[−][src]Module bulletproofs::r1cs
The rank1 constraint system API for programmatically defining constraint systems.
Building a proofofshuffle constraint system
A shuffle is a permutation of a list of \(k\) scalars \(x_i\) into a list of \(k\) scalars \(y_i\).
Algebraically it can be expressed as a statement that for a free variable \(z\), the roots of the two polynomials in terms of \(z\) are the same up to a permutation:
\[ \prod_i (x_i  z) = \prod_i (y_i  z) \]
The prover can commit to blinded scalars \(x_i\) and \(y_i\) then receive a random challenge \(z\), and build a proof that the above relation holds.
Kshuffle requires \( 2*(K1) \) multipliers.
For K > 1
:
(x_0  z)⊗⊗(y_0  z) // mulx[0], muly[0]
 
(x_1  z)⊗ ⊗(y_1  z) // mulx[1], muly[1]
 
... ...
 
(x_{k2}  z)⊗ ⊗(y_{k2}  z) // mulx[k2], muly[k2]
/ \
(x_{k1}  z)_/ \_(y_{k1}  z)
Connect the left and right sides of the shuffle statement:
mulx_out[0] = muly_out[0]
For i == [0, k3]
:
mulx_left[i] = x_i  z
mulx_right[i] = mulx_out[i+1]
muly_left[i] = y_i  z
muly_right[i] = muly_out[i+1]
The last multipliers connect the two last variables (on each side)
mulx_left[k2] = x_{k2}  z
mulx_right[k2] = x_{k1}  z
muly_left[k2] = y_{k2}  z
muly_right[k2] = y_{k1}  z
For K = 1
:
Connect x_0
to y_0
directly. Since there is only one permuatation of a 1element list, we can omit the challenge entirely as it cancels out.
x_0 = y_0
Code for creating constraints for a proofofshuffle constraint system:
extern crate bulletproofs; extern crate curve25519_dalek; extern crate merlin; extern crate rand; use bulletproofs::r1cs::*; use bulletproofs::{BulletproofGens, PedersenGens}; use curve25519_dalek::ristretto::CompressedRistretto; use curve25519_dalek::scalar::Scalar; use merlin::Transcript; use rand::thread_rng; // Shuffle gadget (documented in markdown file) /// A proofofshuffle. struct ShuffleProof(R1CSProof); impl ShuffleProof { fn gadget<CS: RandomizableConstraintSystem>(cs: &mut CS, x: Vec<Variable>, y: Vec<Variable>) > Result<(),R1CSError> { assert_eq!(x.len(), y.len()); let k = x.len(); if k == 1 { cs.constrain(y[0]  x[0]); return Ok(()); } cs.specify_randomized_constraints(move cs { let z = cs.challenge_scalar(b"shuffle challenge"); // Make last x multiplier for i = k1 and k2 let (_, _, last_mulx_out) = cs.multiply(x[k  1]  z, x[k  2]  z); // Make multipliers for x from i == [0, k3] let first_mulx_out = (0..k  2).rev().fold(last_mulx_out, prev_out, i { let (_, _, o) = cs.multiply(prev_out.into(), x[i]  z); o }); // Make last y multiplier for i = k1 and k2 let (_, _, last_muly_out) = cs.multiply(y[k  1]  z, y[k  2]  z); // Make multipliers for y from i == [0, k3] let first_muly_out = (0..k  2).rev().fold(last_muly_out, prev_out, i { let (_, _, o) = cs.multiply(prev_out.into(), y[i]  z); o }); // Constrain last x mul output and last y mul output to be equal cs.constrain(first_mulx_out  first_muly_out); Ok(()) }) } }
In this example, ShuffleProof::gadget()
is private function that adds constraints to the constraint system that enforce that \(y\) (the outputs) are a valid reordering of \(x\) (the inputs).
First, the function gets a challenge scalar \(z\) by calling the ConstraintSystem::challenge_scalar
. This challenge is generated from commitments to highlevel variables that were passed to the ConstraintSystem
when it was created. As noted in the challenge_scalar
documentation, making sure that the challenge circuit is sound requires analysis. In this example, the challenge circuit is sound because the challenge is bound to all of the shuffle inputs and outputs, since the inputs and outputs are highlevel variables.
After a check for the lengths of \(x\) and \(y\), the function then makes
multipliers to create polynomials in terms of the challenge scalar \(z\).
It starts with the last multipliers, representing (x_{k1}  z) * (x_{k2}  z)
and (y_{k1}  z) * (y_{k2}  z)
. The outputs
to these last multipliers than become an input to the next multiplier.
This continues recursively until it reaches x_0
and y_0
.
Then, it adds a constraint that mulx_out[0]
= muly_out[0]
,
which constrains that the two polynomials in terms of challenge scalar
\(z\) are equal to each other. This is true if and only if \(y\) is a valid
reordering of \(x\).
Constructing a proof
The prover prepares the input and output scalar lists, as well as the generators (which are needed to make commitments and to make the proof) and a transcript (which is needed to generate challenges). The prove
function takes the list of scalar inputs and outputs, makes commitments to them, and creates a proof that the committed outputs are a valid reordering of the committed inputs.
For simplicity, in this example the prove
function does not take a list of blinding factors for the inputs and outputs, so it is not possible to make a proof for existing committed points. However, it is possible to modify the function to take in a list of blinding factors instead of generating them internally. Also, in this example the prove
function does not return the list of blinding factors generated, so it is not possible for the prover to open the commitments in the future. This can also be easily modified.
impl ShuffleProof { /// Attempt to construct a proof that `output` is a permutation of `input`. /// /// Returns a tuple `(proof, input_commitments  output_commitments)`. pub fn prove<'a, 'b>( pc_gens: &'b PedersenGens, bp_gens: &'b BulletproofGens, transcript: &'a mut Transcript, input: &[Scalar], output: &[Scalar], ) > Result<(ShuffleProof, Vec<CompressedRistretto>, Vec<CompressedRistretto>), R1CSError> { // Apply a domain separator with the shuffle parameters to the transcript let k = input.len(); transcript.commit_bytes(b"domsep", b"ShuffleProof"); transcript.commit_bytes(b"k", Scalar::from(k as u64).as_bytes()); let mut prover = Prover::new(&pc_gens, transcript); // Construct blinding factors using an RNG. // Note: a nonexample implementation would want to operate on existing commitments. let mut blinding_rng = rand::thread_rng(); let (input_commitments, input_vars): (Vec<_>, Vec<_>) = input.into_iter() .map(v { prover.commit(*v, Scalar::random(&mut blinding_rng)) }) .unzip(); let (output_commitments, output_vars): (Vec<_>, Vec<_>) = output.into_iter() .map(v { prover.commit(*v, Scalar::random(&mut blinding_rng)) }) .unzip(); ShuffleProof::gadget(&mut prover, input_vars, output_vars)?; let proof = prover.prove(&bp_gens)?; Ok((ShuffleProof(proof), input_commitments, output_commitments)) } }
Verifiying a proof
The verifier receives a proof, and a list of committed inputs and outputs, from the prover. It passes these to the verify
function, which verifies that, given a shuffle proof and a list of committed inputs and outputs, the outputs are a valid reordering of the inputs. The verifier receives a Result::ok()
if the proof verified correctly and a Result::error(R1CSError)
otherwise.
impl ShuffleProof { /// Attempt to verify a `ShuffleProof`. pub fn verify<'a, 'b>( &self, pc_gens: &'b PedersenGens, bp_gens: &'b BulletproofGens, transcript: &'a mut Transcript, input_commitments: &Vec<CompressedRistretto>, output_commitments: &Vec<CompressedRistretto>, ) > Result<(), R1CSError> { // Apply a domain separator with the shuffle parameters to the transcript let k = input_commitments.len(); transcript.commit_bytes(b"domsep", b"ShuffleProof"); transcript.commit_bytes(b"k", Scalar::from(k as u64).as_bytes()); let mut verifier = Verifier::new(transcript); let input_vars: Vec<_> = input_commitments.iter().map(commitment { verifier.commit(*commitment) }).collect(); let output_vars: Vec<_> = output_commitments.iter().map(commitment { verifier.commit(*commitment) }).collect(); ShuffleProof::gadget(&mut verifier, input_vars, output_vars)?; verifier.verify(&self.0, &pc_gens, &bp_gens) } }
Using the ShuffleProof
Here, we use the ShuffleProof
gadget by first constructing the common inputs to the prove
and verify
functions: the Pedersen and Bulletproofs generators. Next, the prover makes the other inputs to the prove
function: the list of scalar inputs, the list of scalar outputs, and the prover transcript. The prover calls the prove
function, and gets a proof and a list of committed points that represent the commitments to the input and output scalars.
The prover passes the proof and the commitments to the verifier. The verifier then makes the other inputs to the verify
function: the verifier transcript. Note that the starting transcript state provides domain seperation between different applications, and must be the same for the prover and verifer transcript; if they are not, then the prover and verifier will receive different challenge scalars and the proof will not verify correctly. The verifier then calls the verify
function, and gets back a Result
representing whether or not the proof verified correctly.
Because only the prover knows the scalar values of the inputs and outputs, and the verifier only sees the committed inputs and outputs and not the scalar values themselves, the verifier has no knowledge of what the underlying inputs and outputs are. Therefore, the only information the verifier learns from this protocol is whether or not the committed outputs are a valid shuffle of the committed inputs  this is why it is a zeroknowledge proof.
// Construct generators. 1024 Bulletproofs generators is enough for 512size shuffles. let pc_gens = PedersenGens::default(); let bp_gens = BulletproofGens::new(1024, 1); // Putting the prover code in its own scope means we can't // accidentally reuse prover data in the test. let (proof, in_commitments, out_commitments) = { let inputs = [ Scalar::from(0u64), Scalar::from(1u64), Scalar::from(2u64), Scalar::from(3u64), ]; let outputs = [ Scalar::from(2u64), Scalar::from(3u64), Scalar::from(0u64), Scalar::from(1u64), ]; let mut prover_transcript = Transcript::new(b"ShuffleProofTest"); ShuffleProof::prove( &pc_gens, &bp_gens, &mut prover_transcript, &inputs, &outputs, ) .expect("error during proving") }; let mut verifier_transcript = Transcript::new(b"ShuffleProofTest"); assert!( proof .verify(&pc_gens, &bp_gens, &mut verifier_transcript, &in_commitments, &out_commitments) .is_ok() );
Modules
constraint_system  Definition of the constraint system trait. 
linear_combination  Definition of linear combinations. 
notes  Constraint system proof protocol 
proof  Definition of the proof struct. 
prover  
verifier 
Structs
LinearCombination  Represents a linear combination of

Prover  A 
R1CSProof  A proof of some statement specified by a

Verifier  A 
Enums
R1CSError  Represents an error during the proving or verifying of a constraint system. 
Variable  Represents a variable in a constraint system. 
Traits
ConstraintSystem  The interface for a constraint system, abstracting over the prover and verifier's roles. 
RandomizableConstraintSystem  An extension to the constraint system trait that permits randomized constraints.
Gadgets that do not use randomization should use trait bound 
RandomizedConstraintSystem  Represents a constraint system in the second phase: when the challenges can be sampled to create randomized constraints. 