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# Binary format for R1CS
---eip:title: r1cs binary formatauthor: Jordi Baylina <jordi@baylina.cat>discussions-to:status: drafttype: Standards Trackcategory: ERCcreated: 2019-09-24requires:---
## Simple Summary
This standard defines a standard format for a binery representation of a r1cs constraint system.
## Abstract
## Motivation
The zero knowledge primitives, requires the definition of a statment that wants to be proved. This statment can be expressed as a deterministric program or an algebraic circuit. Lots of primitives like zkSnarks, bulletProofs or aurora, requires to convert this statment to a rank-one constraint system.
This standard specifies a format for a r1cs and allows the to connect a set of tools that compiles a program or a circuit to r1cs that can be used for the zksnarks or bulletproofs primitives.
## Specification
### General considerations
The standard extension is `.r1cs`
A deterministic program (or circuit) is a program that generates a set of deterministic values given an input. All those values are labeled from l_{0} to l_{n_labels}
This file defines a map beween l_{i} -> w_{j} and defines a series a R1CS of the form
$$\left\{ \begin{array}{rclclcl}(a_{0,0}w_0 + a_{0,1}w_1 + ... + a_{0,n}w_{n}) &\cdot& (b_{0,0} w_0 + b_{0,1} w_1 + ... + b_{0,n} w_{n}) &-& (c_{0,0} w_0 + c_{0,1} w_1 + ... + c_{0,n}w_{n}) &=& 0 \\(a_{1,0}w_0 + a_{1,1}w_1 + ... + a_{1,n}w_{n}) &\cdot& (b_{1,0} w_0 + b_{1,1} w_1 + ... + b_{1,n} w_{n}) &-& (c_{1,0} w_0 + c_{1,1}w_1 + ... + c_{1,n}w_{n}) &=& 0 \\...\\(a_{m-1,0}w_0 + a_{m-1,1}w_1 + ... + a_{m-1,n}w_{n}) &\cdot& (b_{m-1,0} w_0 + b_{m-1,1} w_1 + ... + b_{m-1,n} w_{n}) &-& (c_{m-1,0} w_0 + c_{m-1,1}w_1 + ... + c_{m-1,n}w_{n}) &=& 0 \end{array} \right.$$
Wire 0 must be always mapped to label 0 and it's an input forced to value "1" implicitly
### Format of the file
````
┏━━━━┳━━━━━━━━━━━━━━━━━┓ ┃ 4 │ 72 31 63 73 ┃ Magic "r1cs" ┗━━━━┻━━━━━━━━━━━━━━━━━┛ ┏━━━━┳━━━━━━━━━━━━━━━━━┓ ┃ 4 │ 01 00 00 00 ┃ Version 1 ┗━━━━┻━━━━━━━━━━━━━━━━━┛ ┏━━━━┳━━━━━━━━━━━━━━━━━┓ ┃ 4 │ 03 00 00 00 ┃ Number of Sections ┗━━━━┻━━━━━━━━━━━━━━━━━┛ ┏━━━━┳━━━━━━━━━━━━━━━━━┳━━━━━┳━━━━━━━━━━━━━━━━━━━━━━━━┓ ┃ 4 │ sectionType ┃ 8 │ SectionSize ┃ ┗━━━━┻━━━━━━━━━━━━━━━━━┻━━━━━┻━━━━━━━━━━━━━━━━━━━━━━━━┛ ┏━━━━━━━━━━━━━━━━━━━━━┓ ┃ ┃ ┃ ┃ ┃ ┃ ┃ Section Content ┃ ┃ ┃ ┃ ┃ ┃ ┃ ┗━━━━━━━━━━━━━━━━━━━━━┛
┏━━━━┳━━━━━━━━━━━━━━━━━┳━━━━━┳━━━━━━━━━━━━━━━━━━━━━━━━┓ ┃ 4 │ sectionType ┃ 8 │ SectionSize ┃ ┗━━━━┻━━━━━━━━━━━━━━━━━┻━━━━━┻━━━━━━━━━━━━━━━━━━━━━━━━┛ ┏━━━━━━━━━━━━━━━━━━━━━┓ ┃ ┃ ┃ ┃ ┃ ┃ ┃ Section Content ┃ ┃ ┃ ┃ ┃ ┃ ┃ ┗━━━━━━━━━━━━━━━━━━━━━┛
... ... ...````
#### Magic Number
Size: 4 bytesThe file start with a constant 4 bytes (magic number) "r1cs"
```0x72 0x31 0x63 0x73```
#### Version
Size: 4 bytesFormat: Little-Endian
For this standard it's fixed to
```0x01 0x00 0x00 0x00```
#### Number of Sections
Size: 4 bytesFormat: Little-Endian
Number of sections contained in the file
#### SectionType
Size: 4 bytesFormat: Little-Endian
Type of the section.
Currently there are 3 types of sections defined:
* 0x00000001 : Header Section* 0x00000002 : Constraint Section* 0x00000003 : Wire2LabelId Map Section
If the file contain other types, the format is valid, but they MUST be ignored.
Any order of the section must be accepted.
Example:```0x01 0x00 0x00 0x00```
#### SectionSize
Size: `ws` bytesFormat: Little-Endian
Size in bytes of the section
### Header Section
Section Type: 0x01```` ┏━━━━┳━━━━━━━━━━━━━━━━━┓ ┃ 4 │ 20 00 00 00 ┃ Field Size in bytes (fs) ┗━━━━┻━━━━━━━━━━━━━━━━━┛ ┏━━━━┳━━━━━━━━━━━━━━━━━━━━━━━━━━━━━━━━━━━━━━━━━━━━━━━━━━━━━━━━━━━━━━━━━━━━━━━━━━━┓ ┃ fs │ 010000f0 93f5e143 9170b979 48e83328 5d588181 b64550b8 29a031e1 724e6430 ┃ Prime size ┗━━━━┻━━━━━━━━━━━━━━━━━━━━━━━━━━━━━━━━━━━━━━━━━━━━━━━━━━━━━━━━━━━━━━━━━━━━━━━━━━━┛ ┏━━━━┳━━━━━━━━━━━━━━━━━┓ ┃ 32 │ 01 00 00 00 ┃ nWires ┗━━━━┻━━━━━━━━━━━━━━━━━┛ ┏━━━━┳━━━━━━━━━━━━━━━━━┓ ┃ 32 │ 01 00 00 00 ┃ nPubOut ┗━━━━┻━━━━━━━━━━━━━━━━━┛ ┏━━━━┳━━━━━━━━━━━━━━━━━┓ ┃ 32 │ 01 00 00 00 ┃ nPubIn ┗━━━━┻━━━━━━━━━━━━━━━━━┛ ┏━━━━┳━━━━━━━━━━━━━━━━━┓ ┃ 32 │ 01 00 00 00 ┃ nPrvIn ┗━━━━┻━━━━━━━━━━━━━━━━━┛ ┏━━━━┳━━━━━━━━━━━━━━━━━━━━━━━━━━━━━┓ ┃ 64 │ 01 00 00 00 00 00 00 00 ┃ nLabels ┗━━━━┻━━━━━━━━━━━━━━━━━━━━━━━━━━━━━┛ ┏━━━━┳━━━━━━━━━━━━━━━━━┓ ┃ 32 │ 01 00 00 00 ┃ mConstraints ┗━━━━┻━━━━━━━━━━━━━━━━━┛
````
#### field Size (fs)
Size: 4 bytesFormat: Little-Endian
Size in bytes of a field element. Mast be a multiple of 8.
Example:```0x00 0x0 0x00 0x00```
#### Prime
Prime Number of the field
Example:```0x010000f0_93f5e143_9170b979_48e83328_5d588181_b64550b8_29a031e1_724e6430```
#### Number of wires
Size: 4 bytesFormat: Little-Endian
Total Number of wires including ONE signal (Index 0).
#### Number of public outputs
Size: 4 bytesFormat: Little-Endian
Total Number of wires public output wires. They should be starting at idx 1
#### Number of public inputs
Size: 4 bytesFormat: Little-Endian
Total Number of wires public input wires. They should be starting just after the public output
#### Number of private inputs
Size: 4 bytesFormat: Little-Endian
Total Number of wires private input wires. They should be starting just after the public inputs
#### Number of Labels
Size: 8 bytesFormat: Little-Endian
Total Number of wires private input wires. They should be starting just after the public inputs
#### Number of constraints (m)
Size: 4 bytesFormat: Little-Endian
Total Number of constraints
### Constraints section
Section Type: 0x02
```` ┏━━━━┳━━━━━━━━━━━━━━━━━┓ ╲ ┃ 32 │ nA ┃ ╲ ┣━━━━╋━━━━━━━━━━━━━━━━━╋━━━━┳━━━━━━━━━━━━━━━━━━━━━━━━┓ ╲ ┃ 32 │ wireId_1 ┃ fs │ a_{0,wireId_1} ┃ │ ┣━━━━╋━━━━━━━━━━━━━━━━━╋━━━━╋━━━━━━━━━━━━━━━━━━━━━━━━┫ │ ┃ 32 │ wireId_2 ┃ fs │ a_{0,wireId_2} ┃ │ ┗━━━━┻━━━━━━━━━━━━━━━━━┻━━━━┻━━━━━━━━━━━━━━━━━━━━━━━━┛ │ ... ... │ ┏━━━━┳━━━━━━━━━━━━━━━━━┳━━━━┳━━━━━━━━━━━━━━━━━━━━━━━━┓ │ ┃ 32 │ wireId_nA ┃ fs │ a_{0,wireId_nA} ┃ │ ┗━━━━┻━━━━━━━━━━━━━━━━━┻━━━━┻━━━━━━━━━━━━━━━━━━━━━━━━┛ │ ┏━━━━┳━━━━━━━━━━━━━━━━━┓ │ ┃ 32 │ nB ┃ │ ┣━━━━╋━━━━━━━━━━━━━━━━━╋━━━━┳━━━━━━━━━━━━━━━━━━━━━━━━┓ │ ┃ 32 │ wireId_1 ┃ fs │ b_{0,wireId_1} ┃ │ ┣━━━━╋━━━━━━━━━━━━━━━━━╋━━━━╋━━━━━━━━━━━━━━━━━━━━━━━━┫ ╲ ┃ 32 │ wireId_2 ┃ fs │ b_{0,wireId_2} ┃ ╲ ┗━━━━┻━━━━━━━━━━━━━━━━━┻━━━━┻━━━━━━━━━━━━━━━━━━━━━━━━┛ ╱ Constraint_0 ... ... ╱ ┏━━━━┳━━━━━━━━━━━━━━━━━┳━━━━┳━━━━━━━━━━━━━━━━━━━━━━━━┓ │ ┃ 32 │ wireId_nB ┃ fs │ b_{0,wireId_nB} ┃ │ ┗━━━━┻━━━━━━━━━━━━━━━━━┻━━━━┻━━━━━━━━━━━━━━━━━━━━━━━━┛ │ ┏━━━━┳━━━━━━━━━━━━━━━━━┓ │ ┃ 32 │ nC ┃ │ ┣━━━━╋━━━━━━━━━━━━━━━━━╋━━━━┳━━━━━━━━━━━━━━━━━━━━━━━━┓ │ ┃ 32 │ wireId_1 ┃ fs │ c_{0,wireId_1} ┃ │ ┣━━━━╋━━━━━━━━━━━━━━━━━╋━━━━╋━━━━━━━━━━━━━━━━━━━━━━━━┫ │ ┃ 32 │ wireId_2 ┃ fs │ c_{0,wireId_2} ┃ │ ┗━━━━┻━━━━━━━━━━━━━━━━━┻━━━━┻━━━━━━━━━━━━━━━━━━━━━━━━┛ │ ... ... │ ┏━━━━┳━━━━━━━━━━━━━━━━━┳━━━━┳━━━━━━━━━━━━━━━━━━━━━━━━┓ │ ┃ 32 │ wireId_nC ┃ fs │ c_{0,wireId_nC} ┃ ╱ ┗━━━━┻━━━━━━━━━━━━━━━━━┻━━━━┻━━━━━━━━━━━━━━━━━━━━━━━━┛ ╱ ╱
┏━━━━┳━━━━━━━━━━━━━━━━━┓ ╲ ┃ 32 │ nA ┃ ╲ ┣━━━━╋━━━━━━━━━━━━━━━━━╋━━━━┳━━━━━━━━━━━━━━━━━━━━━━━━┓ ╲ ┃ 32 │ wireId_1 ┃ fs │ a_{1,wireId_1} ┃ │ ┣━━━━╋━━━━━━━━━━━━━━━━━╋━━━━╋━━━━━━━━━━━━━━━━━━━━━━━━┫ │ ┃ 32 │ wireId_2 ┃ fs │ a_{1,wireId_2} ┃ │ ┗━━━━┻━━━━━━━━━━━━━━━━━┻━━━━┻━━━━━━━━━━━━━━━━━━━━━━━━┛ │ ... ... │ ┏━━━━┳━━━━━━━━━━━━━━━━━┳━━━━┳━━━━━━━━━━━━━━━━━━━━━━━━┓ │ ┃ 32 │ wireId_nA ┃ fs │ a_{1,wireId_nA} ┃ │ ┗━━━━┻━━━━━━━━━━━━━━━━━┻━━━━┻━━━━━━━━━━━━━━━━━━━━━━━━┛ │ ┏━━━━┳━━━━━━━━━━━━━━━━━┓ │ ┃ 32 │ nB ┃ │ ┣━━━━╋━━━━━━━━━━━━━━━━━╋━━━━┳━━━━━━━━━━━━━━━━━━━━━━━━┓ │ ┃ 32 │ wireId_1 ┃ fs │ b_{1,wireId_1} ┃ │ ┣━━━━╋━━━━━━━━━━━━━━━━━╋━━━━╋━━━━━━━━━━━━━━━━━━━━━━━━┫ ╲ ┃ 32 │ wireId_2 ┃ fs │ b_{1,wireId_2} ┃ ╲ ┗━━━━┻━━━━━━━━━━━━━━━━━┻━━━━┻━━━━━━━━━━━━━━━━━━━━━━━━┛ ╱ Constraint_1 ... ... ╱ ┏━━━━┳━━━━━━━━━━━━━━━━━┳━━━━┳━━━━━━━━━━━━━━━━━━━━━━━━┓ │ ┃ 32 │ wireId_nB ┃ fs │ b_{1,wireId_nB} ┃ │ ┗━━━━┻━━━━━━━━━━━━━━━━━┻━━━━┻━━━━━━━━━━━━━━━━━━━━━━━━┛ │ ┏━━━━┳━━━━━━━━━━━━━━━━━┓ │ ┃ 32 │ nC ┃ │ ┣━━━━╋━━━━━━━━━━━━━━━━━╋━━━━┳━━━━━━━━━━━━━━━━━━━━━━━━┓ │ ┃ 32 │ wireId_1 ┃ fs │ c_{1,wireId_1} ┃ │ ┣━━━━╋━━━━━━━━━━━━━━━━━╋━━━━╋━━━━━━━━━━━━━━━━━━━━━━━━┫ │ ┃ 32 │ wireId_2 ┃ fs │ c_{1,wireId_2} ┃ │ ┗━━━━┻━━━━━━━━━━━━━━━━━┻━━━━┻━━━━━━━━━━━━━━━━━━━━━━━━┛ │ ... ... │ ┏━━━━┳━━━━━━━━━━━━━━━━━┳━━━━┳━━━━━━━━━━━━━━━━━━━━━━━━┓ │ ┃ 32 │ wireId_nC ┃ fs │ c_{1,wireId_nC} ┃ ╱ ┗━━━━┻━━━━━━━━━━━━━━━━━┻━━━━┻━━━━━━━━━━━━━━━━━━━━━━━━┛ ╱ ╱
... ... ...
┏━━━━┳━━━━━━━━━━━━━━━━━┓ ╲ ┃ 32 │ nA ┃ ╲ ┣━━━━╋━━━━━━━━━━━━━━━━━╋━━━━┳━━━━━━━━━━━━━━━━━━━━━━━━┓ ╲ ┃ 32 │ wireId_1 ┃ fs │ a_{m-1,wireId_1} ┃ │ ┣━━━━╋━━━━━━━━━━━━━━━━━╋━━━━╋━━━━━━━━━━━━━━━━━━━━━━━━┫ │ ┃ 32 │ wireId_2 ┃ fs │ a_{m-1,wireId_2} ┃ │ ┗━━━━┻━━━━━━━━━━━━━━━━━┻━━━━┻━━━━━━━━━━━━━━━━━━━━━━━━┛ │ ... ... │ ┏━━━━┳━━━━━━━━━━━━━━━━━┳━━━━┳━━━━━━━━━━━━━━━━━━━━━━━━┓ │ ┃ 32 │ wireId_nA ┃ fs │ a_{m-1,wireId_nA} ┃ │ ┗━━━━┻━━━━━━━━━━━━━━━━━┻━━━━┻━━━━━━━━━━━━━━━━━━━━━━━━┛ │ ┏━━━━┳━━━━━━━━━━━━━━━━━┓ │ ┃ 32 │ nB ┃ │ ┣━━━━╋━━━━━━━━━━━━━━━━━╋━━━━┳━━━━━━━━━━━━━━━━━━━━━━━━┓ │ ┃ 32 │ wireId_1 ┃ fs │ b_{m-1,wireId_1} ┃ │ ┣━━━━╋━━━━━━━━━━━━━━━━━╋━━━━╋━━━━━━━━━━━━━━━━━━━━━━━━┫ ╲ ┃ 32 │ wireId_2 ┃ fs │ b_{m-1,wireId_2} ┃ ╲ ┗━━━━┻━━━━━━━━━━━━━━━━━┻━━━━┻━━━━━━━━━━━━━━━━━━━━━━━━┛ ╱ Constraint_{m-1} ... ... ╱ ┏━━━━┳━━━━━━━━━━━━━━━━━┳━━━━┳━━━━━━━━━━━━━━━━━━━━━━━━┓ │ ┃ 32 │ wireId_nB ┃ fs │ b_{m-1,wireId_nB} ┃ │ ┗━━━━┻━━━━━━━━━━━━━━━━━┻━━━━┻━━━━━━━━━━━━━━━━━━━━━━━━┛ │ ┏━━━━┳━━━━━━━━━━━━━━━━━┓ │ ┃ 32 │ nC ┃ │ ┣━━━━╋━━━━━━━━━━━━━━━━━╋━━━━┳━━━━━━━━━━━━━━━━━━━━━━━━┓ │ ┃ 32 │ wireId_1 ┃ fs │ c_{m-1,wireId_1} ┃ │ ┣━━━━╋━━━━━━━━━━━━━━━━━╋━━━━╋━━━━━━━━━━━━━━━━━━━━━━━━┫ │ ┃ 32 │ wireId_2 ┃ fs │ c_{m-1,wireId_2} ┃ │ ┗━━━━┻━━━━━━━━━━━━━━━━━┻━━━━┻━━━━━━━━━━━━━━━━━━━━━━━━┛ │ ... ... │ ┏━━━━┳━━━━━━━━━━━━━━━━━┳━━━━┳━━━━━━━━━━━━━━━━━━━━━━━━┓ │ ┃ 32 │ wireId_nC ┃ fs │ c_{m-1,wireId_nC} ┃ ╱ ┗━━━━┻━━━━━━━━━━━━━━━━━┻━━━━┻━━━━━━━━━━━━━━━━━━━━━━━━┛ ╱ ╱````
#### Constraints
Each constraint contains 3 linear combinations A, B, C.
The constraint is such that:```A*B-C = 0```
#### Linear combination
Each linear combination is of the form:
$$a_{j,0}w_0 + a_{j,1}w_1 + ... + a_{j,n}w_{n}$$
#### Number of nonZero Factors
Size: 4 bytesFormat: Little-Endian
Total number of non Zero factors in the linear compination.
The factors MUST be sorted in ascending order.
#### Factor
For each factor we have the index of the factor and the value of the factor.
#### WireId of the factor
Size: 4 bytesFormat: Little-Endian
WireId of the nonZero Factor
#### Value of the factor
This is the factor that multiplies the associated wire in the linear convination.
For example, to represent the linear combination:
$$5w_4 +8w_5 + 260w_{886}$$
The linear combination would be represented as:
```` ┏━━━━━━━━━━━━━━━━━┓ ┃ 03 00 00 00 ┃ ┣━━━━━━━━━━━━━━━━━╋━━━━━━━━━━━━━━━━━━━━━━━━━━━━━━━━━━━━━━━━━━━━━━━━━━━━━━━━━━━━━━━━━━━━━━━━━┓ ┃ 04 00 00 00 ┃ 05000000 00000000 00000000 00000000 00000000 00000000 00000000 00000000 ┃ ┣━━━━━━━━━━━━━━━━━╋━━━━━━━━━━━━━━━━━━━━━━━━━━━━━━━━━━━━━━━━━━━━━━━━━━━━━━━━━━━━━━━━━━━━━━━━━┫ ┃ 05 00 00 00 ┃ 08000000 00000000 00000000 00000000 00000000 00000000 00000000 00000000 ┃ ┣━━━━━━━━━━━━━━━━━╋━━━━━━━━━━━━━━━━━━━━━━━━━━━━━━━━━━━━━━━━━━━━━━━━━━━━━━━━━━━━━━━━━━━━━━━━━┫ ┃ 76 03 00 00 ┃ 04010000 00000000 00000000 00000000 00000000 00000000 00000000 00000000 ┃ ┗━━━━━━━━━━━━━━━━━┻━━━━━━━━━━━━━━━━━━━━━━━━━━━━━━━━━━━━━━━━━━━━━━━━━━━━━━━━━━━━━━━━━━━━━━━━━┛````
### WireId2LabelId Map Section
Section Type: 0x03
````┏━━┳━━━━━━━━━━━━━━━━━━━┳━━┳━━━━━━━━━━━━━━━━━━━┓ ┏━━┳━━━━━━━━━━━━━━━━━━━┓┃64│ labelId of Wire_0 ┃64│ labelId of Wire_1 ┃ ... ┃64│ labelId of Wire_n ┃┗━━┻━━━━━━━━━━━━━━━━━━━┻━━┻━━━━━━━━━━━━━━━━━━━┛ ┗━━┻━━━━━━━━━━━━━━━━━━━┛````
## Rationale
Variable size for field elements allows to shrink the size of the file and allows to work with any field.
Version allows to update the format.
Have a very good comprasion ratio for sparse r1cs as it's the normal case.
The motivation of having a map between l and w is that this allows optimizers to calculate equivalent r1cs systems but keeping the original values geneated by the circuit.
## Backward Compatibility
N.A.
## Test Cases
### Example
Given this r1cs in a 256 bit Field:
$$\left\{ \begin{array}{rclclcl}(3w_5 + 8w_6) &\cdot& (2w_0 + 20w_2 + 12w_3) &-& (5w_0 + 7w_2) &=& 0 \\(4w_1 + 8w_4 + 3w_5) &\cdot& (6w_6 + 44w_3) && &=& 0 \\(4w_6) &\cdot& (6w_0 + 5w_3 + 11s_2) &-& (600w_6) &=& 0\end{array} \right.$$
And a Wire to label map.
$$w_0 := l_0 \\w_1 := l_3 \\w_2 := l_{10} \\w_3 := l_{11} \\w_4 := l_{12} \\w_5 := l_{15} \\w_6 := l_{324} \\$$
The format will be:
```` ┏━━━━━━━━━━┓ ┃ 72316377 ┃ Magic ┣━━━━━━━━━━┫ ┃ 01000000 ┃ Version ┣━━━━━━━━━━┫ ┃ 03000000 ┃ nSections ┗━━━━━━━━━━┛ ┏━━━━━━━━━━┳━━━━━━━━━━━━━━━━━━━┓ ┃ 01000000 ┃ 40000000 00000000 ┃ SectionType: Header ┗━━━━━━━━━━┻━━━━━━━━━━━━━━━━━━━┛ ┏━━━━━━━━━━┓ ┃ 20000000 ┃ Field Size ┣━━━━━━━━━━┻━━━━━━━━━━━━━━━━━━━━━━━━━━━━━━━━━━━━━━━━━━━━━━━━━━━━━━━━━━━━━┓ ┃ 010000f0 93f5e143 9170b979 48e83328 5d588181 b64550b8 29a031e1 724e6430 ┃ ┣━━━━━━━━━━┳━━━━━━━━━━━━━━━━━━━━━━━━━━━━━━━━━━━━━━━━━━━━━━━━━━━━━━━━━━━━━┛ ┃ 07000000 ┃ # of wires ┣━━━━━━━━━━┫ ┃ 01000000 ┃ # Public Outs ┣━━━━━━━━━━┫ ┃ 02000000 ┃ # Public Ins ┣━━━━━━━━━━┫ ┃ 03000000 ┃ # Private Ins ┣━━━━━━━━━━┻━━━━━━━━┓ ┃ e8030000 00000000 ┃ # Labels ┣━━━━━━━━━━┳━━━━━━━━┛ ┃ 03000000 ┃ # Constraints ┗━━━━━━━━━━┛ ┏━━━━━━━━━━┳━━━━━━━━━━━━━━━━━━━┓ ┃ 02000000 ┃ 88200000 00000000 ┃ SectionType: Constraints ┗━━━━━━━━━━┻━━━━━━━━━━━━━━━━━━━┛ ┏━━━━━━━━━━┓ Constraint 0: (3w_5 + 8w_6) * (2w_0 + 20w_2 + 12w_3) - (5w_0 + 7w_2) = 0 ┃ 02000000 ┃ ┣━━━━━━━━━━╋━━━━━━━━━━━━━━━━━━━━━━━━━━━━━━━━━━━━━━━━━━━━━━━━━━━━━━━━━━━━━━━━━━━━━━━━━┓ ┃ 05000000 ┃ 03000000 00000000 00000000 00000000 00000000 00000000 00000000 00000000 ┃ ┣━━━━━━━━━━╋━━━━━━━━━━━━━━━━━━━━━━━━━━━━━━━━━━━━━━━━━━━━━━━━━━━━━━━━━━━━━━━━━━━━━━━━━┫ ┃ 06000000 ┃ 01000000 00000000 00000000 00000000 00000000 00000000 00000000 00000000 ┃ ┗━━━━━━━━━━┻━━━━━━━━━━━━━━━━━━━━━━━━━━━━━━━━━━━━━━━━━━━━━━━━━━━━━━━━━━━━━━━━━━━━━━━━━┛ ┏━━━━━━━━━━┓ ┃ 03000000 ┃ ┣━━━━━━━━━━╋━━━━━━━━━━━━━━━━━━━━━━━━━━━━━━━━━━━━━━━━━━━━━━━━━━━━━━━━━━━━━━━━━━━━━━━━━┓ ┃ 00000000 ┃ 02000000 00000000 00000000 00000000 00000000 00000000 00000000 00000000 ┃ ┣━━━━━━━━━━╋━━━━━━━━━━━━━━━━━━━━━━━━━━━━━━━━━━━━━━━━━━━━━━━━━━━━━━━━━━━━━━━━━━━━━━━━━┫ ┃ 02000000 ┃ 01140000 00000000 00000000 00000000 00000000 00000000 00000000 00000000 ┃ ┣━━━━━━━━━━╋━━━━━━━━━━━━━━━━━━━━━━━━━━━━━━━━━━━━━━━━━━━━━━━━━━━━━━━━━━━━━━━━━━━━━━━━━┫ ┃ 03000000 ┃ 0C000000 00000000 00000000 00000000 00000000 00000000 00000000 00000000 ┃ ┗━━━━━━━━━━┻━━━━━━━━━━━━━━━━━━━━━━━━━━━━━━━━━━━━━━━━━━━━━━━━━━━━━━━━━━━━━━━━━━━━━━━━━┛ ┏━━━━━━━━━━┓ ┃ 02000000 ┃ ┣━━━━━━━━━━╋━━━━━━━━━━━━━━━━━━━━━━━━━━━━━━━━━━━━━━━━━━━━━━━━━━━━━━━━━━━━━━━━━━━━━━━━━┓ ┃ 00000000 ┃ 05000000 00000000 00000000 00000000 00000000 00000000 00000000 00000000 ┃ ┣━━━━━━━━━━╋━━━━━━━━━━━━━━━━━━━━━━━━━━━━━━━━━━━━━━━━━━━━━━━━━━━━━━━━━━━━━━━━━━━━━━━━━┫ ┃ 02000000 ┃ 07000000 00000000 00000000 00000000 00000000 00000000 00000000 00000000 ┃ ┗━━━━━━━━━━┻━━━━━━━━━━━━━━━━━━━━━━━━━━━━━━━━━━━━━━━━━━━━━━━━━━━━━━━━━━━━━━━━━━━━━━━━━┛
┏━━━━━━━━━━┓ Constraint 1: (4w_1 + 8w_4 + 3w_5) * (6w_6 + 44w_3) = 0 ┃ 03000000 ┃ ┣━━━━━━━━━━╋━━━━━━━━━━━━━━━━━━━━━━━━━━━━━━━━━━━━━━━━━━━━━━━━━━━━━━━━━━━━━━━━━━━━━━━━━┓ ┃ 01000000 ┃ 04000000 00000000 00000000 00000000 00000000 00000000 00000000 00000000 ┃ ┣━━━━━━━━━━╋━━━━━━━━━━━━━━━━━━━━━━━━━━━━━━━━━━━━━━━━━━━━━━━━━━━━━━━━━━━━━━━━━━━━━━━━━┫ ┃ 04000000 ┃ 08000000 00000000 00000000 00000000 00000000 00000000 00000000 00000000 ┃ ┣━━━━━━━━━━╋━━━━━━━━━━━━━━━━━━━━━━━━━━━━━━━━━━━━━━━━━━━━━━━━━━━━━━━━━━━━━━━━━━━━━━━━━┫ ┃ 05000000 ┃ 03000000 00000000 00000000 00000000 00000000 00000000 00000000 00000000 ┃ ┗━━━━━━━━━━┻━━━━━━━━━━━━━━━━━━━━━━━━━━━━━━━━━━━━━━━━━━━━━━━━━━━━━━━━━━━━━━━━━━━━━━━━━┛ ┏━━━━━━━━━━┓ ┃ 02000000 ┃ ┣━━━━━━━━━━╋━━━━━━━━━━━━━━━━━━━━━━━━━━━━━━━━━━━━━━━━━━━━━━━━━━━━━━━━━━━━━━━━━━━━━━━━━┓ ┃ 03000000 ┃ 2C000000 00000000 00000000 00000000 00000000 00000000 00000000 00000000 ┃ ┣━━━━━━━━━━╋━━━━━━━━━━━━━━━━━━━━━━━━━━━━━━━━━━━━━━━━━━━━━━━━━━━━━━━━━━━━━━━━━━━━━━━━━┫ ┃ 06000000 ┃ 06000000 00000000 00000000 00000000 00000000 00000000 00000000 00000000 ┃ ┗━━━━━━━━━━┻━━━━━━━━━━━━━━━━━━━━━━━━━━━━━━━━━━━━━━━━━━━━━━━━━━━━━━━━━━━━━━━━━━━━━━━━━┛ ┏━━━━━━━━━━┓ ┃ 00000000 ┃ ┗━━━━━━━━━━┛
┏━━━━━━━━━━┓ Constraint 2: (4w_6) * (6w_0 + 5w_3 + 11w_2) - (600w_6) = 0 ┃ 01000000 ┃ ┣━━━━━━━━━━╋━━━━━━━━━━━━━━━━━━━━━━━━━━━━━━━━━━━━━━━━━━━━━━━━━━━━━━━━━━━━━━━━━━━━━━━━━┓ ┃ 06000000 ┃ 04000000 00000000 00000000 00000000 00000000 00000000 00000000 00000000 ┃ ┗━━━━━━━━━━┻━━━━━━━━━━━━━━━━━━━━━━━━━━━━━━━━━━━━━━━━━━━━━━━━━━━━━━━━━━━━━━━━━━━━━━━━━┛ ┏━━━━━━━━━━┓ ┃ 03000000 ┃ ┣━━━━━━━━━━╋━━━━━━━━━━━━━━━━━━━━━━━━━━━━━━━━━━━━━━━━━━━━━━━━━━━━━━━━━━━━━━━━━━━━━━━━━┓ ┃ 00000000 ┃ 06000000 00000000 00000000 00000000 00000000 00000000 00000000 00000000 ┃ ┣━━━━━━━━━━╋━━━━━━━━━━━━━━━━━━━━━━━━━━━━━━━━━━━━━━━━━━━━━━━━━━━━━━━━━━━━━━━━━━━━━━━━━┫ ┃ 02000000 ┃ 0B000000 00000000 00000000 00000000 00000000 00000000 00000000 00000000 ┃ ┣━━━━━━━━━━╋━━━━━━━━━━━━━━━━━━━━━━━━━━━━━━━━━━━━━━━━━━━━━━━━━━━━━━━━━━━━━━━━━━━━━━━━━┫ ┃ 03000000 ┃ 05000000 00000000 00000000 00000000 00000000 00000000 00000000 00000000 ┃ ┗━━━━━━━━━━┻━━━━━━━━━━━━━━━━━━━━━━━━━━━━━━━━━━━━━━━━━━━━━━━━━━━━━━━━━━━━━━━━━━━━━━━━━┛ ┏━━━━━━━━━━┓ ┃ 01000000 ┃ ┣━━━━━━━━━━╋━━━━━━━━━━━━━━━━━━━━━━━━━━━━━━━━━━━━━━━━━━━━━━━━━━━━━━━━━━━━━━━━━━━━━━━━━┓ ┃ 06000000 ┃ 58020000 00000000 00000000 00000000 00000000 00000000 00000000 00000000 ┃ ┗━━━━━━━━━━┻━━━━━━━━━━━━━━━━━━━━━━━━━━━━━━━━━━━━━━━━━━━━━━━━━━━━━━━━━━━━━━━━━━━━━━━━━┛
┏━━━━━━━━━━┳━━━━━━━━━━━━━━━━━━━┓ ┃ 03000000 ┃ 38000000 00000000 ┃ Wire to Label Map ┗━━━━━━━━━━┻━━━━━━━━━━━━━━━━━━━┛ ┏━━━━━━━━━━━━━━━━━━━┓ ┃ 00000000 00000000 ┃ ┣━━━━━━━━━━━━━━━━━━━┫ ┃ 03000000 00000000 ┃ ┣━━━━━━━━━━━━━━━━━━━┫ ┃ 0a000000 00000000 ┃ ┣━━━━━━━━━━━━━━━━━━━┫ ┃ 0b000000 00000000 ┃ ┣━━━━━━━━━━━━━━━━━━━┫ ┃ 0c000000 00000000 ┃ ┣━━━━━━━━━━━━━━━━━━━┫ ┃ 0f000000 00000000 ┃ ┣━━━━━━━━━━━━━━━━━━━┫ ┃ 44010000 00000000 ┃ ┗━━━━━━━━━━━━━━━━━━━┛````
And the binary representation in Hex:
````72316377010000000300000001000000 40000000 0000000020000000010000f0 93f5e143 9170b979 48e83328 5d588181 b64550b8 29a031e1 724e643007000000010000000200000003000000e8030000 000000000300000002000000 88200000 000000000200000005000000 03000000 00000000 00000000 00000000 00000000 00000000 00000000 0000000006000000 01000000 00000000 00000000 00000000 00000000 00000000 00000000 000000000300000000000000 02000000 00000000 00000000 00000000 00000000 00000000 00000000 0000000002000000 01140000 00000000 00000000 00000000 00000000 00000000 00000000 0000000003000000 0C000000 00000000 00000000 00000000 00000000 00000000 00000000 000000000200000000000000 05000000 00000000 00000000 00000000 00000000 00000000 00000000 0000000002000000 07000000 00000000 00000000 00000000 00000000 00000000 00000000 000000000300000001000000 04000000 00000000 00000000 00000000 00000000 00000000 00000000 0000000004000000 08000000 00000000 00000000 00000000 00000000 00000000 00000000 0000000005000000 03000000 00000000 00000000 00000000 00000000 00000000 00000000 000000000200000003000000 2C000000 00000000 00000000 00000000 00000000 00000000 00000000 0000000006000000 06000000 00000000 00000000 00000000 00000000 00000000 00000000 00000000000000000100000006000000 04000000 00000000 00000000 00000000 00000000 00000000 00000000 000000000300000000000000 06000000 00000000 00000000 00000000 00000000 00000000 00000000 0000000002000000 0B000000 00000000 00000000 00000000 00000000 00000000 00000000 0000000003000000 05000000 00000000 00000000 00000000 00000000 00000000 00000000 000000000100000006000000 58020000 00000000 00000000 00000000 00000000 00000000 00000000 0000000003000000 38000000 0000000000000000 0000000003000000 000000000a000000 000000000b000000 000000000c000000 000000000f000000 0000000044010000 00000000
````
## Implementation
circom will output this format.
## Copyright
Copyright and related rights waived via [CC0](https://creativecommons.org/publicdomain/zero/1.0/).
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