Buletproofs
Bulletproofs are short zero-knowledge arguments of knowledge that do not require a trusted setup. Argument systems are proof systems with computational soundness.
Bulletproofs are suitable for proving statements on committed values, such as range proofs, verifiable suffles, arithmetic circuits, etc. They rely on the discrete logarithmic assumption and are made non-interactive using the Fiat-Shamir heuristic.
The core algorithm of Bulletproofs is the inner-product algorithm presented by Groth [2].
The algorithm provides an argument of knowledge of two binding vector Pedersen commitments that satisfy a given inner product relation.
Bulletproofs build on the techniques of Bootle et al. [3] to introduce a communication efficient inner-product proof that reduces
overall communication complexity of the argument to only 
where 
is the dimension
of the two vectors of commitments.
Range proofs
Bulletproofs present a protocol for conducting short and aggregatable range proofs. They encode a proof of the range of a committed number in an inner product, using polynomials. Range proofs are proofs that a secret value lies in a certain interval. Range proofs do not leak any information about the secret value, other than the fact that they lie in the interval.
The proof algorithm can be sketched out in 5 steps:
Let 
be a value in 
and 
a vector of bit such that
.
The components of are the binary digits of .
We construct a complementary vector 
and require that 
holds.
- where 
and 
are blinded Pedersen commitments to and 
.
    
    
-
- Verifier sends challenges 
and 
to fix and . -
- where 
and 
are commitments to
the coefficients 
, of a polynomial 
constructed from the existing
values in the protocol.
    
    
    
    
, Â Â Â Â 
-
- Verifier challenges Prover with value 
. -
- Prover sends several commitments that the verifier will then check.
    
    
See Prover.hs for implementation details.
The interaction described is made non-interactive using the Fiat-Shamir Transform wherein all the random challenges made by V are replaced with a hash of the transcript up until that point.
Inner-product range proof
The size of the proof is further reduced by leveraging the compact 
inner product proof.
The inner-product argument in the protocol allows to prove knowledge of vectors 
and 
, whose inner product is and
the commitment 
is a commitment of these two vectors. We can therefore replace sending
(
) with a transfer of (
) and an execution of an inner product argument.
Then, instead of sharing and , which has a communication cost of 
elements, the inner-product
argument transmits only 
elements. In total, the prover sends only 
group elements and 5 elements in 
Aggregating Logarithmic Proofs
We can construct a single proof of range of multiple values, while only incurring an additional space cost of 
for
additional values , as opposed to a multiplicative factor of when creating independent range proofs.
The aggregate range proof makes use of the inner product argument. It uses (
) group elements and 5 elements in .
Usage
Single range proof
import Data.Curve.Weierstrass.SECP256K1 (Fr)
import qualified Bulletproofs.RangeProof as RP
import Bulletproofs.Utils (commit)
testSingleRangeProof :: Integer -> (Fr, Fr) -> IO Bool
testSingleRangeProof upperBound (v, vBlinding) = do
let vCommit = commit v vBlinding
-- Prover
proofE <- runExceptT (RP.generateProof upperBound (v, vBlinding))
-- Verifier
case proofE of
Left err -> panic (show err)
Right proof@RP.RangeProof{..}
-> pure (RP.verifyProof upperBound vCommit proof)Multi range proof
import Data.Curve.Weierstrass.SECP256K1 (Fr)
import qualified Bulletproofs.MultiRangeProof as MRP
import Bulletproofs.Utils (commit)
testMultiRangeProof :: Integer -> [(Fr, Fr)] -> IO Bool
testMultiRangeProof upperBound vsAndvBlindings = do
let vCommits = fmap (uncurry commit) vsAndvBlindings
-- Prover
proofE <- runExceptT (MRP.generateProof upperBound vsAndvBlindings)
-- Verifier
case proofE of
Left err -> panic (show err)
Right proof@RP.RangeProof{..}
-> pure (MRP.verifyProof upperBound vCommits proof)Note that the upper bound 
must be such that 
, where is also a power of 2.
This implementation uses the elliptic curve secp256k1, a Koblitz curve, which
has 128 bit security.
See Range proofs examples for further details.
Zero-knowledge proofs for Arithmetic Circuits
An arithmetic circuit over a field and variables 
is a directed acyclic graph whose vertices are called gates.
Arithmetic circuit can be described alternatively as a list of multiplication gates with a collection of linear consistency equations relating the inputs and outputs of the gates. Any circuit described as an acyclic graph can be efficiently converted into this alternative description.
Bulletproofs present a protocol to generate zero-knowledge argument for arithmetic circuits using the inner product argument,
which allows to get a proof of size 
elements and include committed values as inputs to the arithmetic circuit.
In the protocol, the Prover proves that the hadamard product of and and a set of linear constraints hold.
The input values used to generate the proof are then committed and shared with the Verifier.
import Data.Curve.Weierstrass.SECP256K1 (Fr)
import Data.Field.Galois (rnd)
import Bulletproofs.ArithmeticCircuit
import Bulletproofs.Utils (hadamard, commit)
-- Example:
-- 2 linear constraints (q = 2):
-- aL[0] + aL[1] + aL[2] + aL[3] = v[0]
-- aR[0] + aR[1] + aR[2] + aR[3] = v[1]
--
-- 4 multiplication constraints (implicit) (n = 4):
-- aL[0] * aR[0] = aO[0]
-- aL[1] * aR[1] = aO[1]
-- aL[2] * aR[2] = aO[2]
-- aL[3] * aR[3] = aO[3]
--
-- 2 input values (m = 2)
arithCircuitExample :: ArithCircuit Fr
arithCircuitExample = ArithCircuit
{ weights = GateWeights
{ wL = [[1, 1, 1, 1]
,[0, 0, 0, 0]]
, wR = [[0, 0, 0, 0]
,[1, 1, 1, 1]]
, wO = [[0, 0, 0, 0]
,[0, 0, 0, 0]]
}
, commitmentWeights = [[1, 0]
,[0, 1]]
, cs = [0, 0]
}
testArithCircuitProof :: ([Fr], [Fr]) -> ArithCircuit Fr -> IO Bool
testArithCircuitProof (aL, aR) arithCircuit = do
let m = 2
-- Multiplication constraints
let aO = aL `hadamard` aR
-- Linear constraints
v0 = sum aL
v1 = sum aR
commitBlinders <- replicateM m rnd
let commitments = zipWith commit [v0, v1] commitBlinders
let arithWitness = ArithWitness
{ assignment = Assignment aL aR aO
, commitments = commitments
, commitBlinders = commitBlinders
}
proof <- generateProof arithCircuit arithWitness
pure (verifyProof commitments proof arithCircuit)See Aritmetic circuit example for further details.
References:
-
Bunz B., Bootle J., Boneh D., Poelstra A., Wuille P., Maxwell G. "Bulletproofs: Short Proofs for Confidential Transactions and More". Stanford, UCL, Blockstream, 2017
-
Groth J. "Linear Algebra with Sub-linear Zero-Knowledge Arguments". University College London, 2009
-
Bootle J., Cerully A., Chaidos P., Groth J, Petit C. "Efficient Zero-Knowledge Arguments for Arithmetic Circuits in the Discrete Log Setting". University College London and University of Oxford, 2016.
Notation:
: Hadamard product
:Inner product
: Vector
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
Copyright 2018-2022 Stephen Diehl
Licensed under the Apache License, Version 2.0 (the "License");
you may not use this file except in compliance with the License.
You may obtain a copy of the License at
http://www.apache.org/licenses/LICENSE-2.0
Unless required by applicable law or agreed to in writing, software
distributed under the License is distributed on an "AS IS" BASIS,
WITHOUT WARRANTIES OR CONDITIONS OF ANY KIND, either express or implied.
See the License for the specific language governing permissions and
limitations under the License.