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# libsnark: a C++ library for zkSNARK proofs

## 🚩🚩🚩 This repository is not maintained 🚩🚩🚩

This is the fork of libsnark that was used by Zcash up to the point at which zcashd stopped verifying BCTV14/BN254 proofs in 2019. It is retained purely for historical reference, and we would not recommend using it in any deployment.

## Authors

The libsnark library is developed by the [SCIPR Lab] project and contributors and is released under the MIT License (see the [LICENSE] file).

Copyright (c) 2012-2014 SCIPR Lab and contributors (see [AUTHORS] file).

[TOC]

## Overview

This library implements **zkSNARK** schemes, which are a cryptographic method
for proving/verifying, in zero knowledge, the integrity of computations.

A computation can be expressed as an NP statement, in forms such as the following:

- "The C program
*foo*, when executed, returns exit code 0 if given the input*bar*and some additional input*qux*." - "The Boolean circuit
*foo*is satisfiable by some input*qux*." - "The arithmetic circuit
*foo*accepts the partial assignment*bar*, when extended into some full assignment*qux*." - "The set of constraints
*foo*is satisfiable by the partial assignment*bar*, when extended into some full assignment*qux*."

A prover who knows the witness for the NP statement (i.e., a satisfying input/assignment) can produce a short proof attesting to the truth of the NP statement. This proof can be verified by anyone, and offers the following properties.

**Zero knowledge:**the verifier learns nothing from the proof beside the truth of the statement (i.e., the value*qux*, in the above examples, remains secret).**Succinctness:**the proof is short and easy to verify.**Non-interactivity:**the proof is a string (i.e. it does not require back-and-forth interaction between the prover and the verifier).**Soundness:**the proof is computationally sound (i.e., it is infeasible to fake a proof of a false NP statement). Such a proof system is also called an*argument*.**Proof of knowledge:**the proof attests not just that the NP statement is true, but also that the prover knows why (e.g., knows a valid*qux*).

These properties are summarized by the *zkSNARK* acronym, which stands for *Zero-Knowledge Succinct Non-interactive ARgument of Knowledge* (though zkSNARKs are also knows as
*succinct non-interactive computationally-sound zero-knowledge proofs of knowledge*).
For formal definitions and theoretical discussions about these, see

```
BCCT12], \[BCIOP13], and the references therein.
The libsnark library currently provides a C++ implementation of:
1. General-purpose proof systems:
1. A preprocessing zkSNARK for the NP-complete language "R1CS"
(_Rank-1 Constraint Systems_), which is a language that is similar to arithmetic
circuit satisfiability.
2. A preprocessing SNARK for a language of arithmetic circuits, "BACS"
(_Bilinear Arithmetic Circuit Satisfiability_). This simplifies the writing
of NP statements when the additional flexibility of R1CS is not needed.
Internally, it reduces to R1CS.
3. A preprocessing SNARK for the language "USCS"
(_Unitary-Square Constraint Systems_). This abstracts and implements the core
contribution of \[DFGK14]
4. A preprocessing SNARK for a language of Boolean circuits, "TBCS"
(_Two-input Boolean Circuit Satisfiability_). Internally, it reduces to USCS.
This is much more efficient than going through R1CS.
5. ADSNARK, a preprocessing SNARKs for proving statements on authenticated
data, as described in \[BBFR15].
6. Proof-Carrying Data (PCD). This uses recursive composition of SNARKs, as
explained in \[BCCT13] and optimized in \[BCTV14b].
2. Gadget libraries (gadgetlib1 and gadgetlib2) for constructing R1CS
instances out of modular "gadget" classes.
3. Examples of applications that use the above proof systems to prove
statements about:
1. Several toy examples.
2. Execution of TinyRAM machine code, as explained in \[BCTV14a] and
\[BCGTV13]. (Such machine code can be obtained, e.g., by compiling from C.)
This is easily adapted to any other Random Access Machine that satisfies a
simple load-store interface.
3. A scalable for TinyRAM using Proof-Carrying Data, as explained in \[BCTV14b]
4. Zero-knowldge cluster MapReduce, as explained in \[CTV15].
The zkSNARK construction implemented by libsnark follows, extends, and
optimizes the approach described in \[BCTV14], itself an extension of
\[BCGTV13], following the approach of \[BCIOP13] and \[GGPR13]. An alternative
implementation of the basic approach is the _Pinocchio_ system of \[PGHR13].
See these references for discussions of efficiency aspects that arise in
practical use of such constructions, as well as security and trust
considerations.
This scheme is a _preprocessing zkSNARK_ (_ppzkSNARK_): before proofs can be
created and verified, one needs to first decide on a size/circuit/system
representing the NP statements to be proved, and run a _generator_ algorithm to
create corresponding public parameters (a long proving key and a short
verification key).
Using the library involves the following high-level steps:
1. Express the statements to be proved as an R1CS (or any of the other
languages above, such as arithmetic circuits, Boolean circuits, or TinyRAM).
This is done by writing C++ code that constructs an R1CS, and linking this code
together with libsnark
2. Use libsnark's generator algorithm to create the public parameters for this
statement (once and for all).
3. Use libsnark's prover algorithm to create proofs of true statements about
the satisfiability of the R1CS.
4. Use libsnark's verifier algorithm to check proofs for alleged statements.
--------------------------------------------------------------------------------
The NP-complete language R1CS
--------------------------------------------------------------------------------
The ppzkSNARK supports proving/verifying membership in a specific NP-complete
language: R1CS (*rank-1 constraint systems*). An instance of the language is
specified by a set of equations over a prime field F, and each equation looks like:
< A, (1,X) > * < B , (1,X) > = < C, (1,X) >
where A,B,C are vectors over F, and X is a vector of variables.
In particular, arithmetic (as well as boolean) circuits are easily reducible to
this language by converting each gate into a rank-1 constraint. See \[BCGTV13]
Appendix E (and "System of Rank 1 Quadratic Equations") for more details about this.
--------------------------------------------------------------------------------
Elliptic curve choices
--------------------------------------------------------------------------------
The ppzkSNARK can be instantiated with different parameter choices, depending on
which elliptic curve is used. The libsnark library currently provides three
options:
* "edwards":
an instantiation based on an Edwards curve, providing 80 bits of security.
* "bn128":
an instantiation based on a Barreto-Naehrig curve, providing 128
bits of security. The underlying curve implementation is
\[ate-pairing], which has incorporated our patch that changes the
BN curve to one suitable for SNARK applications.
* This implementation uses dynamically-generated machine code for the curve
arithmetic. Some modern systems disallow execution of code on the heap, and
will thus block this implementation.
For example, on Fedora 20 at its default settings, you will get the error
`zmInit ERR:can't protect` when running this code. To solve this,
run `sudo setsebool -P allow_execheap 1` to allow execution,
or use `make CURVE=ALT_BN128` instead.
* "alt_bn128":
an alternative to "bn128", somewhat slower but avoids dynamic code generation.
Note that bn128 requires an x86-64 CPU while the other curve choices
should be architecture-independent; see [portability](#portability).
--------------------------------------------------------------------------------
Gadget libraries
--------------------------------------------------------------------------------
The libsnark library currently provides two libraries for conveniently constructing
R1CS instances out of reusable "gadgets". Both libraries provide a way to construct
gadgets on other gadgets as well as additional explicit equations. In this way,
complex R1CS instances can be built bottom up.
### gadgetlib1
This is a low-level library which expose all features of the preprocessing
zkSNARK for R1CS. Its design is based on templates (as does the ppzkSNARK code)
to efficiently support working on multiple elliptic curves simultaneously. This
library is used for most of the constraint-building in libsnark, both internal
(reductions and Proof-Carrying Data) and examples applications.
### gadgetlib2
This is an alternative library for constructing systems of polynomial equations
and, in particular, also R1CS instances. It is better documented and easier to
use than gadgetlib1, and its interface does not use templates. However, fewer
useful gadgets are provided.
--------------------------------------------------------------------------------
Security
--------------------------------------------------------------------------------
The theoretical security of the underlying mathematical constructions, and the
requisite assumptions, are analyzed in detailed in the aforementioned research
papers.
**
This code is a research-quality proof of concept, and has not
yet undergone extensive review or testing. It is thus not suitable,
as is, for use in critical or production systems.
**
Known issues include the following:
* The ppzkSNARK's generator and prover exhibit data-dependent running times
and memory usage. These form timing and cache-contention side channels,
which may be an issue in some applications.
* Randomness is retrieved from /dev/urandom, but this should be
changed to a carefully considered (depending on system and threat
model) external, high-quality randomness source when creating
long-term proving/verification keys.
--------------------------------------------------------------------------------
Build instructions
--------------------------------------------------------------------------------
The libsnark library relies on the following:
- C++ build environment
- GMP for certain bit-integer arithmetic
- libprocps for reporting memory usage
- GTest for some of the unit tests
So far we have tested these only on Linux, though we have been able to make the library work,
with some features disabled (such as memory profiling or GTest tests), on Windows via Cygwin
and on Mac OS X. (If you succeed in achieving more complete ports of the library, please
let us know!) See also the notes on [portability](#portability) below.
For example, on a fresh install of Ubuntu 14.04, install the following packages:
$ sudo apt-get install build-essential git libgmp3-dev libprocps3-dev libgtest-dev python-markdown libboost-all-dev libssl-dev
Or, on Fedora 20:
$ sudo yum install gcc-c++ make git gmp-devel procps-ng-devel gtest-devel python-markdown
Run the following, to fetch dependencies from their GitHub repos and compile them.
(Not required if you set `CURVE` to other than the default `BN128` and also set `NO_SUPERCOP=1`.)
$ ./prepare-depends.sh
Then, to compile the library, tests, profiling harness and documentation, run:
$ make
To create just the HTML documentation, run
$ make doc
and then view the resulting `README.html` (which contains the very text you are reading now).
To create Doxygen documentation summarizing all files, classes and functions,
with some (currently sparse) comments, install the `doxygen` and `graphviz` packages, then run
$ make doxy
(this may take a few minutes). Then view the resulting [`doxygen/index.html`](doxygen/index.html).
### Using libsnark as a library
To develop an application that uses libsnark, you could add it within the libsnark directory tree and adjust the Makefile, but it is far better to build libsnark as a (shared or static) library. You can then write your code in a separate directory tree, and link it against libsnark.
To build just the shared object library `libsnark.so`, run:
$ make lib
To build just the static library `libsnark.a`, run:
$ make lib STATIC=1
Note that static compilation requires static versions of all libraries it depends on.
It may help to minize these dependencies by appending
`CURVE=ALT_BN128 NO_PROCPS=1 NO_GTEST=1 NO_SUPERCOP=1`. On Fedora 21, the requisite
library RPM dependencies are then:
`boost-static glibc-static gmp-static libstdc++-static openssl-static zlib-static
boost-devel glibc-devel gmp-devel gmp-devel libstdc++-devel openssl-devel openssl-devel`.
To build *and install* the libsnark library:
$ make install PREFIX=/install/path
This will install `libsnark.so` into `/install/path/lib`; so your application should be linked using `-L/install/path/lib -lsnark`. It also installs the requisite headers into `/install/path/include`; so your application should be compiled using `-I/install/path/include`.
In addition, unless you use `NO_SUPERCOP=1`, `libsupercop.a` will be installed and should be linked in using `-lsupercop`.
### Building on Windows using Cygwin
Install Cygwin using the graphical installer, including the `g++`, `libgmp`
and `git` packages. Then disable the dependencies not easily supported under CygWin,
using:
$ make NO_PROCPS=1 NO_GTEST=1 NO_DOCS=1
### Building on Mac OS X
On Mac OS X, install GMP from MacPorts (`port install gmp`). Then disable the
dependencies not easily supported under CygWin, using:
$ make NO_PROCPS=1 NO_GTEST=1 NO_DOCS=1
MacPorts does not write its libraries into standard system folders, so you
might need to explicitly provide the paths to the header files and libraries by
appending `CXXFLAGS=-I/opt/local/include LDFLAGS=-L/opt/local/lib` to the line
above. Similarly, to pass the paths to ate-pairing you would run
`INC_DIR=-I/opt/local/include LIB_DIR=-L/opt/local/lib ./prepare-depends.sh`
instead of `./prepare-depends.sh` above.
--------------------------------------------------------------------------------
Tutorials
--------------------------------------------------------------------------------
libsnark includes a tutorial, and some usage examples, for the high-level API.
* `src/gadgetlib1/examples1` contains a simple example for constructing a
constraint system using gadgetlib1.
* `src/gadgetlib2/examples` contains a tutorial for using gadgetlib2 to express
NP statements as constraint systems. It introduces basic terminology, design
overview, and recommended programming style. It also shows how to invoke
ppzkSNARKs on such constraint systems. The main file, `tutorial.cpp`, builds
into a standalone executable.
* `src/zk_proof_systems/ppzksnark/r1cs_ppzksnark/profiling/profile_r1cs_ppzksnark.cpp`
constructs a simple constraint system and runs the ppzksnark. See below for how to
run it.
--------------------------------------------------------------------------------
Executing profiling example
--------------------------------------------------------------------------------
The command
$ src/zk_proof_systems/ppzksnark/r1cs_ppzksnark/profiling/profile_r1cs_ppzksnark 1000 10 Fr
exercises the ppzkSNARK (first generator, then prover, then verifier) on an
R1CS instance with 1000 equations and an input consisting of 10 field elements.
(If you get the error `zmInit ERR:can't protect`, see the discussion
[above](#elliptic-curve-choices).)
The command
$ src/zk_proof_systems/ppzksnark/r1cs_ppzksnark/profiling/profile_r1cs_ppzksnark 1000 10 bytes
does the same but now the input consists of 10 bytes.
--------------------------------------------------------------------------------
Build options
--------------------------------------------------------------------------------
The following flags change the behavior of the compiled code.
* `make FEATUREFLAGS='-Dname1 -Dname2 ...'`
Override the active conditional #define names (you can see the default at the top of the Makefile).
The next bullets list the most important conditionally-#defined features.
For example, `make FEATUREFLAGS='-DBINARY_OUTPUT'` enables binary output and disables the default
assembly optimizations and Montgomery-representation output.
* define `BINARY_OUTPUT`
In serialization, output raw binary data (instead of decimal, when not set).
* `make CURVE=choice` / define `CURVE_choice` (where `choice` is one of:
ALT_BN128, BN128, EDWARDS, MNT4, MNT6)
Set the default curve to one of the above (see [elliptic curve choices](#elliptic-curve-choices)).
* `make DEBUG=1` / define `DEBUG`
Print additional information for debugging purposes.
* `make LOWMEM=1` / define `LOWMEM`
Limit the size of multi-exponentiation tables, for low-memory platforms.
* `make NO_DOCS=1`
Do not generate HTML documentation, e.g. on platforms where Markdown is not easily available.
* `make NO_PROCPS=1`
Do not link against libprocps. This disables memory profiling.
* `make NO_GTEST=1`
Do not link against GTest. The tutorial and test suite of gadgetlib2 tutorial won't be compiled.
* `make NO_SUPERCOP=1`
Do not link against SUPERCOP for optimized crypto. The ADSNARK executables will not be built.
* `make MULTICORE=1`
Enable parallelized execution of the ppzkSNARK generator and prover, using OpenMP.
This will utilize all cores on the CPU for heavyweight parallelizabe operations such as
FFT and multiexponentiation. The default is single-core.
To override the maximum number of cores used, set the environment variable `OMP_NUM_THREADS`
at runtime (not compile time), e.g., `OMP_NUM_THREADS=8 test_r1cs_sp_ppzkpc`. It defaults
to the autodetected number of cores, but on some devices, dynamic core management confused
OpenMP's autodetection, so setting `OMP_NUM_THREADS` is necessary for full utilization.
* define `NO_PT_COMPRESSION`
Do not use point compression.
This gives much faster serialization times, at the expense of ~2x larger
sizes for serialized keys and proofs.
* define `MONTGOMERY_OUTPUT` (on by default)
Serialize Fp elements as their Montgomery representations. If this
option is disabled then Fp elements are serialized as their
equivalence classes, which is slower but produces human-readable
output.
* `make PROFILE_OP_COUNTS=1` / define `PROFILE_OP_COUNTS`
Collect counts for field and curve operations inside static variables
of the corresponding algebraic objects. This option works for all
curves except bn128.
* define `USE_ASM` (on by default)
Use unrolled assembly routines for F[p] arithmetic and faster heap in
multi-exponentiation. (When not set, use GMP's `mpn_*` routines instead.)
* define `USE_MIXED_ADDITION`
Convert each element of the proving key and verification key to
affine coordinates. This allows using mixed addition formulas in
multiexponentiation and results in slightly faster prover and
verifier runtime at expense of increased proving time.
* `make PERFORMANCE=1`
Enables compiler optimizations such as link-time optimization, and disables debugging aids.
(On some distributions this causes a `plugin needed to handle lto object` link error and `undefined reference`s, which can be remedied by `AR=gcc-ar make ...`.)
Not all combinations are tested together or supported by every part of the codebase.
--------------------------------------------------------------------------------
Portability
--------------------------------------------------------------------------------
libsnark is written in fairly standard C++11.
However, having been developed on Linux on x86-64 CPUs, libsnark has some limitations
with respect to portability. Specifically:
1. libsnark's algebraic data structures assume little-endian byte order.
2. Profiling routines use `clock_gettime` and `readproc` calls, which are Linux-specific.
3. Random-number generation is done by reading from `/dev/urandom`, which is
specific to Unix-like systems.
4. libsnark binary serialization routines (see `BINARY_OUTPUT` above) assume
a fixed machine word size (i.e. sizeof(mp_limb_t) for GMP's limb data type).
Objects serialized in binary on a 64-bit system cannot be de-serialized on
a 32-bit system, and vice versa.
(The decimal serialization routines have no such limitation.)
5. libsnark requires a C++ compiler with good C++11 support. It has been
tested with g++ 4.7, g++ 4.8, and clang 3.4.
6. On x86-64, we by default use highly optimized assembly implementations for some
operations (see `USE_ASM` above). On other architectures we fall back to a
portable C++ implementation, which is slower.
Tested configurations include:
* Debian jessie with g++ 4.7 on x86-64
* Debian jessie with clang 3.4 on x86-64
* Fedora 20/21 with g++ 4.8.2/4.9.2 on x86-64 and i686
* Ubuntu 14.04 LTS with g++ 4.8 on x86-64
* Ubuntu 14.04 LTS with g++ 4.8 on x86-32, for EDWARDS and ALT_BN128 curve choices
* Debian wheezy with g++ 4.7 on ARM little endian (Debian armel port) inside QEMU, for EDWARDS and ALT_BN128 curve choices
* Windows 7 with g++ 4.8.3 under Cygwin 1.7.30 on x86-64 with NO_PROCPS=1, NO_GTEST=1 and NO_DOCS=1, for EDWARDS and ALT_BN128 curve choices
* Mac OS X 10.9.4 (Mavericks) with Apple LLVM version 5.1 (based on LLVM 3.4svn) on x86-64 with NO_PROCPS=1, NO_GTEST=1 and NO_DOCS=1
--------------------------------------------------------------------------------
Directory structure
--------------------------------------------------------------------------------
The directory structure of the libsnark library is as follows:
* src/ --- main C++ source code, containing the following modules:
* algebra/ --- fields and elliptic curve groups
* common/ --- miscellaneous utilities
* gadgetlib1/ --- gadgetlib1, a library to construct R1CS instances
* gadgets/ --- basic gadgets for gadgetlib1
* gadgetlib2/ --- gadgetlib2, a library to construct R1CS instances
* qap/ --- quadratic arithmetic program
* domains/ --- support for fast interpolation/evaluation, by providing
FFTs and Lagrange-coefficient computations for various domains
* relations/ --- interfaces for expressing statement (relations between instances and witnesses) as various NP-complete languages
* constraint_satisfaction_problems/ --- R1CS and USCS languages
* circuit_satisfaction_problems/ --- Boolean and arithmetic circuit satisfiability languages
* ram_computations/ --- RAM computation languages
* zk_proof_systems --- interfaces and implementations of the proof systems
* reductions --- reductions between languages (used internally, but contains many examples of building constraints)
Some of these module directories have the following subdirectories:
* ...
* examples/ --- example code and tutorials for this module
* tests/ --- unit tests for this module
In particular, the top-level API examples are at `src/r1cs_ppzksnark/examples/` and `src/gadgetlib2/examples/`.
* depsrc/ --- created by `prepare_depends.sh` for retrieved sourcecode and local builds of external code
(currently: \[ate-pairing], and its dependency xbyak).
* depinst/ --- created by `prepare_depends.sh` and `Makefile`
for local installation of locally-compiled dependencies.
* doxygen/ --- created by `make doxy` and contains a Doxygen summary of all files, classes etc. in libsnark.
--------------------------------------------------------------------------------
Further considerations
--------------------------------------------------------------------------------
### Multiexponentiation window size
The ppzkSNARK's generator has to solve a fixed-base multi-exponentiation
problem. We use a window-based method in which the optimal window size depends
on the size of the multiexponentiation instance *and* the platform.
On our benchmarking platform (a 3.40 GHz Intel Core i7-4770 CPU), we have
computed for each curve optimal windows, provided as
"fixed_base_exp_window_table" initialization sequences, for each curve; see
`X_init.cpp` for X=edwards,bn128,alt_bn128.
Performance on other platforms may not be optimal (but probably not be far off).
Future releases of the libsnark library will include a tool that generates
optimal window sizes.
--------------------------------------------------------------------------------
References
--------------------------------------------------------------------------------
\[BBFR15] [
_ADSNARK: nearly practical and privacy-preserving proofs on authenticated data_
](https://eprint.iacr.org/2014/617),
Michael Backes, Manuel Barbosa, Dario Fiore, Raphael M. Reischuk,
IEEE Symposium on Security and Privacy (Oakland) 2015
\[BCCT12] [
_From extractable collision resistance to succinct non-Interactive arguments of knowledge, and back again_
](http://eprint.iacr.org/2011/443),
Nir Bitansky, Ran Canetti, Alessandro Chiesa, Eran Tromer,
Innovations in Computer Science (ITCS) 2012
\[BCCT13] [
_Recursive composition and bootstrapping for SNARKs and proof-carrying data_
](http://eprint.iacr.org/2012/095)
Nir Bitansky, Ran Canetti, Alessandro Chiesa, Eran Tromer,
Symposium on Theory of Computing (STOC) 13
\[BCGTV13] [
_SNARKs for C: Verifying Program Executions Succinctly and in Zero Knowledge_
](http://eprint.iacr.org/2013/507),
Eli Ben-Sasson, Alessandro Chiesa, Daniel Genkin, Eran Tromer, Madars Virza,
CRYPTO 2013
\[BCIOP13] [
_Succinct Non-Interactive Arguments via Linear Interactive Proofs_
](http://eprint.iacr.org/2012/718),
Nir Bitansky, Alessandro Chiesa, Yuval Ishai, Rafail Ostrovsky, Omer Paneth,
Theory of Cryptography Conference 2013
\[BCTV14a] [
_Succinct Non-Interactive Zero Knowledge for a von Neumann Architecture_
](http://eprint.iacr.org/2013/879),
Eli Ben-Sasson, Alessandro Chiesa, Eran Tromer, Madars Virza,
USENIX Security 2014
\[BCTV14b] [
_Scalable succinct non-interactive arguments via cycles of elliptic curves_
](https://eprint.iacr.org/2014/595),
Eli Ben-Sasson, Alessandro Chiesa, Eran Tromer, Madars Virza,
CRYPTO 2014
\[CTV15] [
_Cluster computing in zero knowledge_
](https://eprint.iacr.org/2015/377),
Alessandro Chiesa, Eran Tromer, Madars Virza,
Eurocrypt 2015
\[DFGK14] [
Square span programs with applications to succinct NIZK arguments
](https://eprint.iacr.org/2014/718),
George Danezis, Cedric Fournet, Jens Groth, Markulf Kohlweiss,
ASIACCS 2014
\[GGPR13] [
_Quadratic span programs and succinct NIZKs without PCPs_
](http://eprint.iacr.org/2012/215),
Rosario Gennaro, Craig Gentry, Bryan Parno, Mariana Raykova,
EUROCRYPT 2013
\[ate-pairing] [
_High-Speed Software Implementation of the Optimal Ate Pairing over Barreto-Naehrig Curves_
](https://github.com/herumi/ate-pairing),
MITSUNARI Shigeo, TERUYA Tadanori
\[PGHR13] [
_Pinocchio: Nearly Practical Verifiable Computation_
](http://eprint.iacr.org/2013/279),
Bryan Parno, Craig Gentry, Jon Howell, Mariana Raykova,
IEEE Symposium on Security and Privacy (Oakland) 2013
[SCIPR Lab]: http://www.scipr-lab.org/ (Succinct Computational Integrity and Privacy Research Lab)
[LICENSE]: LICENSE (LICENSE file in top directory of libsnark distribution)
[AUTHORS]: AUTHORS (AUTHORS file in top directory of libsnark distribution)
```