arithmetic-circuits crate

[Cesar199999]

Description

This PR adds a new crate, arithmetic-circuits, which contains the structures ArithmeticCircuit and Expression, as well as methods for compiling R1CS constraint systems (ark_relations::r1cs::ConstraintSystem) into arithmetic circuits.
An ArithmeticCircuit is a directed acyclic graph where nodes are either variables, constants, or gates for addition and multiplication over a prime field F. The circuit is internally represented as a list of nodes, each being a variant of the enum Node: Variable, Constant, Add, and Mul. Gates have a left and right input, which are indices for the respective nodes in the array. In order to directly construct an arithmetic circuit, the user must instantiate an empty ArithmeticCircuit struct and mutate it via its public methods (e.g. new_variable, add, mul), each of which returns the index of the new node in the array. For example, an arithmetic circuit expressing the computation 2 * x + y can be constructed as follows:

let mut circuit = ArithmeticCircuit::new();
let two = circuit.constant(F::from(2));
let x = circuit.new_variable();
let y = circuit.new_variable();
let two_x = circuit.mul(two, x);
let result = circuit.add(two_x, y);

Since the construction of an arithmetic circuit can be slightly verbose, the crate also provides the struct Expression, which allows for a more user-friendly way to construct arithmetic circuits. Manually adding nodes to the circuit is not required: expressions can be combined using the usual arithmetic operators, and then converted to a circuit using the to_arithmetic_circuit method. For reference, the previous circuit can be expressed as follows:

let x = Expression::variable("x");
let y = Expression::variable("y");
let circuit = (2 * x + y).to_arithmetic_circuit();

Note that the variables must be labelled uniquely.

Lastly, the crate provides methods for compiling R1CS constraint systems from circom generated .r1cs and .wasm files into arithmetic circuits.

Discussion NP-Eng/ligero#25

[mmagician]

@Pratyush, as per the discussion in NP-Eng/ligero#25 (comment), this PR adds the arithmetic circuit functionality, ready for review when you get some cycles

[Pratyush]

Do you think this would be a better fit in ark-relations? Since r1cs (another satisfiability relation) lives in there?

[Cesar199999]

@Pratyush Sure, that makes sense