# 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).