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@@ -1,21 +1,24 @@
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# zkSNARK for a Pedersen commitment
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-*Ian Goldberg (iang@uwaterloo.ca), August 2019*
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+*Ian Goldberg (iang@uwaterloo.ca), updated November 2019*
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-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.
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+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.
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-The algorithm is still a pretty naive square-and-multiply; there are surely better ones. This circuit ends up with 2045 constraints for a Pedersen commitment over a 254-bit elliptic curve.
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+This circuit ends up with 1531 constraints for a Pedersen commitment over a 254-bit elliptic curve.
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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.
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-The code uses three generators of this curve, which must not have a known DL representation among them. They are G(0,11977228949870389393715360594190192321220966033310912010610740966317727761886),
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-H(1,21803877843449984883423225223478944275188924769286999517937427649571474907279), and
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-C(2,4950745124018817972378217179409499695353526031437053848725554590521829916331).
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+The code uses four generators of this curve, which must not have a known DL representation among them. They are G(0,11977228949870389393715360594190192321220966033310912010610740966317727761886),
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+H(1,21803877843449984883423225223478944275188924769286999517937427649571474907279),
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+C(2,4950745124018817972378217179409499695353526031437053848725554590521829916331), and
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+A(4,1929778687269876629657252589535788315400602403700102541701561325064015752665).
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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.
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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."
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+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).
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+
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Building:
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* Clone the repo
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Ian Goldberg