[](https://circleci.com/gh/adjoint-io/bulletproofs) [](http://hackage.haskell.org/package/bulletproofs) 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 2log₂(n) where n 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 _v_ be a value in _[0, n)_ and **a_L** a vector of bit such that <**a_L**, **2ⁿ**> = _v_. The components of **a_L** are the binary digits of _v_. We construct a complementary vector **a_R** = **a_L** − **1**ⁿ and require that **a_L** ◦ **a_R** = 0 holds. - **P -> V : A, S** - where A and S are blinded Pedersen commitments to **a_L** and **a_R**.                 - **V -> P : y, z** - Verifier sends challenges _y_ and _z_ to fix **A** and **S**. - **P -> V : T₁, T₂** - where T₁ and T₂ are commitments to the coefficients t₁, of a polynomial t constructed from the existing values in the protocol.        &space;=&space;\textbf{a}_L&space;-&space;z&space;\cdot&space;\textbf{1}^n&space;+&space;\textbf{s}_L&space;\cdot&space;x&space;\in&space;\mathbb{Z}^n_p$)        &space;=&space;\textbf{y}^n&space;\circ&space;(\textbf{a}_R&space;+&space;z&space;\cdot&space;\textbf{1}^n&space;+&space;\textbf{s}_R&space;\cdot&space;x&space;)&space;+&space;z^2&space;\cdot&space;\textbf{2}^n&space;\in&space;\mathbb{Z}^n_p&space;$)                 - **V -> P : x** - Verifier challenges Prover with value _x_. - **P -> V : tau, mu, t, l, r** - Prover sends several commitments that the verifier will then check.                 See [Prover.hs](https://github.com/adjoint-io/bulletproofs/blob/master/Bulletproofs/RangeProof/Prover.hs "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 O(log_n) inner product proof. The inner-product argument in the protocol allows to prove knowledge of vectors **l** and **r**, whose inner product is _t_ and the commitment _P_ ∈ _G_ is a commitment of these two vectors. We can therefore replace sending (tau, mu, t, **l**, **r**) with a transfer of (tau, mu, t) and an execution of an inner product argument. Then, instead of sharing **l** and **r**, which has a communication cost of 2n elements, the inner-product argument transmits only 2 [log₂] + 2 elements. In total, the prover sends only 2 [log₂(n)] + 4 group elements and 5 elements in _Z__p Aggregating Logarithmic Proofs ============================== We can construct a single proof of range of multiple values, while only incurring an additional space cost of 2 log₂(m) for _m_ additional values _v_, as opposed to a multiplicative factor of _m_ when creating _m_ independent range proofs. The aggregate range proof makes use of the inner product argument. It uses 2 [log₂ (n * m)] + 4 group elements and 5 elements in Z_p. See [Multi range proof example](https://github.com/adjoint-io/bulletproofs/tree/master#multi-range-proof) Usage ===== Single range proof: ------------------- ```haskell import qualified Bulletproofs.RangeProof as RP testSingleRangeProof :: (Fq, Fq) -> IO Bool testSingleRangeProof (v, vBlinding) = do let vCommit = commit v vBlinding -- n needs to be a power of 2 n = 2 ^ 8 upperBound = 2 ^ n -- 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: ------------------ ```haskell import qualified Bulletproofs.MultiRangeProof as MRP testMultiRangeProof :: [(Fq, Fq)] -> IO Bool testMultiRangeProof vsAndvBlindings = do let vCommits = fmap (uncurry commit) vsAndvBlindings -- n needs to be a power of 2 n = 2 ^ 8 upperBound = 2 ^ n -- 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 ``` The dimension _n_ needs to be a power of 2. This implementation offers support for SECp256k1, a Koblitz curve. Further information about this curve can be found in the Uplink docs: [SECp256k1 curve](https://www.adjoint.io/docs/cryptography.html#id1 "SECp256k1 curve") Zero-knowledge proof for Arithmetic Circuits ============================================ An arithmetic circuit over a field and variables (a₁, ..., a_n) 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 2 log₂(n) + 13 elements and include committed values as inputs to the arithmetic circuit. In the protocol, the Prover proves that the hadamard product of _a_L_ and _a_R_ and a set of linear constraints hold. The input values _v_ used to generate the proof are then committed and shared with the Verifier. ```haskell import qualified Bulletproofs.ArithmeticCircuit -- 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 Fq 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 :: ([Fq], [Fq]) -> ArithCircuit Fq -> IO Bool testArithCircuitProof (aL, aR) arithCircuit = do let n = 4 m = 2 q = 2 -- Multiplication constraints let aO = aL `hadamardp` aR -- Linear constraints v0 = sum aL v1 = sum aR commitBlinders <- replicateM m fqRandom 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 ``` **References**: 1. Bunz B., Bootle J., Boneh D., Poelstra A., Wuille P., Maxwell G. "Bulletproofs: Short Proofs for Confidential Transactions and More". Stanford, UCL, Blockstream, 2017 2. Groth J. "Linear Algebra with Sub-linear Zero-Knowledge Arguments". University College London, 2009 3. 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 - **a**: Vector License ------- ``` Copyright 2018 Adjoint Inc 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. ```