diff --git a/README.md b/README.md new file mode 100644 index 0000000..5fe46d6 --- /dev/null +++ b/README.md @@ -0,0 +1,97 @@ +# go-snark [![Go Report Card](https://goreportcard.com/badge/github.com/arnaucube/go-snark)](https://goreportcard.com/report/github.com/arnaucube/go-snark) + +Not finished, work in progress (doing this in my free time, so I don't have much time). + + + +#### Test +``` +go test ./... -v +``` + +## R1CS to Quadratic Arithmetic Program +- `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 +- Vitalik Buterin blog post about QAP https://medium.com/@VitalikButerin/quadratic-arithmetic-programs-from-zero-to-hero-f6d558cea649 +- Ariel Gabizon in Zcash blog https://z.cash/blog/snark-explain5 +- Lagrange polynomial Wikipedia article https://en.wikipedia.org/wiki/Lagrange_polynomial + +#### Usage +- R1CS to QAP +```go +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) +/* +out: +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]] +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]] +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]] +z: [24 -50 35 -10 1] +*/ +``` + +## 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 + + +#### 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)) +``` diff --git a/bn128/LICENSE b/bn128/LICENSE new file mode 100644 index 0000000..f288702 --- /dev/null +++ b/bn128/LICENSE @@ -0,0 +1,674 @@ + GNU GENERAL PUBLIC LICENSE + Version 3, 29 June 2007 + + Copyright (C) 2007 Free Software Foundation, Inc. + Everyone is permitted to copy and distribute verbatim copies + of this license document, but changing it is not allowed. + + Preamble + + The GNU General Public License is a free, copyleft license for +software and other kinds of works. + + The licenses for most software and other practical works are designed +to take away your freedom to share and change the works. By contrast, +the GNU General Public License is intended to guarantee your freedom to +share and change all versions of a program--to make sure it remains free +software for all its users. We, the Free Software Foundation, use the +GNU General Public License for most of our software; it applies also to +any other work released this way by its authors. You can apply it to +your programs, too. + + When we speak of free software, we are referring to freedom, not +price. Our General Public Licenses are designed to make sure that you +have the freedom to distribute copies of free software (and charge for +them if you wish), that you receive source code or can get it if you +want it, that you can change the software or use pieces of it in new +free programs, and that you know you can do these things. + + To protect your rights, we need to prevent others from denying you +these rights or asking you to surrender the rights. Therefore, you have +certain responsibilities if you distribute copies of the software, or if +you modify it: responsibilities to respect the freedom of others. + + For example, if you distribute copies of such a program, whether +gratis or for a fee, you must pass on to the recipients the same +freedoms that you received. You must make sure that they, too, receive +or can get the source code. And you must show them these terms so they +know their rights. + + Developers that use the GNU GPL protect your rights with two steps: +(1) assert copyright on the software, and (2) offer you this License +giving you legal permission to copy, distribute and/or modify it. + + For the developers' and authors' protection, the GPL clearly explains +that there is no warranty for this free software. For both users' and +authors' sake, the GPL requires that modified versions be marked as +changed, so that their problems will not be attributed erroneously to +authors of previous versions. + + Some devices are designed to deny users access to install or run +modified versions of the software inside them, although the manufacturer +can do so. This is fundamentally incompatible with the aim of +protecting users' freedom to change the software. The systematic +pattern of such abuse occurs in the area of products for individuals to +use, which is precisely where it is most unacceptable. Therefore, we +have designed this version of the GPL to prohibit the practice for those +products. If such problems arise substantially in other domains, we +stand ready to extend this provision to those domains in future versions +of the GPL, as needed to protect the freedom of users. + + Finally, every program is threatened constantly by software patents. +States should not allow patents to restrict development and use of +software on general-purpose computers, but in those that do, we wish to +avoid the special danger that patents applied to a free program could +make it effectively proprietary. To prevent this, the GPL assures that +patents cannot be used to render the program non-free. + + The precise terms and conditions for copying, distribution and +modification follow. + + TERMS AND CONDITIONS + + 0. Definitions. + + "This License" refers to version 3 of the GNU General Public License. + + "Copyright" also means copyright-like laws that apply to other kinds of +works, such as semiconductor masks. + + "The Program" refers to any copyrightable work licensed under this +License. Each licensee is addressed as "you". "Licensees" and +"recipients" may be individuals or organizations. + + To "modify" a work means to copy from or adapt all or part of the work +in a fashion requiring copyright permission, other than the making of an +exact copy. The resulting work is called a "modified version" of the +earlier work or a work "based on" the earlier work. + + A "covered work" means either the unmodified Program or a work based +on the Program. + + To "propagate" a work means to do anything with it that, without +permission, would make you directly or secondarily liable for +infringement under applicable copyright law, except executing it on a +computer or modifying a private copy. Propagation includes copying, +distribution (with or without modification), making available to the +public, and in some countries other activities as well. + + To "convey" a work means any kind of propagation that enables other +parties to make or receive copies. Mere interaction with a user through +a computer network, with no transfer of a copy, is not conveying. + + An interactive user interface displays "Appropriate Legal Notices" +to the extent that it includes a convenient and prominently visible +feature that (1) displays an appropriate copyright notice, and (2) +tells the user that there is no warranty for the work (except to the +extent that warranties are provided), that licensees may convey the +work under this License, and how to view a copy of this License. If +the interface presents a list of user commands or options, such as a +menu, a prominent item in the list meets this criterion. + + 1. Source Code. + + The "source code" for a work means the preferred form of the work +for making modifications to it. "Object code" means any non-source +form of a work. + + A "Standard Interface" means an interface that either is an official +standard defined by a recognized standards body, or, in the case of +interfaces specified for a particular programming language, one that +is widely used among developers working in that language. + + The "System Libraries" of an executable work include anything, other +than the work as a whole, that (a) is included in the normal form of +packaging a Major Component, but which is not part of that Major +Component, and (b) serves only to enable use of the work with that +Major Component, or to implement a Standard Interface for which an +implementation is available to the public in source code form. A +"Major Component", in this context, means a major essential component +(kernel, window system, and so on) of the specific operating system +(if any) on which the executable work runs, or a compiler used to +produce the work, or an object code interpreter used to run it. + + The "Corresponding Source" for a work in object code form means all +the source code needed to generate, install, and (for an executable +work) run the object code and to modify the work, including scripts to +control those activities. However, it does not include the work's +System Libraries, or general-purpose tools or generally available free +programs which are used unmodified in performing those activities but +which are not part of the work. For example, Corresponding Source +includes interface definition files associated with source files for +the work, and the source code for shared libraries and dynamically +linked subprograms that the work is specifically designed to require, +such as by intimate data communication or control flow between those +subprograms and other parts of the work. + + The Corresponding Source need not include anything that users +can regenerate automatically from other parts of the Corresponding +Source. + + The Corresponding Source for a work in source code form is that +same work. + + 2. Basic Permissions. + + All rights granted under this License are granted for the term of +copyright on the Program, and are irrevocable provided the stated +conditions are met. This License explicitly affirms your unlimited +permission to run the unmodified Program. The output from running a +covered work is covered by this License only if the output, given its +content, constitutes a covered work. This License acknowledges your +rights of fair use or other equivalent, as provided by copyright law. + + You may make, run and propagate covered works that you do not +convey, without conditions so long as your license otherwise remains +in force. You may convey covered works to others for the sole purpose +of having them make modifications exclusively for you, or provide you +with facilities for running those works, provided that you comply with +the terms of this License in conveying all material for which you do +not control copyright. Those thus making or running the covered works +for you must do so exclusively on your behalf, under your direction +and control, on terms that prohibit them from making any copies of +your copyrighted material outside their relationship with you. + + Conveying under any other circumstances is permitted solely under +the conditions stated below. Sublicensing is not allowed; section 10 +makes it unnecessary. + + 3. Protecting Users' Legal Rights From Anti-Circumvention Law. + + No covered work shall be deemed part of an effective technological +measure under any applicable law fulfilling obligations under article +11 of the WIPO copyright treaty adopted on 20 December 1996, or +similar laws prohibiting or restricting circumvention of such +measures. + + When you convey a covered work, you waive any legal power to forbid +circumvention of technological measures to the extent such circumvention +is effected by exercising rights under this License with respect to +the covered work, and you disclaim any intention to limit operation or +modification of the work as a means of enforcing, against the work's +users, your or third parties' legal rights to forbid circumvention of +technological measures. + + 4. Conveying Verbatim Copies. + + You may convey verbatim copies of the Program's source code as you +receive it, in any medium, provided that you conspicuously and +appropriately publish on each copy an appropriate copyright notice; +keep intact all notices stating that this License and any +non-permissive terms added in accord with section 7 apply to the code; +keep intact all notices of the absence of any warranty; and give all +recipients a copy of this License along with the Program. + + You may charge any price or no price for each copy that you convey, +and you may offer support or warranty protection for a fee. + + 5. Conveying Modified Source Versions. + + You may convey a work based on the Program, or the modifications to +produce it from the Program, in the form of source code under the +terms of section 4, provided that you also meet all of these conditions: + + a) The work must carry prominent notices stating that you modified + it, and giving a relevant date. + + b) The work must carry prominent notices stating that it is + released under this License and any conditions added under section + 7. This requirement modifies the requirement in section 4 to + "keep intact all notices". + + c) You must license the entire work, as a whole, under this + License to anyone who comes into possession of a copy. This + License will therefore apply, along with any applicable section 7 + additional terms, to the whole of the work, and all its parts, + regardless of how they are packaged. This License gives no + permission to license the work in any other way, but it does not + invalidate such permission if you have separately received it. + + d) If the work has interactive user interfaces, each must display + Appropriate Legal Notices; however, if the Program has interactive + interfaces that do not display Appropriate Legal Notices, your + work need not make them do so. + + A compilation of a covered work with other separate and independent +works, which are not by their nature extensions of the covered work, +and which are not combined with it such as to form a larger program, +in or on a volume of a storage or distribution medium, is called an +"aggregate" if the compilation and its resulting copyright are not +used to limit the access or legal rights of the compilation's users +beyond what the individual works permit. Inclusion of a covered work +in an aggregate does not cause this License to apply to the other +parts of the aggregate. + + 6. Conveying Non-Source Forms. + + You may convey a covered work in object code form under the terms +of sections 4 and 5, provided that you also convey the +machine-readable Corresponding Source under the terms of this License, +in one of these ways: + + a) Convey the object code in, or embodied in, a physical product + (including a physical distribution medium), accompanied by the + Corresponding Source fixed on a durable physical medium + customarily used for software interchange. + + b) Convey the object code in, or embodied in, a physical product + (including a physical distribution medium), accompanied by a + written offer, valid for at least three years and valid for as + long as you offer spare parts or customer support for that product + model, to give anyone who possesses the object code either (1) a + copy of the Corresponding Source for all the software in the + product that is covered by this License, on a durable physical + medium customarily used for software interchange, for a price no + 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. + + + Copyright (C) + + 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 . + +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: + + Copyright (C) + 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 +. + + 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 +. diff --git a/bn128/README.md b/bn128/README.md new file mode 100644 index 0000000..1d4ed9f --- /dev/null +++ b/bn128/README.md @@ -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) +``` diff --git a/bn128/bn128.go b/bn128/bn128.go new file mode 100644 index 0000000..95a4ac8 --- /dev/null +++ b/bn128/bn128.go @@ -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 +} diff --git a/bn128/bn128_test.go b/bn128/bn128_test.go new file mode 100644 index 0000000..f565258 --- /dev/null +++ b/bn128/bn128_test.go @@ -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") +} diff --git a/bn128/fq.go b/bn128/fq.go new file mode 100644 index 0000000..a736f83 --- /dev/null +++ b/bn128/fq.go @@ -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()) +} diff --git a/bn128/fq12.go b/bn128/fq12.go new file mode 100644 index 0000000..490ad97 --- /dev/null +++ b/bn128/fq12.go @@ -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]) +} diff --git a/bn128/fq2.go b/bn128/fq2.go new file mode 100644 index 0000000..a36497e --- /dev/null +++ b/bn128/fq2.go @@ -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]), + } +} diff --git a/bn128/fq6.go b/bn128/fq6.go new file mode 100644 index 0000000..2d401f5 --- /dev/null +++ b/bn128/fq6.go @@ -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]), + } +} diff --git a/bn128/fqn_test.go b/bn128/fqn_test.go new file mode 100644 index 0000000..a22e5eb --- /dev/null +++ b/bn128/fqn_test.go @@ -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)) +} diff --git a/bn128/g1.go b/bn128/g1.go new file mode 100644 index 0000000..4edd043 --- /dev/null +++ b/bn128/g1.go @@ -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) +} diff --git a/bn128/g1_test.go b/bn128/g1_test.go new file mode 100644 index 0000000..e2bf533 --- /dev/null +++ b/bn128/g1_test.go @@ -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())) +} diff --git a/bn128/g2.go b/bn128/g2.go new file mode 100644 index 0000000..f6b7cad --- /dev/null +++ b/bn128/g2.go @@ -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) +} diff --git a/bn128/g2_test.go b/bn128/g2_test.go new file mode 100644 index 0000000..0782ae0 --- /dev/null +++ b/bn128/g2_test.go @@ -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)) +} diff --git a/go.mod b/go.mod new file mode 100644 index 0000000..001be98 --- /dev/null +++ b/go.mod @@ -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 +) diff --git a/go.sum b/go.sum new file mode 100644 index 0000000..9fa27b1 --- /dev/null +++ b/go.sum @@ -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= diff --git a/r1csqap.go b/r1csqap.go new file mode 100644 index 0000000..02684e4 --- /dev/null +++ b/r1csqap.go @@ -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 +} diff --git a/r1csqap_test.go b/r1csqap_test.go new file mode 100644 index 0000000..a46f09e --- /dev/null +++ b/r1csqap_test.go @@ -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) + +}