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)
• Groth-Maller (GM17)
• Pinnochio (PGHR13)
• Bulletproofs (BBBPWM17)
• Sonic (MBKM19)

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

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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