arnaucube
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@ -1,21 +1,674 @@ |
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MIT License |
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Copyright (c) 2018 arnau |
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Permission is hereby granted, free of charge, to any person obtaining a copy |
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of this software and associated documentation files (the "Software"), to deal |
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in the Software without restriction, including without limitation the rights |
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to use, copy, modify, merge, publish, distribute, sublicense, and/or sell |
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copies of the Software, and to permit persons to whom the Software is |
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furnished to do so, subject to the following conditions: |
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The above copyright notice and this permission notice shall be included in all |
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copies or substantial portions of the Software. |
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THE SOFTWARE IS PROVIDED "AS IS", WITHOUT WARRANTY OF ANY KIND, EXPRESS OR |
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IMPLIED, INCLUDING BUT NOT LIMITED TO THE WARRANTIES OF MERCHANTABILITY, |
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FITNESS FOR A PARTICULAR PURPOSE AND NONINFRINGEMENT. IN NO EVENT SHALL THE |
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AUTHORS OR COPYRIGHT HOLDERS BE LIABLE FOR ANY CLAIM, DAMAGES OR OTHER |
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LIABILITY, WHETHER IN AN ACTION OF CONTRACT, TORT OR OTHERWISE, ARISING FROM, |
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OUT OF OR IN CONNECTION WITH THE SOFTWARE OR THE USE OR OTHER DEALINGS IN THE |
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SOFTWARE. |
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GNU GENERAL PUBLIC LICENSE |
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Version 3, 29 June 2007 |
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Copyright (C) 2007 Free Software Foundation, Inc. <https://fsf.org/> |
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Everyone is permitted to copy and distribute verbatim copies |
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of this license document, but changing it is not allowed. |
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Preamble |
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The GNU General Public License is a free, copyleft license for |
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software and other kinds of works. |
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The licenses for most software and other practical works are designed |
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the GNU General Public License is intended to guarantee your freedom to |
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share and change all versions of a program--to make sure it remains free |
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software for all its users. We, the Free Software Foundation, use the |
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GNU General Public License for most of our software; it applies also to |
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any other work released this way by its authors. You can apply it to |
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your programs, too. |
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When we speak of free software, we are referring to freedom, not |
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All rights granted under this License are granted for the term of |
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Conveying under any other circumstances is permitted solely under |
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c) You must license the entire work, as a whole, under this |
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License will therefore apply, along with any applicable section 7 |
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invalidate such permission if you have separately received it. |
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d) If the work has interactive user interfaces, each must display |
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A compilation of a covered work with other separate and independent |
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6. Conveying Non-Source Forms. |
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You may convey a covered work in object code form under the terms |
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a) Convey the object code in, or embodied in, a physical product |
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customarily used for software interchange. |
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b) Convey the object code in, or embodied in, a physical product |
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written offer, valid for at least three years and valid for as |
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model, to give anyone who possesses the object code either (1) a |
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product that is covered by this License, on a durable physical |
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medium customarily used for software interchange, for a price no |
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more than your reasonable cost of physically performing this |
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conveying of source, or (2) access to copy the |
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Corresponding Source from a network server at no charge. |
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c) Convey individual copies of the object code with a copy of the |
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only if you received the object code with such an offer, in accord |
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with subsection 6b. |
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d) Convey the object code by offering access from a designated |
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place (gratis or for a charge), and offer equivalent access to the |
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Corresponding Source in the same way through the same place at no |
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further charge. You need not require recipients to copy the |
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Corresponding Source along with the object code. If the place to |
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copy the object code is a network server, the Corresponding Source |
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clear directions next to the object code saying where to find the |
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e) Convey the object code using peer-to-peer transmission, provided |
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you inform other peers where the object code and Corresponding |
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Source of the work are being offered to the general public at no |
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charge under subsection 6d. |
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A separable portion of the object code, whose source code is excluded |
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from the Corresponding Source as a System Library, need not be |
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included in conveying the object code work. |
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A "User Product" is either (1) a "consumer product", which means any |
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tangible personal property which is normally used for personal, family, |
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into a dwelling. In determining whether a product is a consumer product, |
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doubtful cases shall be resolved in favor of coverage. For a particular |
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product received by a particular user, "normally used" refers to a |
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typical or common use of that class of product, regardless of the status |
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commercial, industrial or non-consumer uses, unless such uses represent |
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the only significant mode of use of the product. |
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"Installation Information" for a User Product means any methods, |
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procedures, authorization keys, or other information required to install |
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and execute modified versions of a covered work in that User Product from |
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a modified version of its Corresponding Source. The information must |
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suffice to ensure that the continued functioning of the modified object |
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code is in no case prevented or interfered with solely because |
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modification has been made. |
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If you convey an object code work under this section in, or with, or |
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specifically for use in, a User Product, and the conveying occurs as |
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part of a transaction in which the right of possession and use of the |
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User Product is transferred to the recipient in perpetuity or for a |
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Corresponding Source conveyed under this section must be accompanied |
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by the Installation Information. But this requirement does not apply |
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if neither you nor any third party retains the ability to install |
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modified object code on the User Product (for example, the work has |
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been installed in ROM). |
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The requirement to provide Installation Information does not include a |
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requirement to continue to provide support service, warranty, or updates |
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for a work that has been modified or installed by the recipient, or for |
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the User Product in which it has been modified or installed. Access to a |
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network may be denied when the modification itself materially and |
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adversely affects the operation of the network or violates the rules and |
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protocols for communication across the network. |
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Corresponding Source conveyed, and Installation Information provided, |
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in accord with this section must be in a format that is publicly |
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documented (and with an implementation available to the public in |
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source code form), and must require no special password or key for |
||||
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unpacking, reading or copying. |
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7. Additional Terms. |
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"Additional permissions" are terms that supplement the terms of this |
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License by making exceptions from one or more of its conditions. |
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Additional permissions that are applicable to the entire Program shall |
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be treated as though they were included in this License, to the extent |
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that they are valid under applicable law. If additional permissions |
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apply only to part of the Program, that part may be used separately |
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under those permissions, but the entire Program remains governed by |
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this License without regard to the additional permissions. |
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When you convey a copy of a covered work, you may at your option |
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remove any additional permissions from that copy, or from any part of |
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it. (Additional permissions may be written to require their own |
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removal in certain cases when you modify the work.) You may place |
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additional permissions on material, added by you to a covered work, |
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for which you have or can give appropriate copyright permission. |
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Notwithstanding any other provision of this License, for material you |
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add to a covered work, you may (if authorized by the copyright holders of |
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that material) supplement the terms of this License with terms: |
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a) Disclaiming warranty or limiting liability differently from the |
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terms of sections 15 and 16 of this License; or |
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b) Requiring preservation of specified reasonable legal notices or |
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author attributions in that material or in the Appropriate Legal |
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Notices displayed by works containing it; or |
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c) Prohibiting misrepresentation of the origin of that material, or |
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requiring that modified versions of such material be marked in |
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|
reasonable ways as different from the original version; or |
||||
|
|
||||
|
d) Limiting the use for publicity purposes of names of licensors or |
||||
|
authors of the material; or |
||||
|
|
||||
|
e) Declining to grant rights under trademark law for use of some |
||||
|
trade names, trademarks, or service marks; or |
||||
|
|
||||
|
f) Requiring indemnification of licensors and authors of that |
||||
|
material by anyone who conveys the material (or modified versions of |
||||
|
it) with contractual assumptions of liability to the recipient, for |
||||
|
any liability that these contractual assumptions directly impose on |
||||
|
those licensors and authors. |
||||
|
|
||||
|
All other non-permissive additional terms are considered "further |
||||
|
restrictions" within the meaning of section 10. If the Program as you |
||||
|
received it, or any part of it, contains a notice stating that it is |
||||
|
governed by this License along with a term that is a further |
||||
|
restriction, you may remove that term. If a license document contains |
||||
|
a further restriction but permits relicensing or conveying under this |
||||
|
License, you may add to a covered work material governed by the terms |
||||
|
of that license document, provided that the further restriction does |
||||
|
not survive such relicensing or conveying. |
||||
|
|
||||
|
If you add terms to a covered work in accord with this section, you |
||||
|
must place, in the relevant source files, a statement of the |
||||
|
additional terms that apply to those files, or a notice indicating |
||||
|
where to find the applicable terms. |
||||
|
|
||||
|
Additional terms, permissive or non-permissive, may be stated in the |
||||
|
form of a separately written license, or stated as exceptions; |
||||
|
the above requirements apply either way. |
||||
|
|
||||
|
8. Termination. |
||||
|
|
||||
|
You may not propagate or modify a covered work except as expressly |
||||
|
provided under this License. Any attempt otherwise to propagate or |
||||
|
modify it is void, and will automatically terminate your rights under |
||||
|
this License (including any patent licenses granted under the third |
||||
|
paragraph of section 11). |
||||
|
|
||||
|
However, if you cease all violation of this License, then your |
||||
|
license from a particular copyright holder is reinstated (a) |
||||
|
provisionally, unless and until the copyright holder explicitly and |
||||
|
finally terminates your license, and (b) permanently, if the copyright |
||||
|
holder fails to notify you of the violation by some reasonable means |
||||
|
prior to 60 days after the cessation. |
||||
|
|
||||
|
Moreover, your license from a particular copyright holder is |
||||
|
reinstated permanently if the copyright holder notifies you of the |
||||
|
violation by some reasonable means, this is the first time you have |
||||
|
received notice of violation of this License (for any work) from that |
||||
|
copyright holder, and you cure the violation prior to 30 days after |
||||
|
your receipt of the notice. |
||||
|
|
||||
|
Termination of your rights under this section does not terminate the |
||||
|
licenses of parties who have received copies or rights from you under |
||||
|
this License. If your rights have been terminated and not permanently |
||||
|
reinstated, you do not qualify to receive new licenses for the same |
||||
|
material under section 10. |
||||
|
|
||||
|
9. Acceptance Not Required for Having Copies. |
||||
|
|
||||
|
You are not required to accept this License in order to receive or |
||||
|
run a copy of the Program. Ancillary propagation of a covered work |
||||
|
occurring solely as a consequence of using peer-to-peer transmission |
||||
|
to receive a copy likewise does not require acceptance. However, |
||||
|
nothing other than this License grants you permission to propagate or |
||||
|
modify any covered work. These actions infringe copyright if you do |
||||
|
not accept this License. Therefore, by modifying or propagating a |
||||
|
covered work, you indicate your acceptance of this License to do so. |
||||
|
|
||||
|
10. Automatic Licensing of Downstream Recipients. |
||||
|
|
||||
|
Each time you convey a covered work, the recipient automatically |
||||
|
receives a license from the original licensors, to run, modify and |
||||
|
propagate that work, subject to this License. You are not responsible |
||||
|
for enforcing compliance by third parties with this License. |
||||
|
|
||||
|
An "entity transaction" is a transaction transferring control of an |
||||
|
organization, or substantially all assets of one, or subdividing an |
||||
|
organization, or merging organizations. If propagation of a covered |
||||
|
work results from an entity transaction, each party to that |
||||
|
transaction who receives a copy of the work also receives whatever |
||||
|
licenses to the work the party's predecessor in interest had or could |
||||
|
give under the previous paragraph, plus a right to possession of the |
||||
|
Corresponding Source of the work from the predecessor in interest, if |
||||
|
the predecessor has it or can get it with reasonable efforts. |
||||
|
|
||||
|
You may not impose any further restrictions on the exercise of the |
||||
|
rights granted or affirmed under this License. For example, you may |
||||
|
not impose a license fee, royalty, or other charge for exercise of |
||||
|
rights granted under this License, and you may not initiate litigation |
||||
|
(including a cross-claim or counterclaim in a lawsuit) alleging that |
||||
|
any patent claim is infringed by making, using, selling, offering for |
||||
|
sale, or importing the Program or any portion of it. |
||||
|
|
||||
|
11. Patents. |
||||
|
|
||||
|
A "contributor" is a copyright holder who authorizes use under this |
||||
|
License of the Program or a work on which the Program is based. The |
||||
|
work thus licensed is called the contributor's "contributor version". |
||||
|
|
||||
|
A contributor's "essential patent claims" are all patent claims |
||||
|
owned or controlled by the contributor, whether already acquired or |
||||
|
hereafter acquired, that would be infringed by some manner, permitted |
||||
|
by this License, of making, using, or selling its contributor version, |
||||
|
but do not include claims that would be infringed only as a |
||||
|
consequence of further modification of the contributor version. For |
||||
|
purposes of this definition, "control" includes the right to grant |
||||
|
patent sublicenses in a manner consistent with the requirements of |
||||
|
this License. |
||||
|
|
||||
|
Each contributor grants you a non-exclusive, worldwide, royalty-free |
||||
|
patent license under the contributor's essential patent claims, to |
||||
|
make, use, sell, offer for sale, import and otherwise run, modify and |
||||
|
propagate the contents of its contributor version. |
||||
|
|
||||
|
In the following three paragraphs, a "patent license" is any express |
||||
|
agreement or commitment, however denominated, not to enforce a patent |
||||
|
(such as an express permission to practice a patent or covenant not to |
||||
|
sue for patent infringement). To "grant" such a patent license to a |
||||
|
party means to make such an agreement or commitment not to enforce a |
||||
|
patent against the party. |
||||
|
|
||||
|
If you convey a covered work, knowingly relying on a patent license, |
||||
|
and the Corresponding Source of the work is not available for anyone |
||||
|
to copy, free of charge and under the terms of this License, through a |
||||
|
publicly available network server or other readily accessible means, |
||||
|
then you must either (1) cause the Corresponding Source to be so |
||||
|
available, or (2) arrange to deprive yourself of the benefit of the |
||||
|
patent license for this particular work, or (3) arrange, in a manner |
||||
|
consistent with the requirements of this License, to extend the patent |
||||
|
license to downstream recipients. "Knowingly relying" means you have |
||||
|
actual knowledge that, but for the patent license, your conveying the |
||||
|
covered work in a country, or your recipient's use of the covered work |
||||
|
in a country, would infringe one or more identifiable patents in that |
||||
|
country that you have reason to believe are valid. |
||||
|
|
||||
|
If, pursuant to or in connection with a single transaction or |
||||
|
arrangement, you convey, or propagate by procuring conveyance of, a |
||||
|
covered work, and grant a patent license to some of the parties |
||||
|
receiving the covered work authorizing them to use, propagate, modify |
||||
|
or convey a specific copy of the covered work, then the patent license |
||||
|
you grant is automatically extended to all recipients of the covered |
||||
|
work and works based on it. |
||||
|
|
||||
|
A patent license is "discriminatory" if it does not include within |
||||
|
the scope of its coverage, prohibits the exercise of, or is |
||||
|
conditioned on the non-exercise of one or more of the rights that are |
||||
|
specifically granted under this License. You may not convey a covered |
||||
|
work if you are a party to an arrangement with a third party that is |
||||
|
in the business of distributing software, under which you make payment |
||||
|
to the third party based on the extent of your activity of conveying |
||||
|
the work, and under which the third party grants, to any of the |
||||
|
parties who would receive the covered work from you, a discriminatory |
||||
|
patent license (a) in connection with copies of the covered work |
||||
|
conveyed by you (or copies made from those copies), or (b) primarily |
||||
|
for and in connection with specific products or compilations that |
||||
|
contain the covered work, unless you entered into that arrangement, |
||||
|
or that patent license was granted, prior to 28 March 2007. |
||||
|
|
||||
|
Nothing in this License shall be construed as excluding or limiting |
||||
|
any implied license or other defenses to infringement that may |
||||
|
otherwise be available to you under applicable patent law. |
||||
|
|
||||
|
12. No Surrender of Others' Freedom. |
||||
|
|
||||
|
If conditions are imposed on you (whether by court order, agreement or |
||||
|
otherwise) that contradict the conditions of this License, they do not |
||||
|
excuse you from the conditions of this License. If you cannot convey a |
||||
|
covered work so as to satisfy simultaneously your obligations under this |
||||
|
License and any other pertinent obligations, then as a consequence you may |
||||
|
not convey it at all. For example, if you agree to terms that obligate you |
||||
|
to collect a royalty for further conveying from those to whom you convey |
||||
|
the Program, the only way you could satisfy both those terms and this |
||||
|
License would be to refrain entirely from conveying the Program. |
||||
|
|
||||
|
13. Use with the GNU Affero General Public License. |
||||
|
|
||||
|
Notwithstanding any other provision of this License, you have |
||||
|
permission to link or combine any covered work with a work licensed |
||||
|
under version 3 of the GNU Affero General Public License into a single |
||||
|
combined work, and to convey the resulting work. The terms of this |
||||
|
License will continue to apply to the part which is the covered work, |
||||
|
but the special requirements of the GNU Affero General Public License, |
||||
|
section 13, concerning interaction through a network will apply to the |
||||
|
combination as such. |
||||
|
|
||||
|
14. Revised Versions of this License. |
||||
|
|
||||
|
The Free Software Foundation may publish revised and/or new versions of |
||||
|
the GNU General Public License from time to time. Such new versions will |
||||
|
be similar in spirit to the present version, but may differ in detail to |
||||
|
address new problems or concerns. |
||||
|
|
||||
|
Each version is given a distinguishing version number. If the |
||||
|
Program specifies that a certain numbered version of the GNU General |
||||
|
Public License "or any later version" applies to it, you have the |
||||
|
option of following the terms and conditions either of that numbered |
||||
|
version or of any later version published by the Free Software |
||||
|
Foundation. If the Program does not specify a version number of the |
||||
|
GNU General Public License, you may choose any version ever published |
||||
|
by the Free Software Foundation. |
||||
|
|
||||
|
If the Program specifies that a proxy can decide which future |
||||
|
versions of the GNU General Public License can be used, that proxy's |
||||
|
public statement of acceptance of a version permanently authorizes you |
||||
|
to choose that version for the Program. |
||||
|
|
||||
|
Later license versions may give you additional or different |
||||
|
permissions. However, no additional obligations are imposed on any |
||||
|
author or copyright holder as a result of your choosing to follow a |
||||
|
later version. |
||||
|
|
||||
|
15. Disclaimer of Warranty. |
||||
|
|
||||
|
THERE IS NO WARRANTY FOR THE PROGRAM, TO THE EXTENT PERMITTED BY |
||||
|
APPLICABLE LAW. EXCEPT WHEN OTHERWISE STATED IN WRITING THE COPYRIGHT |
||||
|
HOLDERS AND/OR OTHER PARTIES PROVIDE THE PROGRAM "AS IS" WITHOUT WARRANTY |
||||
|
OF ANY KIND, EITHER EXPRESSED OR IMPLIED, INCLUDING, BUT NOT LIMITED TO, |
||||
|
THE IMPLIED WARRANTIES OF MERCHANTABILITY AND FITNESS FOR A PARTICULAR |
||||
|
PURPOSE. THE ENTIRE RISK AS TO THE QUALITY AND PERFORMANCE OF THE PROGRAM |
||||
|
IS WITH YOU. SHOULD THE PROGRAM PROVE DEFECTIVE, YOU ASSUME THE COST OF |
||||
|
ALL NECESSARY SERVICING, REPAIR OR CORRECTION. |
||||
|
|
||||
|
16. Limitation of Liability. |
||||
|
|
||||
|
IN NO EVENT UNLESS REQUIRED BY APPLICABLE LAW OR AGREED TO IN WRITING |
||||
|
WILL ANY COPYRIGHT HOLDER, OR ANY OTHER PARTY WHO MODIFIES AND/OR CONVEYS |
||||
|
THE PROGRAM AS PERMITTED ABOVE, BE LIABLE TO YOU FOR DAMAGES, INCLUDING ANY |
||||
|
GENERAL, SPECIAL, INCIDENTAL OR CONSEQUENTIAL DAMAGES ARISING OUT OF THE |
||||
|
USE OR INABILITY TO USE THE PROGRAM (INCLUDING BUT NOT LIMITED TO LOSS OF |
||||
|
DATA OR DATA BEING RENDERED INACCURATE OR LOSSES SUSTAINED BY YOU OR THIRD |
||||
|
PARTIES OR A FAILURE OF THE PROGRAM TO OPERATE WITH ANY OTHER PROGRAMS), |
||||
|
EVEN IF SUCH HOLDER OR OTHER PARTY HAS BEEN ADVISED OF THE POSSIBILITY OF |
||||
|
SUCH DAMAGES. |
||||
|
|
||||
|
17. Interpretation of Sections 15 and 16. |
||||
|
|
||||
|
If the disclaimer of warranty and limitation of liability provided |
||||
|
above cannot be given local legal effect according to their terms, |
||||
|
reviewing courts shall apply local law that most closely approximates |
||||
|
an absolute waiver of all civil liability in connection with the |
||||
|
Program, unless a warranty or assumption of liability accompanies a |
||||
|
copy of the Program in return for a fee. |
||||
|
|
||||
|
END OF TERMS AND CONDITIONS |
||||
|
|
||||
|
How to Apply These Terms to Your New Programs |
||||
|
|
||||
|
If you develop a new program, and you want it to be of the greatest |
||||
|
possible use to the public, the best way to achieve this is to make it |
||||
|
free software which everyone can redistribute and change under these terms. |
||||
|
|
||||
|
To do so, attach the following notices to the program. It is safest |
||||
|
to attach them to the start of each source file to most effectively |
||||
|
state the exclusion of warranty; and each file should have at least |
||||
|
the "copyright" line and a pointer to where the full notice is found. |
||||
|
|
||||
|
<one line to give the program's name and a brief idea of what it does.> |
||||
|
Copyright (C) <year> <name of author> |
||||
|
|
||||
|
This program is free software: you can redistribute it and/or modify |
||||
|
it under the terms of the GNU General Public License as published by |
||||
|
the Free Software Foundation, either version 3 of the License, or |
||||
|
(at your option) any later version. |
||||
|
|
||||
|
This program is distributed in the hope that it will be useful, |
||||
|
but WITHOUT ANY WARRANTY; without even the implied warranty of |
||||
|
MERCHANTABILITY or FITNESS FOR A PARTICULAR PURPOSE. See the |
||||
|
GNU General Public License for more details. |
||||
|
|
||||
|
You should have received a copy of the GNU General Public License |
||||
|
along with this program. If not, see <https://www.gnu.org/licenses/>. |
||||
|
|
||||
|
Also add information on how to contact you by electronic and paper mail. |
||||
|
|
||||
|
If the program does terminal interaction, make it output a short |
||||
|
notice like this when it starts in an interactive mode: |
||||
|
|
||||
|
<program> Copyright (C) <year> <name of author> |
||||
|
This program comes with ABSOLUTELY NO WARRANTY; for details type `show w'. |
||||
|
This is free software, and you are welcome to redistribute it |
||||
|
under certain conditions; type `show c' for details. |
||||
|
|
||||
|
The hypothetical commands `show w' and `show c' should show the appropriate |
||||
|
parts of the General Public License. Of course, your program's commands |
||||
|
might be different; for a GUI interface, you would use an "about box". |
||||
|
|
||||
|
You should also get your employer (if you work as a programmer) or school, |
||||
|
if any, to sign a "copyright disclaimer" for the program, if necessary. |
||||
|
For more information on this, and how to apply and follow the GNU GPL, see |
||||
|
<https://www.gnu.org/licenses/>. |
||||
|
|
||||
|
The GNU General Public License does not permit incorporating your program |
||||
|
into proprietary programs. If your program is a subroutine library, you |
||||
|
may consider it more useful to permit linking proprietary applications with |
||||
|
the library. If this is what you want to do, use the GNU Lesser General |
||||
|
Public License instead of this License. But first, please read |
||||
|
<https://www.gnu.org/licenses/why-not-lgpl.html>. |
||||
@ -0,0 +1,145 @@ |
|||||
|
## Bn128 |
||||
|
**[not finished]** |
||||
|
|
||||
|
This is implemented followng the implementations and info from: |
||||
|
- 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 |
||||
|
- `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 |
||||
|
- `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 |
||||
|
|
||||
|
- [x] Fq, Fq2, Fq6, Fq12 operations |
||||
|
- [x] G1, G2 operations |
||||
|
|
||||
|
|
||||
|
#### Usage |
||||
|
First let's define 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 / Beta |
||||
|
nonResidueFq2, ok := new(big.Int).SetString(nonResidueFq2str, 10) |
||||
|
fq2 := Fq2{fq1, nonResidueFq2} |
||||
|
nonResidueFq6 := iiToBig(9, 1) |
||||
|
|
||||
|
// basic operations of finite field of order 2 |
||||
|
res := fq2.Add(iiToBig(4, 4), iiToBig(3, 4)) |
||||
|
res = fq2.Double(iiToBig(5, 3)) |
||||
|
res = fq2.Sub(iiToBig(5, 3), iiToBig(7, 2)) |
||||
|
res = fq2.Neg(iiToBig(4, 4)) |
||||
|
res = fq2.Mul(iiToBig(4, 4), iiToBig(3, 4)) |
||||
|
res = fq2.Inverse(iiToBig(4, 4)) |
||||
|
res = fq2.Div(iiToBig(4, 4), iiToBig(3, 4)) |
||||
|
res = fq2.Square(iiToBig(4, 4)) |
||||
|
|
||||
|
|
||||
|
// new finite field of order 6 |
||||
|
nonResidueFq6 := iiToBig(9, 1) // TODO |
||||
|
fq6 := Fq6{fq2, nonResidueFq6} |
||||
|
|
||||
|
// define two new values of Finite Field 6, in order to be able to perform the operations |
||||
|
a := [3][2]*big.Int{ |
||||
|
iiToBig(1, 2), |
||||
|
iiToBig(3, 4), |
||||
|
iiToBig(5, 6)} |
||||
|
b := [3][2]*big.Int{ |
||||
|
iiToBig(12, 11), |
||||
|
iiToBig(10, 9), |
||||
|
iiToBig(8, 7)} |
||||
|
|
||||
|
// basic operations of finite field order 6 |
||||
|
res := fq6.Add(a, b) |
||||
|
res = fq6.Sub(a, b) |
||||
|
res = fq6.Mul(a, b) |
||||
|
divRes := fq6.Div(mulRes, b) |
||||
|
|
||||
|
|
||||
|
// new finite field of order 12 |
||||
|
q, ok := new(big.Int).SetString("21888242871839275222246405745257275088696311157297823662689037894645226208583", 10) // i |
||||
|
if !ok { |
||||
|
fmt.Println("error parsing string to big integer") |
||||
|
} |
||||
|
|
||||
|
fq1 := NewFq(q) |
||||
|
nonResidueFq2, ok := new(big.Int).SetString("21888242871839275222246405745257275088696311157297823662689037894645226208582", 10) // i |
||||
|
assert.True(t, ok) |
||||
|
nonResidueFq6 := iiToBig(9, 1) |
||||
|
|
||||
|
fq2 := Fq2{fq1, nonResidueFq2} |
||||
|
fq6 := Fq6{fq2, nonResidueFq6} |
||||
|
fq12 := Fq12{fq6, fq2, nonResidueFq6} |
||||
|
|
||||
|
``` |
||||
|
|
||||
|
- G1 operations |
||||
|
```go |
||||
|
bn128, err := NewBn128() |
||||
|
assert.Nil(t, err) |
||||
|
|
||||
|
r1 := big.NewInt(int64(33)) |
||||
|
r2 := big.NewInt(int64(44)) |
||||
|
|
||||
|
gr1 := bn128.G1.MulScalar(bn128.G1.G, bn128.Fq1.Copy(r1)) |
||||
|
gr2 := bn128.G1.MulScalar(bn128.G1.G, bn128.Fq1.Copy(r2)) |
||||
|
|
||||
|
grsum1 := bn128.G1.Add(gr1, gr2) |
||||
|
r1r2 := bn128.Fq1.Add(r1, r2) |
||||
|
grsum2 := bn128.G1.MulScalar(bn128.G1.G, r1r2) |
||||
|
|
||||
|
a := bn128.G1.Affine(grsum1) |
||||
|
b := bn128.G1.Affine(grsum2) |
||||
|
assert.Equal(t, a, b) |
||||
|
assert.Equal(t, "0x2f978c0ab89ebaa576866706b14787f360c4d6c3869efe5a72f7c3651a72ff00", utils.BytesToHex(a[0].Bytes())) |
||||
|
assert.Equal(t, "0x12e4ba7f0edca8b4fa668fe153aebd908d322dc26ad964d4cd314795844b62b2", utils.BytesToHex(a[1].Bytes())) |
||||
|
``` |
||||
|
|
||||
|
- G2 operations |
||||
|
```go |
||||
|
bn128, err := NewBn128() |
||||
|
assert.Nil(t, err) |
||||
|
|
||||
|
r1 := big.NewInt(int64(33)) |
||||
|
r2 := big.NewInt(int64(44)) |
||||
|
|
||||
|
gr1 := bn128.G2.MulScalar(bn128.G2.G, bn128.Fq1.Copy(r1)) |
||||
|
gr2 := bn128.G2.MulScalar(bn128.G2.G, bn128.Fq1.Copy(r2)) |
||||
|
|
||||
|
grsum1 := bn128.G2.Add(gr1, gr2) |
||||
|
r1r2 := bn128.Fq1.Add(r1, r2) |
||||
|
grsum2 := bn128.G2.MulScalar(bn128.G2.G, r1r2) |
||||
|
|
||||
|
a := bn128.G2.Affine(grsum1) |
||||
|
b := bn128.G2.Affine(grsum2) |
||||
|
assert.Equal(t, a, b) |
||||
|
``` |
||||
@ -0,0 +1,49 @@ |
|||||
|
## ECC ECDSA |
||||
|
- https://en.wikipedia.org/wiki/Elliptic_Curve_Digital_Signature_Algorithm |
||||
|
|
||||
|
- [x] define ECDSA data structure |
||||
|
- [x] ECDSA Sign |
||||
|
- [x] ECDSA Verify signature |
||||
|
|
||||
|
|
||||
|
#### Usage |
||||
|
```go |
||||
|
// define new elliptic curve |
||||
|
ec := ecc.NewEC(big.NewInt(int64(1)), big.NewInt(int64(18)), big.NewInt(int64(19))) |
||||
|
// define new point |
||||
|
g := ecc.Point{big.NewInt(int64(7)), big.NewInt(int64(11))} |
||||
|
|
||||
|
// define new ECDSA system |
||||
|
dsa, err := NewDSA(ec, g) |
||||
|
if err!=nil { |
||||
|
fmt.Println(err) |
||||
|
} |
||||
|
|
||||
|
// define privK&pubK over the elliptic curve |
||||
|
privK := big.NewInt(int64(5)) |
||||
|
pubK, err := dsa.PubK(privK) |
||||
|
if err!=nil { |
||||
|
fmt.Println(err) |
||||
|
} |
||||
|
|
||||
|
// hash value to sign |
||||
|
hashval := big.NewInt(int64(40)) |
||||
|
|
||||
|
// define r |
||||
|
r := big.NewInt(int64(11)) |
||||
|
|
||||
|
// sign hashed value |
||||
|
sig, err := dsa.Sign(hashval, privK, r) |
||||
|
if err!=nil { |
||||
|
fmt.Println(err) |
||||
|
} |
||||
|
|
||||
|
// verify signature |
||||
|
verified, err := dsa.Verify(hashval, sig, pubK) |
||||
|
if err!=nil { |
||||
|
fmt.Println(err) |
||||
|
} |
||||
|
if verified { |
||||
|
fmt.Println("signature correctly verified") |
||||
|
} |
||||
|
``` |
||||
@ -0,0 +1,50 @@ |
|||||
|
## ECC ElGamal |
||||
|
- https://en.wikipedia.org/wiki/ElGamal_encryption |
||||
|
|
||||
|
- [x] ECC ElGamal key generation |
||||
|
- [x] ECC ElGamal Encrypton |
||||
|
- [x] ECC ElGamal Decryption |
||||
|
|
||||
|
|
||||
|
#### Usage |
||||
|
- NewEG, Encryption, Decryption |
||||
|
```go |
||||
|
// define new elliptic curve |
||||
|
ec := ecc.NewEC(big.NewInt(int64(1)), big.NewInt(int64(18)), big.NewInt(int64(19))) |
||||
|
|
||||
|
// define new point |
||||
|
g := ecc.Point{big.NewInt(int64(7)), big.NewInt(int64(11))} |
||||
|
|
||||
|
// define new ElGamal crypto system with the elliptic curve and the point |
||||
|
eg, err := NewEG(ec, g) |
||||
|
if err!=nil { |
||||
|
fmt.Println(err) |
||||
|
} |
||||
|
|
||||
|
// define privK&pubK over the elliptic curve |
||||
|
privK := big.NewInt(int64(5)) |
||||
|
pubK, err := eg.PubK(privK) |
||||
|
if err!=nil { |
||||
|
fmt.Println(err) |
||||
|
} |
||||
|
|
||||
|
// define point to encrypt |
||||
|
m := ecc.Point{big.NewInt(int64(11)), big.NewInt(int64(12))} |
||||
|
|
||||
|
// encrypt |
||||
|
c, err := eg.Encrypt(m, pubK, big.NewInt(int64(15))) |
||||
|
if err!=nil { |
||||
|
fmt.Println(err) |
||||
|
} |
||||
|
|
||||
|
// decrypt |
||||
|
d, err := eg.Decrypt(c, privK) |
||||
|
if err!=nil { |
||||
|
fmt.Println(err) |
||||
|
} |
||||
|
|
||||
|
// check that decryption is correct |
||||
|
if !m.Equal(d) { |
||||
|
fmt.Println("decrypted not equal to original") |
||||
|
} |
||||
|
``` |
||||
@ -0,0 +1,56 @@ |
|||||
|
## Paillier cryptosystem & Homomorphic Addition |
||||
|
- https://en.wikipedia.org/wiki/Paillier_cryptosystem |
||||
|
- https://en.wikipedia.org/wiki/Homomorphic_encryption |
||||
|
|
||||
|
- [x] GenerateKeyPair |
||||
|
- [x] Encrypt |
||||
|
- [x] Decrypt |
||||
|
- [x] Homomorphic Addition |
||||
|
|
||||
|
#### Usage |
||||
|
- Encrypt, Decrypt |
||||
|
```go |
||||
|
// key generation |
||||
|
key, err := GenerateKeyPair() |
||||
|
if err!=nil { |
||||
|
fmt.Println(err) |
||||
|
} |
||||
|
|
||||
|
mBytes := []byte("Hi") |
||||
|
m := new(big.Int).SetBytes(mBytes) |
||||
|
|
||||
|
// encryption |
||||
|
c := Encrypt(m, key.PubK) |
||||
|
|
||||
|
// decryption |
||||
|
d := Decrypt(c, key.PubK, key.PrivK) |
||||
|
if m == d { |
||||
|
fmt.Println("ciphertext decrypted correctly") |
||||
|
} |
||||
|
``` |
||||
|
|
||||
|
- Homomorphic Addition |
||||
|
```go |
||||
|
// key generation [Alice] |
||||
|
key, err := GenerateKeyPair() |
||||
|
if err!=nil { |
||||
|
fmt.Println(err) |
||||
|
} |
||||
|
|
||||
|
// define values [Alice] |
||||
|
n1 := big.NewInt(int64(110)) |
||||
|
n2 := big.NewInt(int64(150)) |
||||
|
|
||||
|
// encrypt values [Alice] |
||||
|
c1 := Encrypt(n1, key.PubK) |
||||
|
c2 := Encrypt(n2, key.PubK) |
||||
|
|
||||
|
// compute homomorphic addition [Bob] |
||||
|
c3c4 := HomomorphicAddition(c1, c2, key.PubK) |
||||
|
|
||||
|
// decrypt the result [Alice] |
||||
|
d := Decrypt(c3c4, key.PubK, key.PrivK) |
||||
|
if !bytes.Equal(new(big.Int).Add(n1, n2).Bytes(), d.Bytes()) { |
||||
|
fmt.Println("decrypted result not equal to expected result") |
||||
|
} |
||||
|
``` |
||||
@ -0,0 +1,94 @@ |
|||||
|
## RSA cryptosystem & Blind signature & Homomorphic Multiplication |
||||
|
- https://en.wikipedia.org/wiki/RSA_(cryptosystem)# |
||||
|
- https://en.wikipedia.org/wiki/Blind_signature |
||||
|
- https://en.wikipedia.org/wiki/Homomorphic_encryption |
||||
|
|
||||
|
- [x] GenerateKeyPair |
||||
|
- [x] Encrypt |
||||
|
- [x] Decrypt |
||||
|
- [x] Blind |
||||
|
- [x] Blind Signature |
||||
|
- [x] Unblind Signature |
||||
|
- [x] Verify Signature |
||||
|
- [x] Homomorphic Multiplication |
||||
|
|
||||
|
|
||||
|
#### Usage |
||||
|
- Key generation, Encryption, Decryption |
||||
|
```go |
||||
|
// generate key pair |
||||
|
key, err := GenerateKeyPair() |
||||
|
if err!=nil { |
||||
|
fmt.Println(err) |
||||
|
} |
||||
|
mBytes := []byte("Hi") |
||||
|
m := new(big.Int).SetBytes(mBytes) |
||||
|
|
||||
|
// encrypt message |
||||
|
c := Encrypt(m, key.PubK) |
||||
|
|
||||
|
// decrypt ciphertext |
||||
|
d := Decrypt(c, key.PrivK) |
||||
|
if m == d { |
||||
|
fmt.Println("correctly decrypted") |
||||
|
} |
||||
|
``` |
||||
|
|
||||
|
- Blind signatures |
||||
|
```go |
||||
|
// key generation [Alice] |
||||
|
key, err := GenerateKeyPair() |
||||
|
if err!=nil { |
||||
|
fmt.Println(err) |
||||
|
} |
||||
|
|
||||
|
// create new message [Alice] |
||||
|
mBytes := []byte("Hi") |
||||
|
m := new(big.Int).SetBytes(mBytes) |
||||
|
|
||||
|
// define r value [Alice] |
||||
|
rVal := big.NewInt(int64(101)) |
||||
|
|
||||
|
// blind message [Alice] |
||||
|
mBlinded := Blind(m, rVal, key.PubK) |
||||
|
|
||||
|
// Blind Sign the blinded message [Bob] |
||||
|
sigma := BlindSign(mBlinded, key.PrivK) |
||||
|
|
||||
|
// unblind the blinded signed message, and get the signature of the message [Alice] |
||||
|
mSigned := Unblind(sigma, rVal, key.PubK) |
||||
|
|
||||
|
// verify the signature [Alice/Bob/Trudy] |
||||
|
verified := Verify(m, mSigned, key.PubK) |
||||
|
if !verified { |
||||
|
fmt.Println("signature could not be verified") |
||||
|
} |
||||
|
``` |
||||
|
|
||||
|
- Homomorphic Multiplication |
||||
|
```go |
||||
|
// key generation [Alice] |
||||
|
key, err := GenerateKeyPair() |
||||
|
if err!=nil { |
||||
|
fmt.Println(err) |
||||
|
} |
||||
|
|
||||
|
// define values [Alice] |
||||
|
n1 := big.NewInt(int64(11)) |
||||
|
n2 := big.NewInt(int64(15)) |
||||
|
|
||||
|
// encrypt the values [Alice] |
||||
|
c1 := Encrypt(n1, key.PubK) |
||||
|
c2 := Encrypt(n2, key.PubK) |
||||
|
|
||||
|
// compute homomorphic multiplication with the encrypted values [Bob] |
||||
|
c3c4 := HomomorphicMul(c1, c2, key.PubK) |
||||
|
|
||||
|
// decrypt the result [Alice] |
||||
|
d := Decrypt(c3c4, key.PrivK) |
||||
|
|
||||
|
// check that the result is the expected |
||||
|
if !bytes.Equal(new(big.Int).Mul(n1, n2).Bytes(), d.Bytes()) { |
||||
|
fmt.Println("decrypted result not equal to expected result") |
||||
|
} |
||||
|
``` |
||||
@ -0,0 +1,47 @@ |
|||||
|
## Schnorr signature |
||||
|
- https://en.wikipedia.org/wiki/Schnorr_signature |
||||
|
|
||||
|
- [x] Hash[M || R] (where M is the msg bytes and R is a Point on the ECC, using sha256 hash function) |
||||
|
- [x] Generate Schnorr scheme |
||||
|
- [x] Sign |
||||
|
- [x] Verify signature |
||||
|
|
||||
|
|
||||
|
#### Usage |
||||
|
```go |
||||
|
// define new elliptic curve |
||||
|
ec := ecc.NewEC(big.NewInt(int64(0)), big.NewInt(int64(7)), big.NewInt(int64(11))) |
||||
|
// define new point |
||||
|
g := ecc.Point{big.NewInt(int64(11)), big.NewInt(int64(27))} // Generator |
||||
|
// define new random r |
||||
|
r := big.NewInt(int64(23)) // random r |
||||
|
|
||||
|
// define new Schnorr crypto system using the values |
||||
|
schnorr, sk, err := Gen(ec, g, r) |
||||
|
if err!=nil { |
||||
|
fmt.println(err) |
||||
|
} |
||||
|
|
||||
|
// define message to sign |
||||
|
m := []byte("hola") |
||||
|
|
||||
|
// also we can hash the message, but it's not mandatory, as it will be done inside the schnorr.Sign, but we can perform it now, just to check the function |
||||
|
h := Hash([]byte("hola"), c) |
||||
|
if h.String() != "34719153732582497359642109898768696927847420320548121616059449972754491425079") { |
||||
|
fmt.Println("not correctly hashed") |
||||
|
} |
||||
|
|
||||
|
s, rPoint, err := schnorr.Sign(sk, m) |
||||
|
if err!=nil { |
||||
|
fmt.println(err) |
||||
|
} |
||||
|
|
||||
|
// verify Schnorr signature |
||||
|
verified, err := Verify(schnorr.EC, sk.PubK, m, s, rPoint) |
||||
|
if err!=nil { |
||||
|
fmt.println(err) |
||||
|
} |
||||
|
if verified { |
||||
|
fmt.Println("Schnorr signature correctly verified") |
||||
|
} |
||||
|
``` |
||||
@ -0,0 +1,48 @@ |
|||||
|
## Shamir Secret Sharing |
||||
|
- https://en.wikipedia.org/wiki/Shamir%27s_Secret_Sharing |
||||
|
|
||||
|
- [x] create secret sharing from number of secrets needed, number of shares, random point p, secret to share |
||||
|
- [x] Lagrange Interpolation to restore the secret from the shares |
||||
|
|
||||
|
#### Usage |
||||
|
```go |
||||
|
// define secret to share |
||||
|
k := 123456789 |
||||
|
|
||||
|
// define random prime |
||||
|
p, err := rand.Prime(rand.Reader, bits/2) |
||||
|
if err!=nil { |
||||
|
fmt.Println(err) |
||||
|
} |
||||
|
|
||||
|
// define how many shares want to generate |
||||
|
nShares := big.NewInt(int64(6)) |
||||
|
|
||||
|
// define how many shares are needed to recover the secret |
||||
|
nNeededShares := big.NewInt(int64(3)) |
||||
|
|
||||
|
// create the shares |
||||
|
shares, err := Create( |
||||
|
nNeededShares, |
||||
|
nShares, |
||||
|
p, |
||||
|
big.NewInt(int64(k))) |
||||
|
assert.Nil(t, err) |
||||
|
if err!=nil { |
||||
|
fmt.Println(err) |
||||
|
} |
||||
|
|
||||
|
// select shares to use |
||||
|
var sharesToUse [][]*big.Int |
||||
|
sharesToUse = append(sharesToUse, shares[2]) |
||||
|
sharesToUse = append(sharesToUse, shares[1]) |
||||
|
sharesToUse = append(sharesToUse, shares[0]) |
||||
|
|
||||
|
// recover the secret using Lagrange Interpolation |
||||
|
secr := LagrangeInterpolation(sharesToUse, p) |
||||
|
|
||||
|
// check that the restored secret matches the original secret |
||||
|
if !bytes.Equal(k.Bytes(), secr.Bytes()) { |
||||
|
fmt.Println("reconstructed secret not correspond to original secret") |
||||
|
} |
||||
|
``` |
||||