) |
||||
|
return [3][2]*big.Int{ |
||||
|
fq6.F.Mul(t6, c0), |
||||
|
fq6.F.Mul(t6, c1), |
||||
|
fq6.F.Mul(t6, c2), |
||||
|
} |
||||
|
} |
||||
|
|
||||
|
// Div performs a division on the Fq6
|
||||
|
func (fq6 Fq6) Div(a, b [3][2]*big.Int) [3][2]*big.Int { |
||||
|
return fq6.Mul(a, fq6.Inverse(b)) |
||||
|
} |
||||
|
|
||||
|
// Square performs a square operation on the Fq6
|
||||
|
func (fq6 Fq6) Square(a [3][2]*big.Int) [3][2]*big.Int { |
||||
|
s0 := fq6.F.Square(a[0]) |
||||
|
ab := fq6.F.Mul(a[0], a[1]) |
||||
|
s1 := fq6.F.Add(ab, ab) |
||||
|
s2 := fq6.F.Square( |
||||
|
fq6.F.Add( |
||||
|
fq6.F.Sub(a[0], a[1]), |
||||
|
a[2])) |
||||
|
bc := fq6.F.Mul(a[1], a[2]) |
||||
|
s3 := fq6.F.Add(bc, bc) |
||||
|
s4 := fq6.F.Square(a[2]) |
||||
|
|
||||
|
return [3][2]*big.Int{ |
||||
|
fq6.F.Add( |
||||
|
s0, |
||||
|
fq6.mulByNonResidue(s3)), |
||||
|
fq6.F.Add( |
||||
|
s1, |
||||
|
fq6.mulByNonResidue(s4)), |
||||
|
fq6.F.Sub( |
||||
|
fq6.F.Add( |
||||
|
fq6.F.Add(s1, s2), |
||||
|
s3), |
||||
|
fq6.F.Add(s0, s4)), |
||||
|
} |
||||
|
} |
||||
|
|
||||
|
func (fq6 Fq6) Affine(a [3][2]*big.Int) [3][2]*big.Int { |
||||
|
return [3][2]*big.Int{ |
||||
|
fq6.F.Affine(a[0]), |
||||
|
fq6.F.Affine(a[1]), |
||||
|
fq6.F.Affine(a[2]), |
||||
|
} |
||||
|
} |
||||
|
func (fq6 Fq6) Equal(a, b [3][2]*big.Int) bool { |
||||
|
return fq6.F.Equal(a[0], b[0]) && fq6.F.Equal(a[1], b[1]) && fq6.F.Equal(a[2], b[2]) |
||||
|
} |
||||
|
|
||||
|
func (fq6 Fq6) Copy(a [3][2]*big.Int) [3][2]*big.Int { |
||||
|
return [3][2]*big.Int{ |
||||
|
fq6.F.Copy(a[0]), |
||||
|
fq6.F.Copy(a[1]), |
||||
|
fq6.F.Copy(a[2]), |
||||
|
} |
||||
|
} |
||||
@ -0,0 +1,160 @@ |
|||||
|
package bn128 |
||||
|
|
||||
|
import ( |
||||
|
"math/big" |
||||
|
"testing" |
||||
|
|
||||
|
"github.com/stretchr/testify/assert" |
||||
|
) |
||||
|
|
||||
|
func iToBig(a int) *big.Int { |
||||
|
return big.NewInt(int64(a)) |
||||
|
} |
||||
|
|
||||
|
func iiToBig(a, b int) [2]*big.Int { |
||||
|
return [2]*big.Int{iToBig(a), iToBig(b)} |
||||
|
} |
||||
|
|
||||
|
func iiiToBig(a, b int) [2]*big.Int { |
||||
|
return [2]*big.Int{iToBig(a), iToBig(b)} |
||||
|
} |
||||
|
|
||||
|
func TestFq1(t *testing.T) { |
||||
|
fq1 := NewFq(iToBig(7)) |
||||
|
|
||||
|
res := fq1.Add(iToBig(4), iToBig(4)) |
||||
|
assert.Equal(t, iToBig(1), fq1.Affine(res)) |
||||
|
|
||||
|
res = fq1.Double(iToBig(5)) |
||||
|
assert.Equal(t, iToBig(3), fq1.Affine(res)) |
||||
|
|
||||
|
res = fq1.Sub(iToBig(5), iToBig(7)) |
||||
|
assert.Equal(t, iToBig(5), fq1.Affine(res)) |
||||
|
|
||||
|
res = fq1.Neg(iToBig(5)) |
||||
|
assert.Equal(t, iToBig(2), fq1.Affine(res)) |
||||
|
|
||||
|
res = fq1.Mul(iToBig(5), iToBig(11)) |
||||
|
assert.Equal(t, iToBig(6), fq1.Affine(res)) |
||||
|
|
||||
|
res = fq1.Inverse(iToBig(4)) |
||||
|
assert.Equal(t, iToBig(2), res) |
||||
|
|
||||
|
res = fq1.Square(iToBig(5)) |
||||
|
assert.Equal(t, iToBig(4), res) |
||||
|
} |
||||
|
|
||||
|
func TestFq2(t *testing.T) { |
||||
|
fq1 := NewFq(iToBig(7)) |
||||
|
nonResidueFq2str := "-1" // i/j
|
||||
|
nonResidueFq2, ok := new(big.Int).SetString(nonResidueFq2str, 10) |
||||
|
assert.True(t, ok) |
||||
|
assert.Equal(t, nonResidueFq2.String(), nonResidueFq2str) |
||||
|
|
||||
|
fq2 := Fq2{fq1, nonResidueFq2} |
||||
|
|
||||
|
res := fq2.Add(iiToBig(4, 4), iiToBig(3, 4)) |
||||
|
assert.Equal(t, iiToBig(0, 1), fq2.Affine(res)) |
||||
|
|
||||
|
res = fq2.Double(iiToBig(5, 3)) |
||||
|
assert.Equal(t, iiToBig(3, 6), fq2.Affine(res)) |
||||
|
|
||||
|
res = fq2.Sub(iiToBig(5, 3), iiToBig(7, 2)) |
||||
|
assert.Equal(t, iiToBig(5, 1), fq2.Affine(res)) |
||||
|
|
||||
|
res = fq2.Neg(iiToBig(4, 4)) |
||||
|
assert.Equal(t, iiToBig(3, 3), fq2.Affine(res)) |
||||
|
|
||||
|
res = fq2.Mul(iiToBig(4, 4), iiToBig(3, 4)) |
||||
|
assert.Equal(t, iiToBig(3, 0), fq2.Affine(res)) |
||||
|
|
||||
|
res = fq2.Inverse(iiToBig(4, 4)) |
||||
|
assert.Equal(t, iiToBig(1, 6), fq2.Affine(res)) |
||||
|
|
||||
|
res = fq2.Square(iiToBig(4, 4)) |
||||
|
assert.Equal(t, iiToBig(0, 4), fq2.Affine(res)) |
||||
|
res2 := fq2.Mul(iiToBig(4, 4), iiToBig(4, 4)) |
||||
|
assert.Equal(t, fq2.Affine(res), fq2.Affine(res2)) |
||||
|
assert.True(t, fq2.Equal(res, res2)) |
||||
|
|
||||
|
res = fq2.Square(iiToBig(3, 5)) |
||||
|
assert.Equal(t, iiToBig(5, 2), fq2.Affine(res)) |
||||
|
res2 = fq2.Mul(iiToBig(3, 5), iiToBig(3, 5)) |
||||
|
assert.Equal(t, fq2.Affine(res), fq2.Affine(res2)) |
||||
|
} |
||||
|
|
||||
|
func TestFq6(t *testing.T) { |
||||
|
bn128, err := NewBn128() |
||||
|
assert.Nil(t, err) |
||||
|
|
||||
|
a := [3][2]*big.Int{ |
||||
|
iiToBig(1, 2), |
||||
|
iiToBig(3, 4), |
||||
|
iiToBig(5, 6)} |
||||
|
b := [3][2]*big.Int{ |
||||
|
iiToBig(12, 11), |
||||
|
iiToBig(10, 9), |
||||
|
iiToBig(8, 7)} |
||||
|
|
||||
|
mulRes := bn128.Fq6.Mul(a, b) |
||||
|
divRes := bn128.Fq6.Div(mulRes, b) |
||||
|
assert.Equal(t, bn128.Fq6.Affine(a), bn128.Fq6.Affine(divRes)) |
||||
|
} |
||||
|
|
||||
|
func TestFq12(t *testing.T) { |
||||
|
q, ok := new(big.Int).SetString("21888242871839275222246405745257275088696311157297823662689037894645226208583", 10) // i
|
||||
|
assert.True(t, ok) |
||||
|
fq1 := NewFq(q) |
||||
|
nonResidueFq2, ok := new(big.Int).SetString("21888242871839275222246405745257275088696311157297823662689037894645226208582", 10) // i
|
||||
|
assert.True(t, ok) |
||||
|
nonResidueFq6 := iiToBig(9, 1) |
||||
|
|
||||
|
fq2 := Fq2{fq1, nonResidueFq2} |
||||
|
fq6 := Fq6{fq2, nonResidueFq6} |
||||
|
fq12 := Fq12{fq6, fq2, nonResidueFq6} |
||||
|
|
||||
|
a := [2][3][2]*big.Int{ |
||||
|
{ |
||||
|
iiToBig(1, 2), |
||||
|
iiToBig(3, 4), |
||||
|
iiToBig(5, 6), |
||||
|
}, |
||||
|
{ |
||||
|
iiToBig(7, 8), |
||||
|
iiToBig(9, 10), |
||||
|
iiToBig(11, 12), |
||||
|
}, |
||||
|
} |
||||
|
b := [2][3][2]*big.Int{ |
||||
|
{ |
||||
|
iiToBig(12, 11), |
||||
|
iiToBig(10, 9), |
||||
|
iiToBig(8, 7), |
||||
|
}, |
||||
|
{ |
||||
|
iiToBig(6, 5), |
||||
|
iiToBig(4, 3), |
||||
|
iiToBig(2, 1), |
||||
|
}, |
||||
|
} |
||||
|
|
||||
|
res := fq12.Add(a, b) |
||||
|
assert.Equal(t, |
||||
|
[2][3][2]*big.Int{ |
||||
|
{ |
||||
|
iiToBig(13, 13), |
||||
|
iiToBig(13, 13), |
||||
|
iiToBig(13, 13), |
||||
|
}, |
||||
|
{ |
||||
|
iiToBig(13, 13), |
||||
|
iiToBig(13, 13), |
||||
|
iiToBig(13, 13), |
||||
|
}, |
||||
|
}, |
||||
|
res) |
||||
|
|
||||
|
mulRes := fq12.Mul(a, b) |
||||
|
divRes := fq12.Div(mulRes, b) |
||||
|
assert.Equal(t, fq12.Affine(a), fq12.Affine(divRes)) |
||||
|
} |
||||
@ -0,0 +1,191 @@ |
|||||
|
package bn128 |
||||
|
|
||||
|
import ( |
||||
|
"math/big" |
||||
|
) |
||||
|
|
||||
|
type G1 struct { |
||||
|
F Fq |
||||
|
G [3]*big.Int |
||||
|
} |
||||
|
|
||||
|
func NewG1(f Fq, g [2]*big.Int) G1 { |
||||
|
var g1 G1 |
||||
|
g1.F = f |
||||
|
g1.G = [3]*big.Int{ |
||||
|
g[0], |
||||
|
g[1], |
||||
|
g1.F.One(), |
||||
|
} |
||||
|
return g1 |
||||
|
} |
||||
|
|
||||
|
func (g1 G1) Zero() [2]*big.Int { |
||||
|
return [2]*big.Int{g1.F.Zero(), g1.F.Zero()} |
||||
|
} |
||||
|
func (g1 G1) IsZero(p [3]*big.Int) bool { |
||||
|
return g1.F.IsZero(p[2]) |
||||
|
} |
||||
|
|
||||
|
func (g1 G1) Add(p1, p2 [3]*big.Int) [3]*big.Int { |
||||
|
|
||||
|
// https://en.wikibooks.org/wiki/Cryptography/Prime_Curve/Jacobian_Coordinates
|
||||
|
// https://github.com/zcash/zcash/blob/master/src/snark/libsnark/algebra/curves/alt_bn128/alt_bn128_g1.cpp#L208
|
||||
|
// http://hyperelliptic.org/EFD/g1p/auto-code/shortw/jacobian-0/addition/add-2007-bl.op3
|
||||
|
|
||||
|
if g1.IsZero(p1) { |
||||
|
return p2 |
||||
|
} |
||||
|
if g1.IsZero(p2) { |
||||
|
return p1 |
||||
|
} |
||||
|
|
||||
|
x1 := p1[0] |
||||
|
y1 := p1[1] |
||||
|
z1 := p1[2] |
||||
|
x2 := p2[0] |
||||
|
y2 := p2[1] |
||||
|
z2 := p2[2] |
||||
|
|
||||
|
z1z1 := g1.F.Square(z1) |
||||
|
z2z2 := g1.F.Square(z2) |
||||
|
|
||||
|
u1 := g1.F.Mul(x1, z2z2) |
||||
|
u2 := g1.F.Mul(x2, z1z1) |
||||
|
|
||||
|
t0 := g1.F.Mul(z2, z2z2) |
||||
|
s1 := g1.F.Mul(y1, t0) |
||||
|
|
||||
|
t1 := g1.F.Mul(z1, z1z1) |
||||
|
s2 := g1.F.Mul(y2, t1) |
||||
|
|
||||
|
h := g1.F.Sub(u2, u1) |
||||
|
t2 := g1.F.Add(h, h) |
||||
|
i := g1.F.Square(t2) |
||||
|
j := g1.F.Mul(h, i) |
||||
|
t3 := g1.F.Sub(s2, s1) |
||||
|
r := g1.F.Add(t3, t3) |
||||
|
v := g1.F.Mul(u1, i) |
||||
|
t4 := g1.F.Square(r) |
||||
|
t5 := g1.F.Add(v, v) |
||||
|
t6 := g1.F.Sub(t4, j) |
||||
|
x3 := g1.F.Sub(t6, t5) |
||||
|
t7 := g1.F.Sub(v, x3) |
||||
|
t8 := g1.F.Mul(s1, j) |
||||
|
t9 := g1.F.Add(t8, t8) |
||||
|
t10 := g1.F.Mul(r, t7) |
||||
|
|
||||
|
y3 := g1.F.Sub(t10, t9) |
||||
|
|
||||
|
t11 := g1.F.Add(z1, z2) |
||||
|
t12 := g1.F.Square(t11) |
||||
|
t13 := g1.F.Sub(t12, z1z1) |
||||
|
t14 := g1.F.Sub(t13, z2z2) |
||||
|
z3 := g1.F.Mul(t14, h) |
||||
|
|
||||
|
return [3]*big.Int{x3, y3, z3} |
||||
|
} |
||||
|
|
||||
|
func (g1 G1) Neg(p [3]*big.Int) [3]*big.Int { |
||||
|
return [3]*big.Int{ |
||||
|
p[0], |
||||
|
g1.F.Neg(p[1]), |
||||
|
p[2], |
||||
|
} |
||||
|
} |
||||
|
func (g1 G1) Sub(a, b [3]*big.Int) [3]*big.Int { |
||||
|
return g1.Add(a, g1.Neg(b)) |
||||
|
} |
||||
|
func (g1 G1) Double(p [3]*big.Int) [3]*big.Int { |
||||
|
|
||||
|
// https://en.wikibooks.org/wiki/Cryptography/Prime_Curve/Jacobian_Coordinates
|
||||
|
// http://hyperelliptic.org/EFD/g1p/auto-code/shortw/jacobian-0/doubling/dbl-2009-l.op3
|
||||
|
// https://github.com/zcash/zcash/blob/master/src/snark/libsnark/algebra/curves/alt_bn128/alt_bn128_g1.cpp#L325
|
||||
|
|
||||
|
if g1.IsZero(p) { |
||||
|
return p |
||||
|
} |
||||
|
|
||||
|
a := g1.F.Square(p[0]) |
||||
|
b := g1.F.Square(p[1]) |
||||
|
c := g1.F.Square(b) |
||||
|
|
||||
|
t0 := g1.F.Add(p[0], b) |
||||
|
t1 := g1.F.Square(t0) |
||||
|
t2 := g1.F.Sub(t1, a) |
||||
|
t3 := g1.F.Sub(t2, c) |
||||
|
|
||||
|
d := g1.F.Double(t3) |
||||
|
e := g1.F.Add(g1.F.Add(a, a), a) |
||||
|
f := g1.F.Square(e) |
||||
|
|
||||
|
t4 := g1.F.Double(d) |
||||
|
x3 := g1.F.Sub(f, t4) |
||||
|
|
||||
|
t5 := g1.F.Sub(d, x3) |
||||
|
twoC := g1.F.Add(c, c) |
||||
|
fourC := g1.F.Add(twoC, twoC) |
||||
|
t6 := g1.F.Add(fourC, fourC) |
||||
|
t7 := g1.F.Mul(e, t5) |
||||
|
y3 := g1.F.Sub(t7, t6) |
||||
|
|
||||
|
t8 := g1.F.Mul(p[1], p[2]) |
||||
|
z3 := g1.F.Double(t8) |
||||
|
|
||||
|
return [3]*big.Int{x3, y3, z3} |
||||
|
} |
||||
|
|
||||
|
func (g1 G1) MulScalar(p [3]*big.Int, e *big.Int) [3]*big.Int { |
||||
|
// https://en.wikipedia.org/wiki/Elliptic_curve_point_multiplication#Double-and-add
|
||||
|
// for more possible implementations see g2.go file, at the function g2.MulScalar()
|
||||
|
|
||||
|
q := [3]*big.Int{g1.F.Zero(), g1.F.Zero(), g1.F.Zero()} |
||||
|
d := g1.F.Copy(e) |
||||
|
r := p |
||||
|
for i := d.BitLen() - 1; i >= 0; i-- { |
||||
|
q = g1.Double(q) |
||||
|
if d.Bit(i) == 1 { |
||||
|
q = g1.Add(q, r) |
||||
|
} |
||||
|
} |
||||
|
|
||||
|
return q |
||||
|
} |
||||
|
|
||||
|
func (g1 G1) Affine(p [3]*big.Int) [2]*big.Int { |
||||
|
if g1.IsZero(p) { |
||||
|
return g1.Zero() |
||||
|
} |
||||
|
|
||||
|
zinv := g1.F.Inverse(p[2]) |
||||
|
zinv2 := g1.F.Square(zinv) |
||||
|
x := g1.F.Mul(p[0], zinv2) |
||||
|
|
||||
|
zinv3 := g1.F.Mul(zinv2, zinv) |
||||
|
y := g1.F.Mul(p[1], zinv3) |
||||
|
|
||||
|
return [2]*big.Int{x, y} |
||||
|
} |
||||
|
|
||||
|
func (g1 G1) Equal(p1, p2 [3]*big.Int) bool { |
||||
|
if g1.IsZero(p1) { |
||||
|
return g1.IsZero(p2) |
||||
|
} |
||||
|
if g1.IsZero(p2) { |
||||
|
return g1.IsZero(p1) |
||||
|
} |
||||
|
|
||||
|
z1z1 := g1.F.Square(p1[2]) |
||||
|
z2z2 := g1.F.Square(p2[2]) |
||||
|
|
||||
|
u1 := g1.F.Mul(p1[0], z2z2) |
||||
|
u2 := g1.F.Mul(p2[0], z1z1) |
||||
|
|
||||
|
z1cub := g1.F.Mul(p1[2], z1z1) |
||||
|
z2cub := g1.F.Mul(p2[2], z2z2) |
||||
|
|
||||
|
s1 := g1.F.Mul(p1[1], z2cub) |
||||
|
s2 := g1.F.Mul(p2[1], z1cub) |
||||
|
|
||||
|
return g1.F.Equal(u1, u2) && g1.F.Equal(s1, s2) |
||||
|
} |
||||
@ -0,0 +1,31 @@ |
|||||
|
package bn128 |
||||
|
|
||||
|
import ( |
||||
|
"math/big" |
||||
|
"testing" |
||||
|
|
||||
|
"github.com/arnaucube/cryptofun/utils" |
||||
|
"github.com/stretchr/testify/assert" |
||||
|
) |
||||
|
|
||||
|
func TestG1(t *testing.T) { |
||||
|
bn128, err := NewBn128() |
||||
|
assert.Nil(t, err) |
||||
|
|
||||
|
r1 := big.NewInt(int64(33)) |
||||
|
r2 := big.NewInt(int64(44)) |
||||
|
|
||||
|
gr1 := bn128.G1.MulScalar(bn128.G1.G, bn128.Fq1.Copy(r1)) |
||||
|
gr2 := bn128.G1.MulScalar(bn128.G1.G, bn128.Fq1.Copy(r2)) |
||||
|
|
||||
|
grsum1 := bn128.G1.Add(gr1, gr2) // g*33 + g*44
|
||||
|
r1r2 := bn128.Fq1.Add(r1, r2) // 33 + 44
|
||||
|
grsum2 := bn128.G1.MulScalar(bn128.G1.G, r1r2) // g * (33+44)
|
||||
|
|
||||
|
assert.True(t, bn128.G1.Equal(grsum1, grsum2)) |
||||
|
a := bn128.G1.Affine(grsum1) |
||||
|
b := bn128.G1.Affine(grsum2) |
||||
|
assert.Equal(t, a, b) |
||||
|
assert.Equal(t, "0x2f978c0ab89ebaa576866706b14787f360c4d6c3869efe5a72f7c3651a72ff00", utils.BytesToHex(a[0].Bytes())) |
||||
|
assert.Equal(t, "0x12e4ba7f0edca8b4fa668fe153aebd908d322dc26ad964d4cd314795844b62b2", utils.BytesToHex(a[1].Bytes())) |
||||
|
} |
||||
@ -0,0 +1,221 @@ |
|||||
|
package bn128 |
||||
|
|
||||
|
import ( |
||||
|
"math/big" |
||||
|
) |
||||
|
|
||||
|
type G2 struct { |
||||
|
F Fq2 |
||||
|
G [3][2]*big.Int |
||||
|
} |
||||
|
|
||||
|
func NewG2(f Fq2, g [2][2]*big.Int) G2 { |
||||
|
var g2 G2 |
||||
|
g2.F = f |
||||
|
g2.G = [3][2]*big.Int{ |
||||
|
g[0], |
||||
|
g[1], |
||||
|
g2.F.One(), |
||||
|
} |
||||
|
return g2 |
||||
|
} |
||||
|
|
||||
|
func (g2 G2) Zero() [3][2]*big.Int { |
||||
|
return [3][2]*big.Int{g2.F.Zero(), g2.F.One(), g2.F.Zero()} |
||||
|
} |
||||
|
func (g2 G2) IsZero(p [3][2]*big.Int) bool { |
||||
|
return g2.F.IsZero(p[2]) |
||||
|
} |
||||
|
|
||||
|
func (g2 G2) Add(p1, p2 [3][2]*big.Int) [3][2]*big.Int { |
||||
|
|
||||
|
// https://en.wikibooks.org/wiki/Cryptography/Prime_Curve/Jacobian_Coordinates
|
||||
|
// https://github.com/zcash/zcash/blob/master/src/snark/libsnark/algebra/curves/alt_bn128/alt_bn128_g2.cpp#L208
|
||||
|
// http://hyperelliptic.org/EFD/g2p/auto-code/shortw/jacobian-0/addition/add-2007-bl.op3
|
||||
|
|
||||
|
if g2.IsZero(p1) { |
||||
|
return p2 |
||||
|
} |
||||
|
if g2.IsZero(p2) { |
||||
|
return p1 |
||||
|
} |
||||
|
|
||||
|
x1 := p1[0] |
||||
|
y1 := p1[1] |
||||
|
z1 := p1[2] |
||||
|
x2 := p2[0] |
||||
|
y2 := p2[1] |
||||
|
z2 := p2[2] |
||||
|
|
||||
|
z1z1 := g2.F.Square(z1) |
||||
|
z2z2 := g2.F.Square(z2) |
||||
|
|
||||
|
u1 := g2.F.Mul(x1, z2z2) |
||||
|
u2 := g2.F.Mul(x2, z1z1) |
||||
|
|
||||
|
t0 := g2.F.Mul(z2, z2z2) |
||||
|
s1 := g2.F.Mul(y1, t0) |
||||
|
|
||||
|
t1 := g2.F.Mul(z1, z1z1) |
||||
|
s2 := g2.F.Mul(y2, t1) |
||||
|
|
||||
|
h := g2.F.Sub(u2, u1) |
||||
|
t2 := g2.F.Add(h, h) |
||||
|
i := g2.F.Square(t2) |
||||
|
j := g2.F.Mul(h, i) |
||||
|
t3 := g2.F.Sub(s2, s1) |
||||
|
r := g2.F.Add(t3, t3) |
||||
|
v := g2.F.Mul(u1, i) |
||||
|
t4 := g2.F.Square(r) |
||||
|
t5 := g2.F.Add(v, v) |
||||
|
t6 := g2.F.Sub(t4, j) |
||||
|
x3 := g2.F.Sub(t6, t5) |
||||
|
t7 := g2.F.Sub(v, x3) |
||||
|
t8 := g2.F.Mul(s1, j) |
||||
|
t9 := g2.F.Add(t8, t8) |
||||
|
t10 := g2.F.Mul(r, t7) |
||||
|
|
||||
|
y3 := g2.F.Sub(t10, t9) |
||||
|
|
||||
|
t11 := g2.F.Add(z1, z2) |
||||
|
t12 := g2.F.Square(t11) |
||||
|
t13 := g2.F.Sub(t12, z1z1) |
||||
|
t14 := g2.F.Sub(t13, z2z2) |
||||
|
z3 := g2.F.Mul(t14, h) |
||||
|
|
||||
|
return [3][2]*big.Int{x3, y3, z3} |
||||
|
} |
||||
|
|
||||
|
func (g2 G2) Neg(p [3][2]*big.Int) [3][2]*big.Int { |
||||
|
return [3][2]*big.Int{ |
||||
|
p[0], |
||||
|
g2.F.Neg(p[1]), |
||||
|
p[2], |
||||
|
} |
||||
|
} |
||||
|
|
||||
|
func (g2 G2) Sub(a, b [3][2]*big.Int) [3][2]*big.Int { |
||||
|
return g2.Add(a, g2.Neg(b)) |
||||
|
} |
||||
|
|
||||
|
func (g2 G2) Double(p [3][2]*big.Int) [3][2]*big.Int { |
||||
|
|
||||
|
// https://en.wikibooks.org/wiki/Cryptography/Prime_Curve/Jacobian_Coordinates
|
||||
|
// http://hyperelliptic.org/EFD/g2p/auto-code/shortw/jacobian-0/doubling/dbl-2009-l.op3
|
||||
|
// https://github.com/zcash/zcash/blob/master/src/snark/libsnark/algebra/curves/alt_bn128/alt_bn128_g2.cpp#L325
|
||||
|
|
||||
|
if g2.IsZero(p) { |
||||
|
return p |
||||
|
} |
||||
|
|
||||
|
a := g2.F.Square(p[0]) |
||||
|
b := g2.F.Square(p[1]) |
||||
|
c := g2.F.Square(b) |
||||
|
|
||||
|
t0 := g2.F.Add(p[0], b) |
||||
|
t1 := g2.F.Square(t0) |
||||
|
t2 := g2.F.Sub(t1, a) |
||||
|
t3 := g2.F.Sub(t2, c) |
||||
|
|
||||
|
d := g2.F.Double(t3) |
||||
|
e := g2.F.Add(g2.F.Add(a, a), a) |
||||
|
f := g2.F.Square(e) |
||||
|
|
||||
|
t4 := g2.F.Double(d) |
||||
|
x3 := g2.F.Sub(f, t4) |
||||
|
|
||||
|
t5 := g2.F.Sub(d, x3) |
||||
|
twoC := g2.F.Add(c, c) |
||||
|
fourC := g2.F.Add(twoC, twoC) |
||||
|
t6 := g2.F.Add(fourC, fourC) |
||||
|
t7 := g2.F.Mul(e, t5) |
||||
|
y3 := g2.F.Sub(t7, t6) |
||||
|
|
||||
|
t8 := g2.F.Mul(p[1], p[2]) |
||||
|
z3 := g2.F.Double(t8) |
||||
|
|
||||
|
return [3][2]*big.Int{x3, y3, z3} |
||||
|
} |
||||
|
|
||||
|
func (g2 G2) MulScalar(p [3][2]*big.Int, e *big.Int) [3][2]*big.Int { |
||||
|
// https://en.wikipedia.org/wiki/Elliptic_curve_point_multiplication#Double-and-add
|
||||
|
|
||||
|
q := [3][2]*big.Int{g2.F.Zero(), g2.F.Zero(), g2.F.Zero()} |
||||
|
d := g2.F.F.Copy(e) // d := e
|
||||
|
r := p |
||||
|
|
||||
|
/* |
||||
|
here are three possible implementations: |
||||
|
*/ |
||||
|
|
||||
|
/* index decreasing: */ |
||||
|
for i := d.BitLen() - 1; i >= 0; i-- { |
||||
|
q = g2.Double(q) |
||||
|
if d.Bit(i) == 1 { |
||||
|
q = g2.Add(q, r) |
||||
|
} |
||||
|
} |
||||
|
|
||||
|
/* index increasing: */ |
||||
|
// for i := 0; i <= d.BitLen(); i++ {
|
||||
|
// if d.Bit(i) == 1 {
|
||||
|
// q = g2.Add(q, r)
|
||||
|
// }
|
||||
|
// r = g2.Double(r)
|
||||
|
// }
|
||||
|
|
||||
|
// foundone := false
|
||||
|
// for i := d.BitLen(); i >= 0; i-- {
|
||||
|
// if foundone {
|
||||
|
// q = g2.Double(q)
|
||||
|
// }
|
||||
|
// if d.Bit(i) == 1 {
|
||||
|
// foundone = true
|
||||
|
// q = g2.Add(q, r)
|
||||
|
// }
|
||||
|
// }
|
||||
|
|
||||
|
return q |
||||
|
} |
||||
|
|
||||
|
func (g2 G2) Affine(p [3][2]*big.Int) [3][2]*big.Int { |
||||
|
if g2.IsZero(p) { |
||||
|
return g2.Zero() |
||||
|
} |
||||
|
|
||||
|
zinv := g2.F.Inverse(p[2]) |
||||
|
zinv2 := g2.F.Square(zinv) |
||||
|
x := g2.F.Mul(p[0], zinv2) |
||||
|
|
||||
|
zinv3 := g2.F.Mul(zinv2, zinv) |
||||
|
y := g2.F.Mul(p[1], zinv3) |
||||
|
|
||||
|
return [3][2]*big.Int{ |
||||
|
g2.F.Affine(x), |
||||
|
g2.F.Affine(y), |
||||
|
g2.F.One(), |
||||
|
} |
||||
|
} |
||||
|
|
||||
|
func (g2 G2) Equal(p1, p2 [3][2]*big.Int) bool { |
||||
|
if g2.IsZero(p1) { |
||||
|
return g2.IsZero(p2) |
||||
|
} |
||||
|
if g2.IsZero(p2) { |
||||
|
return g2.IsZero(p1) |
||||
|
} |
||||
|
|
||||
|
z1z1 := g2.F.Square(p1[2]) |
||||
|
z2z2 := g2.F.Square(p2[2]) |
||||
|
|
||||
|
u1 := g2.F.Mul(p1[0], z2z2) |
||||
|
u2 := g2.F.Mul(p2[0], z1z1) |
||||
|
|
||||
|
z1cub := g2.F.Mul(p1[2], z1z1) |
||||
|
z2cub := g2.F.Mul(p2[2], z2z2) |
||||
|
|
||||
|
s1 := g2.F.Mul(p1[1], z2cub) |
||||
|
s2 := g2.F.Mul(p2[1], z1cub) |
||||
|
|
||||
|
return g2.F.Equal(u1, u2) && g2.F.Equal(s1, s2) |
||||
|
} |
||||
@ -0,0 +1,24 @@ |
|||||
|
package bn128 |
||||
|
|
||||
|
import ( |
||||
|
"math/big" |
||||
|
"testing" |
||||
|
|
||||
|
"github.com/stretchr/testify/assert" |
||||
|
) |
||||
|
|
||||
|
func TestG2(t *testing.T) { |
||||
|
bn128, err := NewBn128() |
||||
|
assert.Nil(t, err) |
||||
|
|
||||
|
r1 := big.NewInt(int64(33)) |
||||
|
r2 := big.NewInt(int64(44)) |
||||
|
|
||||
|
gr1 := bn128.G2.Affine(bn128.G2.MulScalar(bn128.G2.G, r1)) |
||||
|
gr2 := bn128.G2.Affine(bn128.G2.MulScalar(bn128.G2.G, r2)) |
||||
|
|
||||
|
grsum1 := bn128.G2.Affine(bn128.G2.Add(gr1, gr2)) |
||||
|
r1r2 := bn128.Fq1.Affine(bn128.Fq1.Add(r1, r2)) |
||||
|
grsum2 := bn128.G2.Affine(bn128.G2.MulScalar(bn128.G2.G, r1r2)) |
||||
|
assert.True(t, bn128.G2.Equal(grsum1, grsum2)) |
||||
|
} |
||||
@ -0,0 +1,6 @@ |
|||||
|
module github.com/arnaucube/go-snark |
||||
|
|
||||
|
require ( |
||||
|
github.com/arnaucube/cryptofun v0.0.0-20181124004321-9b11ae8280bd |
||||
|
github.com/stretchr/testify v1.2.2 |
||||
|
) |
||||
@ -0,0 +1,8 @@ |
|||||
|
github.com/arnaucube/cryptofun v0.0.0-20181124004321-9b11ae8280bd h1:NDpNBTFeHNE2IHya+msmKlCzIPGzn8qN3Z2jtegFYT0= |
||||
|
github.com/arnaucube/cryptofun v0.0.0-20181124004321-9b11ae8280bd/go.mod h1:PZE8kKpHPD1UMrS3mTfAMmEEinGtijSwjxLRqRcD64A= |
||||
|
github.com/davecgh/go-spew v1.1.1 h1:vj9j/u1bqnvCEfJOwUhtlOARqs3+rkHYY13jYWTU97c= |
||||
|
github.com/davecgh/go-spew v1.1.1/go.mod h1:J7Y8YcW2NihsgmVo/mv3lAwl/skON4iLHjSsI+c5H38= |
||||
|
github.com/pmezard/go-difflib v1.0.0 h1:4DBwDE0NGyQoBHbLQYPwSUPoCMWR5BEzIk/f1lZbAQM= |
||||
|
github.com/pmezard/go-difflib v1.0.0/go.mod h1:iKH77koFhYxTK1pcRnkKkqfTogsbg7gZNVY4sRDYZ/4= |
||||
|
github.com/stretchr/testify v1.2.2 h1:bSDNvY7ZPG5RlJ8otE/7V6gMiyenm9RtJ7IUVIAoJ1w= |
||||
|
github.com/stretchr/testify v1.2.2/go.mod h1:a8OnRcib4nhh0OaRAV+Yts87kKdq0PP7pXfy6kDkUVs= |
||||
@ -0,0 +1,145 @@ |
|||||
|
package sn |
||||
|
|
||||
|
import ( |
||||
|
"math/big" |
||||
|
) |
||||
|
|
||||
|
func Transpose(matrix [][]*big.Float) [][]*big.Float { |
||||
|
var r [][]*big.Float |
||||
|
for i := 0; i < len(matrix[0]); i++ { |
||||
|
var row []*big.Float |
||||
|
for j := 0; j < len(matrix); j++ { |
||||
|
row = append(row, matrix[j][i]) |
||||
|
} |
||||
|
r = append(r, row) |
||||
|
} |
||||
|
return r |
||||
|
} |
||||
|
|
||||
|
func ArrayOfBigZeros(num int) []*big.Float { |
||||
|
bigZero := big.NewFloat(float64(0)) |
||||
|
var r []*big.Float |
||||
|
for i := 0; i < num; i++ { |
||||
|
r = append(r, bigZero) |
||||
|
} |
||||
|
return r |
||||
|
} |
||||
|
|
||||
|
func PolMul(a, b []*big.Float) []*big.Float { |
||||
|
r := ArrayOfBigZeros(len(a) + len(b) - 1) |
||||
|
for i := 0; i < len(a); i++ { |
||||
|
for j := 0; j < len(b); j++ { |
||||
|
r[i+j] = new(big.Float).Add( |
||||
|
r[i+j], |
||||
|
new(big.Float).Mul(a[i], b[j])) |
||||
|
} |
||||
|
} |
||||
|
return r |
||||
|
} |
||||
|
|
||||
|
func max(a, b int) int { |
||||
|
if a > b { |
||||
|
return a |
||||
|
} |
||||
|
return b |
||||
|
} |
||||
|
|
||||
|
func PolAdd(a, b []*big.Float) []*big.Float { |
||||
|
r := ArrayOfBigZeros(max(len(a), len(b))) |
||||
|
for i := 0; i < len(a); i++ { |
||||
|
r[i] = new(big.Float).Add(r[i], a[i]) |
||||
|
} |
||||
|
for i := 0; i < len(b); i++ { |
||||
|
r[i] = new(big.Float).Add(r[i], b[i]) |
||||
|
} |
||||
|
return r |
||||
|
} |
||||
|
|
||||
|
func PolSub(a, b []*big.Float) []*big.Float { |
||||
|
r := ArrayOfBigZeros(max(len(a), len(b))) |
||||
|
for i := 0; i < len(a); i++ { |
||||
|
r[i] = new(big.Float).Add(r[i], a[i]) |
||||
|
} |
||||
|
for i := 0; i < len(b); i++ { |
||||
|
bneg := new(big.Float).Mul(b[i], big.NewFloat(float64(-1))) |
||||
|
r[i] = new(big.Float).Add(r[i], bneg) |
||||
|
} |
||||
|
return r |
||||
|
|
||||
|
} |
||||
|
|
||||
|
func FloatPow(a *big.Float, e int) *big.Float { |
||||
|
if e == 0 { |
||||
|
return big.NewFloat(float64(1)) |
||||
|
} |
||||
|
result := new(big.Float).Copy(a) |
||||
|
for i := 0; i < e-1; i++ { |
||||
|
result = new(big.Float).Mul(result, a) |
||||
|
} |
||||
|
return result |
||||
|
} |
||||
|
|
||||
|
func PolEval(v []*big.Float, x *big.Float) *big.Float { |
||||
|
r := big.NewFloat(float64(0)) |
||||
|
for i := 0; i < len(v); i++ { |
||||
|
xi := FloatPow(x, i) |
||||
|
elem := new(big.Float).Mul(v[i], xi) |
||||
|
r = new(big.Float).Add(r, elem) |
||||
|
} |
||||
|
return r |
||||
|
} |
||||
|
|
||||
|
func NewPolZeroAt(pointPos, totalPoints int, height *big.Float) []*big.Float { |
||||
|
fac := 1 |
||||
|
for i := 1; i < totalPoints+1; i++ { |
||||
|
if i != pointPos { |
||||
|
fac = fac * (pointPos - i) |
||||
|
} |
||||
|
} |
||||
|
facBig := big.NewFloat(float64(fac)) |
||||
|
hf := new(big.Float).Quo(height, facBig) |
||||
|
r := []*big.Float{hf} |
||||
|
for i := 1; i < totalPoints+1; i++ { |
||||
|
if i != pointPos { |
||||
|
ineg := big.NewFloat(float64(-i)) |
||||
|
b1 := big.NewFloat(float64(1)) |
||||
|
r = PolMul(r, []*big.Float{ineg, b1}) |
||||
|
} |
||||
|
} |
||||
|
return r |
||||
|
} |
||||
|
|
||||
|
func LagrangeInterpolation(v []*big.Float) []*big.Float { |
||||
|
// https://en.wikipedia.org/wiki/Lagrange_polynomial
|
||||
|
var r []*big.Float |
||||
|
for i := 0; i < len(v); i++ { |
||||
|
r = PolAdd(r, NewPolZeroAt(i+1, len(v), v[i])) |
||||
|
} |
||||
|
//
|
||||
|
return r |
||||
|
} |
||||
|
|
||||
|
func R1CSToQAP(a, b, c [][]*big.Float) ([][]*big.Float, [][]*big.Float, [][]*big.Float, []*big.Float) { |
||||
|
aT := Transpose(a) |
||||
|
bT := Transpose(b) |
||||
|
cT := Transpose(c) |
||||
|
var alpha [][]*big.Float |
||||
|
for i := 0; i < len(aT); i++ { |
||||
|
alpha = append(alpha, LagrangeInterpolation(aT[i])) |
||||
|
} |
||||
|
var beta [][]*big.Float |
||||
|
for i := 0; i < len(bT); i++ { |
||||
|
beta = append(beta, LagrangeInterpolation(bT[i])) |
||||
|
} |
||||
|
var gamma [][]*big.Float |
||||
|
for i := 0; i < len(cT); i++ { |
||||
|
gamma = append(gamma, LagrangeInterpolation(cT[i])) |
||||
|
} |
||||
|
z := []*big.Float{big.NewFloat(float64(1))} |
||||
|
for i := 1; i < len(aT[0])+1; i++ { |
||||
|
ineg := big.NewFloat(float64(-i)) |
||||
|
b1 := big.NewFloat(float64(1)) |
||||
|
z = PolMul(z, []*big.Float{ineg, b1}) |
||||
|
} |
||||
|
return alpha, beta, gamma, z |
||||
|
} |
||||
@ -0,0 +1,112 @@ |
|||||
|
package sn |
||||
|
|
||||
|
import ( |
||||
|
"fmt" |
||||
|
"math/big" |
||||
|
"testing" |
||||
|
|
||||
|
"github.com/stretchr/testify/assert" |
||||
|
) |
||||
|
|
||||
|
func TestTranspose(t *testing.T) { |
||||
|
b0 := big.NewFloat(float64(0)) |
||||
|
b1 := big.NewFloat(float64(1)) |
||||
|
bFive := big.NewFloat(float64(5)) |
||||
|
a := [][]*big.Float{ |
||||
|
[]*big.Float{b0, b1, b0, b0, b0, b0}, |
||||
|
[]*big.Float{b0, b0, b0, b1, b0, b0}, |
||||
|
[]*big.Float{b0, b1, b0, b0, b1, b0}, |
||||
|
[]*big.Float{bFive, b0, b0, b0, b0, b1}, |
||||
|
} |
||||
|
aT := Transpose(a) |
||||
|
assert.Equal(t, aT, [][]*big.Float{ |
||||
|
[]*big.Float{b0, b0, b0, bFive}, |
||||
|
[]*big.Float{b1, b0, b1, b0}, |
||||
|
[]*big.Float{b0, b0, b0, b0}, |
||||
|
[]*big.Float{b0, b1, b0, b0}, |
||||
|
[]*big.Float{b0, b0, b1, b0}, |
||||
|
[]*big.Float{b0, b0, b0, b1}, |
||||
|
}) |
||||
|
} |
||||
|
|
||||
|
func TestPol(t *testing.T) { |
||||
|
b0 := big.NewFloat(float64(0)) |
||||
|
b1 := big.NewFloat(float64(1)) |
||||
|
// b1neg := big.NewFloat(float64(-1))
|
||||
|
// b2 := big.NewFloat(float64(2))
|
||||
|
b2neg := big.NewFloat(float64(-2)) |
||||
|
b3 := big.NewFloat(float64(3)) |
||||
|
b4 := big.NewFloat(float64(4)) |
||||
|
b5 := big.NewFloat(float64(5)) |
||||
|
b6 := big.NewFloat(float64(6)) |
||||
|
b16 := big.NewFloat(float64(16)) |
||||
|
|
||||
|
a := []*big.Float{b1, b0, b5} |
||||
|
b := []*big.Float{b3, b0, b1} |
||||
|
|
||||
|
// polynomial multiplication
|
||||
|
c := PolMul(a, b) |
||||
|
assert.Equal(t, c, []*big.Float{b3, b0, b16, b0, b5}) |
||||
|
|
||||
|
// polynomial addition
|
||||
|
c = PolAdd(a, b) |
||||
|
assert.Equal(t, c, []*big.Float{b4, b0, b6}) |
||||
|
|
||||
|
// polynomial substraction
|
||||
|
c = PolSub(a, b) |
||||
|
assert.Equal(t, c, []*big.Float{b2neg, b0, b4}) |
||||
|
|
||||
|
// FloatPow
|
||||
|
p := FloatPow(big.NewFloat(float64(5)), 3) |
||||
|
assert.Equal(t, p, big.NewFloat(float64(125))) |
||||
|
p = FloatPow(big.NewFloat(float64(5)), 0) |
||||
|
assert.Equal(t, p, big.NewFloat(float64(1))) |
||||
|
|
||||
|
// NewPolZeroAt
|
||||
|
r := NewPolZeroAt(3, 4, b4) |
||||
|
assert.Equal(t, PolEval(r, big.NewFloat(3)), b4) |
||||
|
r = NewPolZeroAt(2, 4, b3) |
||||
|
assert.Equal(t, PolEval(r, big.NewFloat(2)), b3) |
||||
|
} |
||||
|
|
||||
|
func TestLagrangeInterpolation(t *testing.T) { |
||||
|
b0 := big.NewFloat(float64(0)) |
||||
|
b5 := big.NewFloat(float64(5)) |
||||
|
a := []*big.Float{b0, b0, b0, b5} |
||||
|
alpha := LagrangeInterpolation(a) |
||||
|
|
||||
|
assert.Equal(t, PolEval(alpha, big.NewFloat(4)), b5) |
||||
|
aux, _ := PolEval(alpha, big.NewFloat(3)).Int64() |
||||
|
assert.Equal(t, aux, int64(0)) |
||||
|
|
||||
|
} |
||||
|
|
||||
|
func TestR1CSToQAP(t *testing.T) { |
||||
|
b0 := big.NewFloat(float64(0)) |
||||
|
b1 := big.NewFloat(float64(1)) |
||||
|
b5 := big.NewFloat(float64(5)) |
||||
|
a := [][]*big.Float{ |
||||
|
[]*big.Float{b0, b1, b0, b0, b0, b0}, |
||||
|
[]*big.Float{b0, b0, b0, b1, b0, b0}, |
||||
|
[]*big.Float{b0, b1, b0, b0, b1, b0}, |
||||
|
[]*big.Float{b5, b0, b0, b0, b0, b1}, |
||||
|
} |
||||
|
b := [][]*big.Float{ |
||||
|
[]*big.Float{b0, b1, b0, b0, b0, b0}, |
||||
|
[]*big.Float{b0, b1, b0, b0, b0, b0}, |
||||
|
[]*big.Float{b1, b0, b0, b0, b0, b0}, |
||||
|
[]*big.Float{b1, b0, b0, b0, b0, b0}, |
||||
|
} |
||||
|
c := [][]*big.Float{ |
||||
|
[]*big.Float{b0, b0, b0, b1, b0, b0}, |
||||
|
[]*big.Float{b0, b0, b0, b0, b1, b0}, |
||||
|
[]*big.Float{b0, b0, b0, b0, b0, b1}, |
||||
|
[]*big.Float{b0, b0, b1, b0, b0, b0}, |
||||
|
} |
||||
|
alpha, beta, gamma, z := R1CSToQAP(a, b, c) |
||||
|
fmt.Println(alpha) |
||||
|
fmt.Println(beta) |
||||
|
fmt.Println(gamma) |
||||
|
fmt.Println(z) |
||||
|
|
||||
|
} |
||||