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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.
- pairing - Optimised bilinear pairings over elliptic curves
- galois-field - Finite field arithmetic
- galois-fft - Finite field polynomial arithmetic based on fast Fourier transforms
- elliptic-curve - Elliptic curve operations
- bulletproofs - Bulletproofs proof system
- arithmoi - Number theory operations
- semirings - Algebraic semirings
- poly - Efficient polynomial arithmetic
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:
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](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 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"
This is experimental code meant for research-grade projects only. Please do not use this code in production until it has matured significantly.
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