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Bulletproofs are short non-interactive zero-knowledge proofs that require no trusted setup

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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.

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.

,

- 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.

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

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 .

**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 panic (show err) Right [email protected]{..} -> 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 panic (show err) Right [email protected]{..} -> 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.

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

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

This is experimental code meant for research-grade projects only. Please do not use this code in production until it has matured significantly.

Copyright 2018-2019 Adjoint IncLicensed 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.