# 03/03/2022:

BLS12-377 / BW6 <- Justin talks about Zexe , 2chain of curve

of order of order <-- circuits in , means I can express group operations efficiently,

is in

## Benchmark

2 Circuits for https://github.com/nikkolasg/grothan/blob/main/src/lib.rs

- Multiplication of two GT elements
- 66 constraints
- Scalar multiplication of one scalar and GT elements
- 39409 constraints
- Enforcing equality:
- 12 constraints

## IPP verifcation

- U
- Z

Per iteration of the loop

- 2 GT multiplication * 3 ⇒ 6 Gt MUL
- 6 Scalr mul

In total:

6 * log(n) * GT_MUL + 6 * log(n) * Scalar_mul

for n = 2**30

math.log2(n) * (6*gtmul + 6*scmul)
**7031880.0 constraints**

## Sumcheck verification

Justin says 1000 for non native arithmetic multiplication

3 Fr multiplication per round of sumcheck

P(x1,x2,x3) = x1x2 + x2x3 + x1x3

P(x,r2,r3) → P(0,r2,r3) + P(1,r2,r3) → 2Fr + 2Fr + 1 ?? →

Prover sends (r2 + r3)x1 + r2r3, already in affine form so 2 Fr operation per evaluation

in total 2 * 2 + 1 = 5 Fr operation per “round” + hash(3 elements) ← 1000 constraints using Poseidon

**Using non native field arithmetic**, supposing 1000 constraints for Fr operations, it’s (1000*5+3000)*log(n) = 240000 constraints for n=2**30

**Using sub Groth16 proof:**

- Need to do scalar multiplication on ALL the public inputs → log(n) elements in
- 500*log(n) constraints for doing
- Need to do 3 pairings → 150 000 constraints

## Concrete scheme / curves

Spartan:

- Operations for GKR/Sumcheck
- Operations over this is for polymnomial commitment

2 choices of “curves”:

- Either we use a 2chain :
- BLS12-377/BW6 →
- BW6: One curve where circuit native in → native operations for Gt, “Non Native Arithmetic” for
- BLS12-377: ANother curve where circuit is native in
- Do a Groth16 proof for sumcheck, over
- Do a Groth16 proof for polynomial commitment + verification of first proof over
- OR we use a regular “pairing curve”, like BLS12-381:
- We do NNA operations for
- F_r operatiopn can do natively
- Problematic because 1-2 order of magnitude
- Potential tricks: , that means we can embed one element directly in one , could potentialy write many operations, like add/mul and only doing the modulo at the end