# 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:

This library presents a low-level interface for building zkSNARK proving systems
from higher-level compilers. This system depends on the following cryptographic
dependenices.

## 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
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 and
as:

The tower of finite fields we work with is defined as:

### Arithmetic circuits

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 be a finite field and a map that takes
arguments as inputs from and outputs l elements in . 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 , , respectively denote the input, witness and output size and
be the number of all inputs and outputs of the circuit
A tuple , is said to be a valid
assignment for an arithmetic circuit C if .

### Quadratic Arithmetic Programs (QAP)

QAPs are encodings of arithmetic circuits that allow the prover to construct a
proof of knowledge of a valid assignment for a given
circuit .

A quadratic arithmetic program (QAP) contains three sets of polynomials in
:

, ,

and a target polynomial .

In this setting, an assignment is valid for a circuit if and
only if the target polynomial divides the polynomial:

Logical circuits can be written in terms of the addition, multiplication and
negation operations.

## DSL and Circuit Builder Monad

Any arithmetic circuit can be built using a domain specific language to
construct circuits that lives inside Lang.hs.

```
type ExprM f a = State (ArithCircuit f, Int) a
execCircuitBuilder :: ExprM f a -> ArithCircuit f
```

```
-- | Binary arithmetic operations
add, sub, mul :: Expr Wire f f -> Expr Wire f f -> Expr Wire f f
```

```
-- | 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
```

```
-- | Negate expression
not_ :: Expr Wire f Bool -> Expr Wire f Bool
```

```
-- | Compare two expressions
eq :: Expr Wire f f -> Expr Wire f f -> Expr Wire f Bool
```

```
-- | Convert wire to expression
deref :: Wire -> Expr Wire f f
```

```
-- | Return compilation of expression into an intermediate wire
e :: Num f => Expr Wire f f -> ExprM f Wire
```

```
-- | Conditional statement on expressions
cond :: Expr Wire f Bool -> Expr Wire f ty -> Expr Wire f ty -> Expr Wire f ty
```

```
-- | 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.

```
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.

```
dotOutput :: Text
dotOutput = arithCircuitToDot (execCircuitBuilder program)
```

## Example

We'll keep taking the program constructed with our DSL as example and will
use the library pairing that
provides a field of points of the BN254 curve and precomputes primitive roots of
unity for binary powers that divide .

```
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 and
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.

```
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.

```
assignment :: QapSet Fr
assignment = generateAssignment program inputs
```

The verifier can check the divisibility property of by for the given circuit.

```
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

```
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of this software and associated documentation files (the "Software"), to deal
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to use, copy, modify, merge, publish, distribute, sublicense, and/or sell
copies of the Software, and to permit persons to whom the Software is
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The above copyright notice and this permission notice shall be included in all
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THE SOFTWARE IS PROVIDED "AS IS", WITHOUT WARRANTY OF ANY KIND,
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IN NO EVENT SHALL THE AUTHORS OR COPYRIGHT HOLDERS BE LIABLE FOR ANY CLAIM,
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