> For the complete documentation index, see [llms.txt](https://calnix.gitbook.io/zk-notes/llms.txt). Markdown versions of documentation pages are available by appending `.md` to page URLs; this page is available as [Markdown](https://calnix.gitbook.io/zk-notes/groth16-explained/2.-r1cs-representation.md).
# 2. R1CS representation
### R1CS representation
In this step, we simply convert our system of constraints into vector form, `Cw = Aw * Bw`.
`w` here is the witness vector: `[1 out x y v1 v2]`.
**What are the dimensions of matrices C, A and B?**
* 3 rows: number of constraints
* 6 columns: number of witness elements
Therefore, they have the dimension 3x6.
**Cw = Aw \* Bw:**
Essentially this is the matrix form of our earlier system of constraints:
```solidity
// Cw = Aw * Bw
v1 = y * y
v2 = x * x
out -v1 +2 = (4v2 * y)
```
where matrix C encompasses all the LHS elements, and A and B deal with the RHS elements.
To be perfectly clear:
Notice that the number 2 in last row for matrix **C**, it represents the `+2` in the third constraint.
From this, it should be clear the purpose of having 1 in the witness vector - to allow for representation of addition of constants.
***
*For detailed understanding of R1CS rules:*
***
#### Why does R1CS require exactly one multiplication per row?
* R1CS requires exactly 1 multiplication per row, so that the constraints can be expressed in the matrix form: `Cw = Aw * Bw`, where each row is a single constraint.
* Adhering to the `Cw = Aw * Bw` form is useful to us as it allows us to apply bilinear pairings.
* If there are multiple multiplications per row, the resulting form would be: `Cw = Aw * Bw * ...` We would not be able to constructively apply bilinear pairings in this case.
> If we have more than 1 multiplication per row, we would not be able to apply a bilinear mapping between G1 and G2 to G12. Without a bilinear mapping we cannot turn this into a succinct ZKP.
***
**CHECKPOINT**
Given a circuit encoded as a rank 1 constraint system, it is possible to create a proof.
* The prover would generate `Cw`, `Aw`, `Bw`, as the proof.
* The verifier will verify if `Cw = Aw * Bw`, to check the validity of the proof
However, it would not be **succinct**. Imagine you are dealing with a complex circuit, with numerous constraints. This means that each of the matrices comprise of **thousands of rows**.
Thus, calculating `Cw = Aw * Bw` would be time-consuming and intensive with O(n), where n is the number of constraints.
So not very succinct at all. This is why we look to convert the R1CS form into a QAP.
*Read more:*
* [https://www.rareskills.io/post/rank-1-constraint-system](https://www.rareskills.io/post/rank-1-constraint-system*)
* [https://www.rareskills.io/post/r1cs-zkp](https://www.rareskills.io/post/r1cs-zkp*)
* [https://www.rareskills.io/post/bilinear-pairing](https://www.rareskills.io/post/bilinear-pairing*)
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