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# Arithmetic Circuits
An *arithmetic circuit* is a low-level representation of a program that consists
of gates computing arithmetic operations of addition and multiplication, with
wires connecting the gates.
This form allows us to express arbitrarily complex programs with a set of
*private inputs* and *public inputs* whose execution can be publicly verified
without revealing the private inputs. This construction relies on recent
advances in zero-knowledge proving systems:
* Groth16 ([G16](https://eprint.iacr.org/2016/260.pdf))
* Groth-Maller ([GM17](https://eprint.iacr.org/2017/540.pdf))
* Pinnochio ([PGHR13](https://eprint.iacr.org/2013/279.pdf))
* Bulletproofs ([BBBPWM17](https://web.stanford.edu/~buenz/pubs/bulletproofs.pdf))
* Sonic ([MBKM19](https://eprint.iacr.org/2019/099))
This library presents a low-level interface for building zkSNARK proving systems
from higher-level compilers. This system depends on the following cryptographic
dependenices.
* [pairing](https://www.github.com/adjoint-io/pairing) - Optimised bilinear
pairings over elliptic curves
* [galois-field](https://www.github.com/adjoint-io/galois-field) - Finite field
arithmetic
* [galois-fft](https://www.github.com/adjoint-io/galois-fft) - Finite field
polynomial arithmetic based on fast Fourier transforms
* [elliptic-curve](https://www.github.com/adjoint-io/elliptic-curve) - Elliptic
curve operations
* [bulletproofs](https://www.github.com/adjoint-io/bulletproofs) - Bulletproofs
proof system
* [arithmoi](https://www.github.com/adjoint-io/arithmoi) - Number theory
operations
* [semirings](https://www.github.com/adjoint-io/semirings) - Algebraic semirings
* [poly](https://www.github.com/adjoint-io/poly) - Efficient polynomial
arithmetic
## Theory
### Towers of Finite Fields
This library can build proof systems polymorphically over a variety of pairing
friendly curves. By default we use the [BN254](https://github.com/adjoint-io/elliptic-curve/blob/master/src/Data/Curve/Weierstrass/BN254.hs)
with an efficient implementation of the optimal Ate pairing.
The Barreto-Naehrig (BN) family of curves achieve high security and efficiency
with pairings due to an optimum embedding degree and high 2-adicity. We have
implemented the optimal Ate pairing over the BN254 curve we define <img src="/tex/d5c18a8ca1894fd3a7d25f242cbe8890.svg?invert_in_darkmode&sanitize=true" align=middle width=7.928106449999989pt height=14.15524440000002pt/> and <img src="/tex/89f2e0d2d24bcf44db73aab8fc03252c.svg?invert_in_darkmode&sanitize=true" align=middle width=7.87295519999999pt height=14.15524440000002pt/>
as:
* <img src="/tex/f041cced24931de6e6fbe77cf7fdc920.svg?invert_in_darkmode&sanitize=true" align=middle width=221.83192679999993pt height=26.76175259999998pt/>
* <img src="/tex/c0ea121dfdf29950f20f29c86635f197.svg?invert_in_darkmode&sanitize=true" align=middle width=221.77679534999993pt height=26.76175259999998pt/>
* <img src="/tex/71d71cc186d8238f79d664af6a48289f.svg?invert_in_darkmode&sanitize=true" align=middle width=184.01870685pt height=21.18721440000001pt/>
The tower of finite fields we work with is defined as:
* <img src="/tex/66eff114f6c4385dd25cca62457ad776.svg?invert_in_darkmode&sanitize=true" align=middle width=134.80114229999998pt height=26.76175259999998pt/>
* <img src="/tex/661afe19025c1f97ae61a41f91d325e9.svg?invert_in_darkmode&sanitize=true" align=middle width=181.79879189999997pt height=26.76175259999998pt/>
* <img src="/tex/ad84179c73fb68d93d2a147cce2cb19c.svg?invert_in_darkmode&sanitize=true" align=middle width=152.75024985pt height=26.76175259999998pt/>
### Arithmetic circuits
<p align="center">
<img src="./.assets/circuit.png" alt="Arithmetic Circuit" height=300 align="left" />
</p>
An arithmetic circuit over a finite field is a directed acyclic graph with gates
as vertices and wires and edges. It consists of a list of multiplication gates
together with a set of linear consistency equations relating the inputs and
outputs of the gates.
Let <img src="/tex/2d4c6ac334688c42fb4089749e372345.svg?invert_in_darkmode&sanitize=true" align=middle width=10.045686749999991pt height=22.648391699999998pt/> be a finite field and <img src="/tex/c8a92e9bd23d2e4841e72114a69462d7.svg?invert_in_darkmode&sanitize=true" align=middle width=124.1115777pt height=27.91243950000002pt/> a map that takes <img src="/tex/5e27bca98285ab8eccf4d53506baeaec.svg?invert_in_darkmode&sanitize=true" align=middle width=39.42918209999999pt height=22.831056599999986pt/>
arguments as inputs from <img src="/tex/2d4c6ac334688c42fb4089749e372345.svg?invert_in_darkmode&sanitize=true" align=middle width=10.045686749999991pt height=22.648391699999998pt/> and outputs l elements in <img src="/tex/2d4c6ac334688c42fb4089749e372345.svg?invert_in_darkmode&sanitize=true" align=middle width=10.045686749999991pt height=22.648391699999998pt/>. The function C is an arithmetic circuit if the
value of the inputs that pass through wires to gates are only manipulated according to arithmetic operations + or x (allowing
constant gates).
Let <img src="/tex/55a049b8f161ae7cfeb0197d75aff967.svg?invert_in_darkmode&sanitize=true" align=middle width=9.86687624999999pt height=14.15524440000002pt/>, <img src="/tex/2ad9d098b937e46f9f58968551adac57.svg?invert_in_darkmode&sanitize=true" align=middle width=9.47111549999999pt height=22.831056599999986pt/>, <img src="/tex/2f2322dff5bde89c37bcae4116fe20a8.svg?invert_in_darkmode&sanitize=true" align=middle width=5.2283516999999895pt height=22.831056599999986pt/> respectively denote the input, witness and output size and
<img src="/tex/dc35769e37858254d0d77fab2d83bcf4.svg?invert_in_darkmode&sanitize=true" align=middle width=114.45175665pt height=24.65753399999998pt/> be the number of all inputs and outputs of the circuit
A tuple <img src="/tex/2c4cf6568afbabb029a579215dfa6e3e.svg?invert_in_darkmode&sanitize=true" align=middle width=120.09967859999999pt height=27.6567522pt/>, is said to be a valid
assignment for an arithmetic circuit C if <img src="/tex/86c8263cd0afa711b888c85bcb5b03a3.svg?invert_in_darkmode&sanitize=true" align=middle width=241.74771059999995pt height=24.65753399999998pt/>.
### Quadratic Arithmetic Programs (QAP)
QAPs are encodings of arithmetic circuits that allow the prover to construct a
proof of knowledge of a valid assignment <img src="/tex/6e1ad13b9c0521871bb453942700c519.svg?invert_in_darkmode&sanitize=true" align=middle width=120.09967859999999pt height=27.6567522pt/> for a given
circuit <img src="/tex/9b325b9e31e85137d1de765f43c0f8bc.svg?invert_in_darkmode&sanitize=true" align=middle width=12.92464304999999pt height=22.465723500000017pt/>.
A quadratic arithmetic program (QAP) <img src="/tex/9000cff3d46536c190fabb076ebe7cbb.svg?invert_in_darkmode&sanitize=true" align=middle width=38.70549539999999pt height=24.65753399999998pt/> contains three sets of polynomials in
<img src="/tex/a420c52aa24a502d60aef830b3b45f9f.svg?invert_in_darkmode&sanitize=true" align=middle width=28.57312259999999pt height=24.65753399999998pt/>:
<img src="/tex/cf792d8b490521d817a643a4adea6f28.svg?invert_in_darkmode&sanitize=true" align=middle width=184.37011065pt height=24.65753399999998pt/>, <img src="/tex/258504deb4909bd9a3058fffbdb20262.svg?invert_in_darkmode&sanitize=true" align=middle width=185.47456784999997pt height=24.65753399999998pt/>, <img src="/tex/36b962f85e4e4de2a88919a5e00acbe1.svg?invert_in_darkmode&sanitize=true" align=middle width=184.38600014999997pt height=24.65753399999998pt/>
and a target polynomial <img src="/tex/083da1124b81d709f20f2575ae9138c3.svg?invert_in_darkmode&sanitize=true" align=middle width=34.06973294999999pt height=24.65753399999998pt/>.
In this setting, an assignment <img src="/tex/d1dd493c98f06e9ef29b5fdc411e29f8.svg?invert_in_darkmode&sanitize=true" align=middle width=78.31669229999999pt height=24.65753399999998pt/> is valid for a circuit <img src="/tex/9b325b9e31e85137d1de765f43c0f8bc.svg?invert_in_darkmode&sanitize=true" align=middle width=12.92464304999999pt height=22.465723500000017pt/> if and
only if the target polynomial <img src="/tex/083da1124b81d709f20f2575ae9138c3.svg?invert_in_darkmode&sanitize=true" align=middle width=34.06973294999999pt height=24.65753399999998pt/> divides the polynomial:
<img src="/tex/d6b98fe1452c7c16cfd1ad84a2e7331b.svg?invert_in_darkmode&sanitize=true" align=middle width=613.0190187pt height=26.438629799999987pt/>
Logical circuits can be written in terms of the addition, multiplication and
negation operations.
* <img src="/tex/cc073b29543f2694def1c15e2d40cb07.svg?invert_in_darkmode&sanitize=true" align=middle width=110.71153004999998pt height=24.65753399999998pt/>
* <img src="/tex/bf01d477a12bededfd9d65617fedacc9.svg?invert_in_darkmode&sanitize=true" align=middle width=117.37814879999999pt height=24.65753399999998pt/>
* <img src="/tex/e5079841837a9e98336245a5ecdb2538.svg?invert_in_darkmode&sanitize=true" align=middle width=151.35072315pt height=24.65753399999998pt/>
* <img src="/tex/065fdfe9ba3a8e9e839cea21d15eb1cd.svg?invert_in_darkmode&sanitize=true" align=middle width=221.2135992pt height=24.65753399999998pt/>
* <img src="/tex/85a105718538df533c60d0d34caeac95.svg?invert_in_darkmode&sanitize=true" align=middle width=187.1858703pt height=24.65753399999998pt/>
## DSL and Circuit Builder Monad
Any arithmetic circuit can be built using a domain specific language to
construct circuits that lives inside [Lang.hs](src/Circuit/Lang.hs).
```haskell ignore
type ExprM f a = State (ArithCircuit f, Int) a
execCircuitBuilder :: ExprM f a -> ArithCircuit f
```
```haskell ignore
-- | Binary arithmetic operations
add, sub, mul :: Expr Wire f f -> Expr Wire f f -> Expr Wire f f
```
```haskell ignore
-- | Binary logic operations
-- Have to use underscore or similar to avoid shadowing @and@ and @or@
-- from Prelude/Protolude.
and_, or_, xor_ :: Expr Wire f Bool -> Expr Wire f Bool -> Expr Wire f Bool
```
```haskell ignore
-- | Negate expression
not_ :: Expr Wire f Bool -> Expr Wire f Bool
```
```haskell ignore
-- | Compare two expressions
eq :: Expr Wire f f -> Expr Wire f f -> Expr Wire f Bool
```
```haskell ignore
-- | Convert wire to expression
deref :: Wire -> Expr Wire f f
```
```haskell ignore
-- | Return compilation of expression into an intermediate wire
e :: Num f => Expr Wire f f -> ExprM f Wire
```
```haskell ignore
-- | Conditional statement on expressions
cond :: Expr Wire f Bool -> Expr Wire f ty -> Expr Wire f ty -> Expr Wire f ty
```
```haskell ignore
-- | Return compilation of expression into an output wire
ret :: Num f => Expr Wire f f -> ExprM f Wire
```
The following program represents the image of the
arithmetic circuit [above](#arithmetic-circuits-1).
```haskell ignore
program :: ArithCircuit Fr
program = execCircuitBuilder (do
i0 <- fmap deref input
i1 <- fmap deref input
i2 <- fmap deref input
let r0 = mul i0 i1
r1 = mul r0 (add i0 i2)
ret r1)
```
The output of an arithmetic circuit can be converted to a DOT graph and save it
as SVG.
```haskell ignore
dotOutput :: Text
dotOutput = arithCircuitToDot (execCircuitBuilder program)
```
<p>
<img src=".assets/arithmetic-circuit-example.svg" width="250"/>
</p>
## Example
We'll keep taking the program constructed with our DSL as example and will
use the library [pairing](https://www.github.com/adjoint-io/pairing) that
provides a field of points of the BN254 curve and precomputes primitive roots of
unity for binary powers that divide <img src="/tex/580e7a6446bf50562e34247c545883a2.svg?invert_in_darkmode&sanitize=true" align=middle width=36.18335654999999pt height=21.18721440000001pt/>.
```haskell
import Protolude
import qualified Data.Map as Map
import Data.Pairing.BN254 (Fr, getRootOfUnity)
import Circuit.Arithmetic
import Circuit.Expr
import Circuit.Lang
import Fresh (evalFresh, fresh)
import QAP
program :: ArithCircuit Fr
program = execCircuitBuilder (do
i0 <- fmap deref input
i1 <- fmap deref input
i2 <- fmap deref input
let r0 = mul i0 i1
r1 = mul r0 (add i0 i2)
ret r1)
```
We need to generate the roots of the circuit to construct polynomials <img src="/tex/083da1124b81d709f20f2575ae9138c3.svg?invert_in_darkmode&sanitize=true" align=middle width=34.06973294999999pt height=24.65753399999998pt/> and
<img src="/tex/52be0087c9da1f0683ccc50761e8bcab.svg?invert_in_darkmode&sanitize=true" align=middle width=35.01719264999999pt height=24.65753399999998pt/> that satisfy the divisibility property and encode the circuit to a QAP to
allow the prover to construct a proof of a valid assignment.
We also need to give values to the three input wires to this arithmetic circuit.
```haskell
roots :: [[Fr]]
roots = evalFresh (generateRoots (fmap (fromIntegral . (+ 1)) fresh) program)
qap :: QAP Fr
qap = arithCircuitToQAPFFT getRootOfUnity roots program
inputs :: Map.Map Int Fr
inputs = Map.fromList [(0, 7), (1, 5), (2, 4)]
```
A prover can now generate a valid assignment.
```haskell
assignment :: QapSet Fr
assignment = generateAssignment program inputs
```
The verifier can check the divisibility property of <img src="/tex/52be0087c9da1f0683ccc50761e8bcab.svg?invert_in_darkmode&sanitize=true" align=middle width=35.01719264999999pt height=24.65753399999998pt/> by <img src="/tex/083da1124b81d709f20f2575ae9138c3.svg?invert_in_darkmode&sanitize=true" align=middle width=34.06973294999999pt height=24.65753399999998pt/> for the given circuit.
```haskell
main :: IO ()
main = do
if verifyAssignment qap assignment
then putText "Valid assignment"
else putText "Invalid assignment"
```
## Disclaimer
This is experimental code meant for research-grade projects only. Please do not
use this code in production until it has matured significantly.
## License
```
Copyright (c) 2017-2020 Adjoint Inc.
Permission is hereby granted, free of charge, to any person obtaining a copy
of this software and associated documentation files (the "Software"), to deal
in the Software without restriction, including without limitation the rights
to use, copy, modify, merge, publish, distribute, sublicense, and/or sell
copies of the Software, and to permit persons to whom the Software is
furnished to do so, subject to the following conditions:
The above copyright notice and this permission notice shall be included in all
copies or substantial portions of the Software.
THE SOFTWARE IS PROVIDED "AS IS", WITHOUT WARRANTY OF ANY KIND,
EXPRESS OR IMPLIED, INCLUDING BUT NOT LIMITED TO THE WARRANTIES OF
MERCHANTABILITY, FITNESS FOR A PARTICULAR PURPOSE AND NONINFRINGEMENT.
IN NO EVENT SHALL THE AUTHORS OR COPYRIGHT HOLDERS BE LIABLE FOR ANY CLAIM,
DAMAGES OR OTHER LIABILITY, WHETHER IN AN ACTION OF CONTRACT, TORT OR
OTHERWISE, ARISING FROM, OUT OF OR IN CONNECTION WITH THE SOFTWARE OR THE USE
OR OTHER DEALINGS IN THE SOFTWARE.
```