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Bulletproofs++ implementation on Go

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# Bulletproofs++ implementation

[![License: MIT](https://img.shields.io/badge/License-MIT-yellow.svg)](https://opensource.org/licenses/MIT)

⚠️ __Please note - this crypto library has not been audited, so use it at your own risk.__

## Abstract

Present Go library contains the implementation of Bulletproofs++ weight norm linear argument protocol, arithmetic
circuit
protocol and reciprocal range proof protocol described in Distributed
Lab's [Bulletproofs++ Construction and Examples](https://distributedlab.com/whitepaper/Bulletproofs-Construction-and-Examples.pdf).

Explore the [circuit_test.go](./circuit_test.go) to check the examples of circuit prove and verification.
It contains several circuits:

- Prove that we know such `x, y` that `x+y=r` and `x*y=z` for public `r, z`. This example encoded directly into the BP++
  circuits.
- Prove the range for 4-bits value: `x` is in `[0..2^n)` range (simple binary range proof).

Also, [reciprocal_test.go](./reciprocal_test.go) contains example of proving that value lies in [0, 16^n) range.

Implemented protocol has 2G points advantage over existing BP and BP+ protocols on proving of one 64-bit value and this
advantage will increase for more values per proof.

| Protocol | G  | F |
|----------|----|---|
| BP       | 16 | 5 |
| BP+      | 15 | 3 |
| Our BP++ | 13 | 3 |

## Reciprocal range proofs

Check the following snippet as an example of usage of range proof protocol:

```go
package main

import (
	"github.com/cloudflare/bn256"
	"github.com/distributed-lab/bulletproofs"
	"math/big"
)

func main() {
	// The uint64 in 16-base system will be encoded in 16 digits. 
	// The 16 base is selected as the most optimal base for this case.

// Our private value is 0xab4f0540ab4f0540. 
	x := uint64(0xab4f0540ab4f0540)
	X := new(big.Int).SetUint64(x)

// Let's encode it as a list of digits:
	digits := bulletproofs.UInt64Hex(x) // [0 4 5 0 15 4 11 10 0 4 5 0 15 4 11 10]

// Public poles multiplicities i-th element corresponds to the 'i-digit' multiplicity (the count of 'i-digit' in digits list)
	m := bulletproofs.HexMapping(digits) // [4 0 0 0 4 2 0 0 0 0 2 2 0 0 0 2]

Nd := 16 // digits size
	Np := 16 // base size

var G *bn256.G1
	// Length of our base points vector should be a power ot 2 to be used in WNLA protocol. 
	// So cause the real HVec size in circuit is `Nd+10` the nearest length is 32   
	var GVec []*bn256.G1 // len = 16
	var HVec []*bn256.G1 // len = 32

public := &bulletproofs.ReciprocalPublic{
		G:    G,
		GVec: GVec[:Nd],
		HVec: HVec[:Nd+10],
		Nd:   Nd,
		Np:   Np,

// Remaining points that will be used in WNLA protocol
		GVec_: GVec[Nd:],
		HVec_: HVec[Nd+10:],
	}

private := &bulletproofs.ReciprocalPrivate{
		X:      x,                // Committed value
		M:      m,                // Corresponding multiplicities
		Digits: digits,           // Corresponding digits
		S:      MustRandScalar(), // Blinding value (secret) used for committing value as: x*G + Sx*H
	}

VCom := public.CommitValue(private.X, private.Sx) // Value commitment: x*G + Sx*H

// Use NewKeccakFS or your own implementation for the Fiat-Shamir heuristics.
	proof := bulletproofs.ProveRange(public, bulletproofs.NewKeccakFS(), private)

// If err is nil -> proof is valid.
	if err := bulletproofs.VerifyRange(public, VCom, bulletproofs.NewKeccakFS(), proof); err != nil {
		panic(err)
	}
}

```

## Weight norm linear argument (WNLA)

The [wnla.go](./wnla.go) contains the implementation of **weight norm linear argument** protocol. This is a fundamental
basis for arithmetic circuit protocol. It uses the Fiat-Shamir heuristics from [fs.go](./fs.go) to generate challenges
and make protocol non-interactive.

Check the following snippet with an example of WNLA usage:

```go
package main

import (
	"github.com/distributed-lab/bulletproofs"
	"math/big"
)

func main() {
	public := bulletproofs.NewWeightNormLinearPublic(4, 2)

// Private
	l := []*big.Int{big.NewInt(4), big.NewInt(5), big.NewInt(10), big.NewInt(1)}
	n := []*big.Int{big.NewInt(2), big.NewInt(1)}

proof := bulletproofs.ProveWNLA(public, public.Commit(l, n), bulletproofs.NewKeccakFS(), l, n)
	if err := bulletproofs.VerifyWNLA(public, proof, public.Commit(l, n), bulletproofs.NewKeccakFS()); err != nil {
		panic(err)
	}
}

```

## Arithmetic circuit

The [circuit.go](./circuit.go) contains the implementation of BP++ arithmetic circuit protocol.
It runs the WNLA protocol as the final stages of proving/verification. Uses the Fiat-Shamir heuristics
from [fs.go](./fs.go) to generate challenges
and make protocol non-interactive.

Check the following snippet with an example of arithmetic circuit protocol usage:

```go
package main

import (
	"github.com/cloudflare/bn256"
	"github.com/distributed-lab/bulletproofs"
)

func main() {
	public := &bulletproofs.ArithmeticCircuitPublic{
		Nm,
		Nl, // Nl = Nv * K
		Nv, // Size on any v witness vector
		Nw, // Nw = Nm + Nm + No
		No,
		K, // count of v witnesses vectors
		G,

// points that will be used directly in circuit protocol
		GVec[:Nm],   // Nm
		HVec[:Nv+9], // Nv+9

// Circuit definition 
		Wm, // Nm * Nw
		Wl, // Nl * Nw
		Am, // Nm
		Al, // Nl
		Fl,
		Fm,

// Partition function
		F: func(typ bulletproofs.PartitionType, index int) *int {
			// define
			return nil
		},

// points that will be used in WNLA protocol to make vectors 2^n len
		HVec[Nv+9:], // 2^x - (Nv+9) dimension
		GVec[Nm:],   // 2^y - Nm dimension
	}

private := &bulletproofs.ArithmeticCircuitPrivate{
		V,  // witness vectors v, dimension k*Nv
		Sv, // witness blinding values, dimension k
		Wl, // Nm
		Wr, // Nm
		Wo, // No
	}

// Commitments to the v witness vectors
	V := make([]*bn256.G1, public.K)
	for i := range V {
		V[i] = public.CommitCircuit(private.V[i], private.Sv[i], public.G, public.HVec)
	}

proof := bulletproofs.ProveCircuit(public, bulletproofs.NewKeccakFS(), private)

if err := bulletproofs.VerifyCircuit(public, V, bulletproofs.NewKeccakFS(), proof); err != nil {
		panic(err)
	}
}
```

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