# zkSNARK for a Pedersen commitment *Ian Goldberg (iang@uwaterloo.ca), updated March 2020* I spent a day learning how to use [libsnark](https://github.com/scipr-lab/libsnark), and thought an interesting first project would be to create a zkSNARK for knowledge of a preimage for a Pedersen commitment. I spent another day reimplementing it with a better scalar multiplication algorithm. A few months later, I did a little more work on the algorithm, further reducing the cost of scalar multiplication (with a constant base point) to 3 constraints per bit, and implementing new features like scalar multiplication of non-constant points. This circuit ends up with 1531 constraints for a Pedersen commitment over a 254-bit elliptic curve. It uses libsnark's BN128 implementation, which has an order (not modulus) of r=21888242871839275222246405745257275088548364400416034343698204186575808495617. Then using [Sage](http://www.sagemath.org/), the [findcurve](sage/findcurve) script in this repo searches for an elliptic curve with _modulus_ r, and with both prime order and whose twist has prime order. (You do not need to run the findcurve script yourself.) The resulting curve (over F_r) is E: y^2 = x^3 - 3*x + b, where b=7950939520449436327800262930799465135910802758673292356620796789196167463969. The order of this curve is the prime 21888242871839275222246405745257275088760161411100494528458776273921456643749. The code uses four generators of this curve, which must not have a known DL representation among them. They are G(0,11977228949870389393715360594190192321220966033310912010610740966317727761886), H(1,21803877843449984883423225223478944275188924769286999517937427649571474907279), C(2,4950745124018817972378217179409499695353526031437053848725554590521829916331), and A(4,1929778687269876629657252589535788315400602403700102541701561325064015752665). If you switch to a different underlying curve for the zkSNARKs than BN128, you will need to find a new E and new generators, and change the precomputed values in [ecgadget.hpp](ecgadget.hpp) to match. (Update 30 March 2020: MNT4 and MNT6 are now also supported.) The code produces a zkSNARK for the statment "I know values _a_ and _b_ such that _a_*G + _b_*H equals the given Pedersen commitment P." Also supported: scalar multiplication by a constant point (768 constraints), scalar multiplication by a public point (3048 constraints with a private precomputation table, or 1530 with a public one, but the public version is considerably slower to verify). As an application of the latter, the included verifenc program provides a zkSNARK for verified ElGamal encryption of a private key (corresponding to a given public key) to a given receiver's public key. As an application of scalar multiplication by constant points, the included ratchetcommit program provides a zkSNARK for commitments of private values, where those private values are the output of a two-hash ratchet. Building: * Clone the repo * git submodule update --init --recursive * Ensure you have the [build dependencies for libsnark](https://github.com/scipr-lab/libsnark/blob/master/README.md#user-content-build-instructions) installed. * cd libsnark * mkdir build && cd build && cmake -DCURVE=BN128 .. * make * cd ../.. * make Thanks to Christian Lundkvist and Sam Mayo for the very helpful [libsnark tutorial](https://github.com/christianlundkvist/libsnark-tutorial).