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Jordi Baylina
2019-12-16 21:37:14 +01:00
parent ec0e7f421b
commit e62c1cdbc3
19 changed files with 956 additions and 314 deletions
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doc/r1cs_bin_format.md
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@@ -29,23 +29,26 @@ This standard specifies a format for a r1cs and allows the to connect a set of t
### General considerations
All integers are represented in Little Endian Fix size format
The standard extension is `.r1cs`
The constraint is in the form
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
$$
\left\{ \begin{array}{rclclcl}
(a_{0,0}s_0 + a_{0,1}s_1 + ... + a_{0,n-1}s_{n-1}) &\cdot& (b_{0,0} s_0 + b_{0,1} s_1 + ... + b_{0,n-1} s_{n-1}) &-& (c_{0,0} s_0 + c_{0,1} s_1 + ... + c_{0,n-1}s_{n-1}) &=& 0 \\
(a_{1,0}s_0 + a_{1,1}s_1 + ... + a_{1,n-1}s_{n-1}) &\cdot& (b_{1,0} s_0 + b_{1,1} s_1 + ... + b_{1,n-1} s_{n-1}) &-& (c_{1,0} s_0 + c_{1,1}s_1 + ... + c_{1,n-1}s_{n-1}) &=& 0 \\
(a_{0,0}w_0 + a_{0,1}w_1 + ... + a_{0,n}w_{n}) &\cdot& (b_{0,0} w_0 + b_{0,1} w_1 + ... + b_{0,n} w_{n}) &-& (c_{0,0} w_0 + c_{0,1} w_1 + ... + c_{0,n}w_{n}) &=& 0 \\
(a_{1,0}w_0 + a_{1,1}w_1 + ... + a_{1,n}w_{n}) &\cdot& (b_{1,0} w_0 + b_{1,1} w_1 + ... + b_{1,n} w_{n}) &-& (c_{1,0} w_0 + c_{1,1}w_1 + ... + c_{1,n}w_{n}) &=& 0 \\
...\\
(a_{m-1,0}s_0 + a_{m-1,1}s_1 + ... + a_{m-1,n-1}s_{n-1}) &\cdot& (b_{m-1,0} s_0 + b_{m-1,1} s_1 + ... + b_{m-1,n-1} s_{n-1}) &-& (c_{m-1,0} s_0 + c_{m-1,1}s_1 + ... + c_{m-1,n-1}s_{n-1}) &=& 0
(a_{m-1,0}w_0 + a_{m-1,1}w_1 + ... + a_{m-1,n}w_{n}) &\cdot& (b_{m-1,0} w_0 + b_{m-1,1} w_1 + ... + b_{m-1,n} w_{n}) &-& (c_{m-1,0} w_0 + c_{m-1,1}w_1 + ... + c_{m-1,n}w_{n}) &=& 0
\end{array} \right.
$$
Wire 0 must be always mapped to label 0 and it's an input forced to value "1" implicitly
### Format
### Format of the file
````
@@ -55,150 +58,50 @@ $$
┏━━━━┳━━━━━━━━━━━━━━━━━┓
┃ 4 │ 01 00 00 00 ┃ Version 1
┗━━━━┻━━━━━━━━━━━━━━━━━┛
┏━━━━┳━━━━━━━━━━━━━━━━━┓
┃ 4 │ nW ┃
┃ 4 │ 03 00 00 00 ┃ Number of Sections
┗━━━━┻━━━━━━━━━━━━━━━━━┛
┏━━━━┳━━━━━━━━━━━━━━━━━┳━━━━━┳━━━━━━━━━━━━━━━━━━━━━━━━┓
┃ 4 │ sectionType ┃ 8 │ SectionSize ┃
┗━━━━┻━━━━━━━━━━━━━━━━━┻━━━━━┻━━━━━━━━━━━━━━━━━━━━━━━━┛
┏━━━━━━━━━━━━━━━━━━━━━┓
┃ ┃
┃ ┃
┃ ┃
┃ Section Content ┃
┃ ┃
┃ ┃
┃ ┃
┗━━━━━━━━━━━━━━━━━━━━━┛
┏━━━━┳━━━━━━━━━━━━━━━━━┓
┃ nW │ 01 00 00 00 ┃ nWires
┗━━━━┻━━━━━━━━━━━━━━━━━┛
┏━━━━┳━━━━━━━━━━━━━━━━━┓
┃ nW │ 01 00 00 00 ┃ nPubOut
┗━━━━┻━━━━━━━━━━━━━━━━━┛
┏━━━━┳━━━━━━━━━━━━━━━━━┓
┃ nW │ 01 00 00 00 ┃ nPubIn
┗━━━━┻━━━━━━━━━━━━━━━━━┛
┏━━━━┳━━━━━━━━━━━━━━━━━┓
┃ nW │ 01 00 00 00 ┃ nPrvIn
┗━━━━┻━━━━━━━━━━━━━━━━━┛
┏━━━━┳━━━━━━━━━━━━━━━━━┳━━━━━┳━━━━━━━━━━━━━━━━━━━━━━━━┓
┃ 4 │ sectionType ┃ 8 │ SectionSize ┃
┗━━━━┻━━━━━━━━━━━━━━━━━┻━━━━━┻━━━━━━━━━━━━━━━━━━━━━━━━┛
┏━━━━━━━━━━━━━━━━━━━━━┓
┃ ┃
┃ ┃
┃ ┃
┃ Section Content ┃
┃ ┃
┃ ┃
┃ ┃
┗━━━━━━━━━━━━━━━━━━━━━┛
┏━━━━┳━━━━━━━━━━━━━━━━━┓
┃ nW │m := NConstraints┃
┗━━━━┻━━━━━━━━━━━━━━━━━┛
┏━━━━┳━━━━━━━━━━━━━━━━━┓ ╲
┃ nW │ nA ┃ ╲
┣━━━━╋━━━━━━━━━━━━━━━━━╋━━━━━┳━━━━━━━━━━━━━━━━━━━━━━━━┓ ╲
┃ nW │ idx_1 ┃ V │ a_{0,idx_1} ┃ │
┣━━━━╋━━━━━━━━━━━━━━━━━╋━━━━━╋━━━━━━━━━━━━━━━━━━━━━━━━┫ │
┃ nW │ idx_2 ┃ V │ a_{0,idx_2} ┃ │
┗━━━━┻━━━━━━━━━━━━━━━━━┻━━━━━┻━━━━━━━━━━━━━━━━━━━━━━━━┛ │
... ... │
┏━━━━┳━━━━━━━━━━━━━━━━━┳━━━━━┳━━━━━━━━━━━━━━━━━━━━━━━━┓ │
┃ nW │ idx_nA ┃ V │ a_{0,idx_nA} ┃ │
┗━━━━┻━━━━━━━━━━━━━━━━━┻━━━━━┻━━━━━━━━━━━━━━━━━━━━━━━━┛ │
┏━━━━┳━━━━━━━━━━━━━━━━━┓ │
┃ nW │ nB ┃ │
┣━━━━╋━━━━━━━━━━━━━━━━━╋━━━━━┳━━━━━━━━━━━━━━━━━━━━━━━━┓ │
┃ nW │ idx_1 ┃ V │ b_{0,idx_1} ┃ │
┣━━━━╋━━━━━━━━━━━━━━━━━╋━━━━━╋━━━━━━━━━━━━━━━━━━━━━━━━┫ ╲
┃ nW │ idx_2 ┃ V │ b_{0,idx_2} ┃ ╲
┗━━━━┻━━━━━━━━━━━━━━━━━┻━━━━━┻━━━━━━━━━━━━━━━━━━━━━━━━┛ ╱ Constraint_0
... ... ╱
┏━━━━┳━━━━━━━━━━━━━━━━━┳━━━━━┳━━━━━━━━━━━━━━━━━━━━━━━━┓ │
┃ nW │ idx_nB ┃ V │ b_{0,idx_nB} ┃ │
┗━━━━┻━━━━━━━━━━━━━━━━━┻━━━━━┻━━━━━━━━━━━━━━━━━━━━━━━━┛ │
┏━━━━┳━━━━━━━━━━━━━━━━━┓ │
┃ nW │ nC ┃ │
┣━━━━╋━━━━━━━━━━━━━━━━━╋━━━━━┳━━━━━━━━━━━━━━━━━━━━━━━━┓ │
┃ nW │ idx_1 ┃ V │ c_{0,idx_1} ┃ │
┣━━━━╋━━━━━━━━━━━━━━━━━╋━━━━━╋━━━━━━━━━━━━━━━━━━━━━━━━┫ │
┃ nW │ idx_2 ┃ V │ c_{0,idx_2} ┃ │
┗━━━━┻━━━━━━━━━━━━━━━━━┻━━━━━┻━━━━━━━━━━━━━━━━━━━━━━━━┛ │
... ... │
┏━━━━┳━━━━━━━━━━━━━━━━━┳━━━━━┳━━━━━━━━━━━━━━━━━━━━━━━━┓ │
┃ nW │ idx_nB ┃ V │ c_{0,idx_nC} ┃ ╱
┗━━━━┻━━━━━━━━━━━━━━━━━┻━━━━━┻━━━━━━━━━━━━━━━━━━━━━━━━┛ ╱
╱
┏━━━━┳━━━━━━━━━━━━━━━━━┓ ╲
┃ nW │ nA ┃ ╲
┣━━━━╋━━━━━━━━━━━━━━━━━╋━━━━━┳━━━━━━━━━━━━━━━━━━━━━━━━┓ ╲
┃ nW │ idx_1 ┃ V │ a_{1,idx_1} ┃ │
┣━━━━╋━━━━━━━━━━━━━━━━━╋━━━━━╋━━━━━━━━━━━━━━━━━━━━━━━━┫ │
┃ nW │ idx_2 ┃ V │ a_{1,idx_2} ┃ │
┗━━━━┻━━━━━━━━━━━━━━━━━┻━━━━━┻━━━━━━━━━━━━━━━━━━━━━━━━┛ │
... ... │
┏━━━━┳━━━━━━━━━━━━━━━━━┳━━━━━┳━━━━━━━━━━━━━━━━━━━━━━━━┓ │
┃ nW │ idx_nA ┃ V │ a_{1,idx_nA} ┃ │
┗━━━━┻━━━━━━━━━━━━━━━━━┻━━━━━┻━━━━━━━━━━━━━━━━━━━━━━━━┛ │
┏━━━━┳━━━━━━━━━━━━━━━━━┓ │
┃ nW │ nB ┃ │
┣━━━━╋━━━━━━━━━━━━━━━━━╋━━━━━┳━━━━━━━━━━━━━━━━━━━━━━━━┓ │
┃ nW │ idx_1 ┃ V │ b_{1,idx_1} ┃ │
┣━━━━╋━━━━━━━━━━━━━━━━━╋━━━━━╋━━━━━━━━━━━━━━━━━━━━━━━━┫ ╲
┃ nW │ idx_2 ┃ V │ b_{1,idx_2} ┃ ╲
┗━━━━┻━━━━━━━━━━━━━━━━━┻━━━━━┻━━━━━━━━━━━━━━━━━━━━━━━━┛ ╱ Constraint_1
... ... ╱
┏━━━━┳━━━━━━━━━━━━━━━━━┳━━━━━┳━━━━━━━━━━━━━━━━━━━━━━━━┓ │
┃ nW │ idx_nB ┃ V │ b_{1,idx_nB} ┃ │
┗━━━━┻━━━━━━━━━━━━━━━━━┻━━━━━┻━━━━━━━━━━━━━━━━━━━━━━━━┛ │
┏━━━━┳━━━━━━━━━━━━━━━━━┓ │
┃ nW │ nC ┃ │
┣━━━━╋━━━━━━━━━━━━━━━━━╋━━━━━┳━━━━━━━━━━━━━━━━━━━━━━━━┓ │
┃ nW │ idx_1 ┃ V │ c_{1,idx_1} ┃ │
┣━━━━╋━━━━━━━━━━━━━━━━━╋━━━━━╋━━━━━━━━━━━━━━━━━━━━━━━━┫ │
┃ nW │ idx_2 ┃ V │ c_{1,idx_2} ┃ │
┗━━━━┻━━━━━━━━━━━━━━━━━┻━━━━━┻━━━━━━━━━━━━━━━━━━━━━━━━┛ │
... ... │
┏━━━━┳━━━━━━━━━━━━━━━━━┳━━━━━┳━━━━━━━━━━━━━━━━━━━━━━━━┓ │
┃ nW │ idx_nB ┃ V │ c_{1,idx_nC} ┃ ╱
┗━━━━┻━━━━━━━━━━━━━━━━━┻━━━━━┻━━━━━━━━━━━━━━━━━━━━━━━━┛ ╱
╱
...
...
...
┏━━━━┳━━━━━━━━━━━━━━━━━┓ ╲
┃ nW │ nA ┃ ╲
┣━━━━╋━━━━━━━━━━━━━━━━━╋━━━━━┳━━━━━━━━━━━━━━━━━━━━━━━━┓ ╲
┃ nW │ idx_1 ┃ V │ a_{m-1,idx_1} ┃ │
┣━━━━╋━━━━━━━━━━━━━━━━━╋━━━━━╋━━━━━━━━━━━━━━━━━━━━━━━━┫ │
┃ nW │ idx_2 ┃ V │ a_{m-1,idx_2} ┃ │
┗━━━━┻━━━━━━━━━━━━━━━━━┻━━━━━┻━━━━━━━━━━━━━━━━━━━━━━━━┛ │
... ... │
┏━━━━┳━━━━━━━━━━━━━━━━━┳━━━━━┳━━━━━━━━━━━━━━━━━━━━━━━━┓ │
┃ nW │ idx_nA ┃ V │ a_{m-1,idx_nA} ┃ │
┗━━━━┻━━━━━━━━━━━━━━━━━┻━━━━━┻━━━━━━━━━━━━━━━━━━━━━━━━┛ │
┏━━━━┳━━━━━━━━━━━━━━━━━┓ │
┃ nW │ nB ┃ │
┣━━━━╋━━━━━━━━━━━━━━━━━╋━━━━━┳━━━━━━━━━━━━━━━━━━━━━━━━┓ │
┃ nW │ idx_1 ┃ V │ b_{m-1,idx_1} ┃ │
┣━━━━╋━━━━━━━━━━━━━━━━━╋━━━━━╋━━━━━━━━━━━━━━━━━━━━━━━━┫ ╲
┃ nW │ idx_2 ┃ V │ b_{m-1,idx_2} ┃ ╲
┗━━━━┻━━━━━━━━━━━━━━━━━┻━━━━━┻━━━━━━━━━━━━━━━━━━━━━━━━┛ ╱ Constraint_{m-1}
... ... ╱
┏━━━━━━━━━━━━━━━━━━━━━━┳━━━━━┳━━━━━━━━━━━━━━━━━━━━━━━━┓ │
┃ nW idx_nB ┃ V │ b_{m-1,idx_nB} ┃ │
┗━━━━━━━━━━━━━━━━━━━━━━┻━━━━━┻━━━━━━━━━━━━━━━━━━━━━━━━┛ │
┏━━━━┳━━━━━━━━━━━━━━━━━┓ │
┃ nW │ nC ┃ │
┣━━━━╋━━━━━━━━━━━━━━━━━╋━━━━━┳━━━━━━━━━━━━━━━━━━━━━━━━┓ │
┃ nW │ idx_1 ┃ V │ c_{m-1,idx_1} ┃ │
┣━━━━╋━━━━━━━━━━━━━━━━━╋━━━━━╋━━━━━━━━━━━━━━━━━━━━━━━━┫ │
┃ nW │ idx_2 ┃ V │ c_{m-1,idx_2} ┃ │
┗━━━━┻━━━━━━━━━━━━━━━━━┻━━━━━┻━━━━━━━━━━━━━━━━━━━━━━━━┛ │
... ... │
┏━━━━┳━━━━━━━━━━━━━━━━━┳━━━━━┳━━━━━━━━━━━━━━━━━━━━━━━━┓ │
┃ nW │ idx_nB ┃ V │ c_{m-1,idx_nC} ┃ ╱
┗━━━━┻━━━━━━━━━━━━━━━━━┻━━━━━┻━━━━━━━━━━━━━━━━━━━━━━━━┛ ╱
╱ ╱
...
...
...
````
### Magic Number
#### Magic Number
Size: 4 bytes
The file start with a constant 4 byts (magic number) "r1cs"
The file start with a constant 4 bytes (magic number) "r1cs"
```
0x72 0x31 0x63 0x73
```
### Version
#### Version
Size: 4 bytes
Format: Little-Endian
@@ -209,58 +112,277 @@ For this standard it's fixed to
0x01 0x00 0x00 0x00
```
### Word With (nW)
#### Number of Sections
Size: 4 bytes
Format: Little-Endian
This is the standard word size in bytes used to specify lenghts and indexes in the file.
Number of sections contained in the file
The format of this field is little endian.
#### SectionType
In most of the cases this will be 4 (32bit values)
Size: 4 bytes
Format: 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:
```
0x04 0x00 0x00 0x00
0x01 0x00 0x00 0x00
```
### Number of wires
#### SectionSize
Size: nW bytes
Size: `ws` bytes
Format: Little-Endian
Size in bytes of the section
### Header Section
Section Type: 0x01
````
┏━━━━┳━━━━━━━━━━━━━━━━━┳━━━━━━━━━━━━━━━━━━━━━━━━━━━━━━┓
┃ 4 │ FieldDefSize ┃ FieldDef ┃ field Id
┗━━━━┻━━━━━━━━━━━━━━━━━┻━━━━━━━━━━━━━━━━━━━━━━━━━━━━━━┛
┏━━━━┳━━━━━━━━━━━━━━━━━┓
┃ 4 │ 00 00 00 00 ┃ bigInt Format
┗━━━━┻━━━━━━━━━━━━━━━━━┛
┏━━━━┳━━━━━━━━━━━━━━━━━┓
┃ 4 │ is ┃ Id size ( Normally 4 (32bits))
┗━━━━┻━━━━━━━━━━━━━━━━━┛
┏━━━━┳━━━━━━━━━━━━━━━━━┓
┃ is │ 01 00 00 00 ┃ nWires
┗━━━━┻━━━━━━━━━━━━━━━━━┛
┏━━━━┳━━━━━━━━━━━━━━━━━┓
┃ is │ 01 00 00 00 ┃ nPubOut
┗━━━━┻━━━━━━━━━━━━━━━━━┛
┏━━━━┳━━━━━━━━━━━━━━━━━┓
┃ is │ 01 00 00 00 ┃ nPubIn
┗━━━━┻━━━━━━━━━━━━━━━━━┛
┏━━━━┳━━━━━━━━━━━━━━━━━┓
┃ is │ 01 00 00 00 ┃ nPrvIn
┗━━━━┻━━━━━━━━━━━━━━━━━┛
┏━━━━┳━━━━━━━━━━━━━━━━━┓
┃ is │ 01 00 00 00 ┃ nLabels
┗━━━━┻━━━━━━━━━━━━━━━━━┛
┏━━━━┳━━━━━━━━━━━━━━━━━┓
┃ is │ 01 00 00 00 ┃ mConstraints
┗━━━━┻━━━━━━━━━━━━━━━━━┛
````
#### fieldDefSize
Size: 4 bytes
Format: Little-Endian
Size of the field Definition
Example:
```
0x00 0x0 0x00 0x00
```
#### fieldDef
Field dfinition the first 4 bytes are the type in LE. 0x0000001 Ar prime fields.
For the prime fields, the next bytes are the prime in variable length LE base 256 format.
NOTE: This number is independent of the bigInt Format defined next
#### bigInt Format
Size: 4 bytes
Format: Little-Endian
0 Means that the Big Int are variable size LE.
That is the First byte indicates the size and the remaining bytes are the number in little enfian (LSB first) base 256.
Numbers from 1 to 16383 are fixed size Litle endian format base 256.
Example:
```
0x00 0x00 0x00 0x00
```
#### Id Size (is)
Size: 4 bytes
Format: Little-Endian
Size of the identifiers for wires, labels and constraints. In small circuits this is going to be 4 (32 bits)
but can be increaset to 8 for bigger circiuits.
The only possible numbers are 4 or 8
#### Number of wires
Size: `is` bytes
Format: Little-Endian
Total Number of wires including ONE signal (Index 0).
### Number of public outputs
#### Number of public outputs
Size: nW bytes
Size: `is` bytes
Format: Little-Endian
Total Number of wires public output wires. They should be starting at idx 1
### Number of public inputs
#### Number of public inputs
Size: nW bytes
Size: `is` bytes
Format: Little-Endian
Total Number of wires public input wires. They should be starting just after the public output
### Number of private inputs
#### Number of private inputs
Size: nW bytes
Size: `is` bytes
Format: Little-Endian
Total Number of wires private input wires. They should be starting just after the public inputs
### Number of constraints
#### Number of constraints (m)
Size: nW bytes
Size: `ìs` bytes
Format: Little-Endian
Total Number of constraints
### Constraints
### Constraints section
Section Type: 0x02
````
┏━━━━┳━━━━━━━━━━━━━━━━━┓ ╲
┃ is │ nA ┃ ╲
┣━━━━╋━━━━━━━━━━━━━━━━━╋━━━━━┳━━━━━━━━━━━━━━━━━━━━━━━━┓ ╲
┃ is │ wireId_1 ┃ V │ a_{0,wireId_1} ┃ │
┣━━━━╋━━━━━━━━━━━━━━━━━╋━━━━━╋━━━━━━━━━━━━━━━━━━━━━━━━┫ │
┃ is │ wireId_2 ┃ V │ a_{0,wireId_2} ┃ │
┗━━━━┻━━━━━━━━━━━━━━━━━┻━━━━━┻━━━━━━━━━━━━━━━━━━━━━━━━┛ │
... ... │
┏━━━━┳━━━━━━━━━━━━━━━━━┳━━━━━┳━━━━━━━━━━━━━━━━━━━━━━━━┓ │
┃ is │ wireId_nA ┃ V │ a_{0,wireId_nA} ┃ │
┗━━━━┻━━━━━━━━━━━━━━━━━┻━━━━━┻━━━━━━━━━━━━━━━━━━━━━━━━┛ │
┏━━━━┳━━━━━━━━━━━━━━━━━┓ │
┃ is │ nB ┃ │
┣━━━━╋━━━━━━━━━━━━━━━━━╋━━━━━┳━━━━━━━━━━━━━━━━━━━━━━━━┓ │
┃ is │ wireId_1 ┃ V │ b_{0,wireId_1} ┃ │
┣━━━━╋━━━━━━━━━━━━━━━━━╋━━━━━╋━━━━━━━━━━━━━━━━━━━━━━━━┫ ╲
┃ is │ wireId_2 ┃ V │ b_{0,wireId_2} ┃ ╲
┗━━━━┻━━━━━━━━━━━━━━━━━┻━━━━━┻━━━━━━━━━━━━━━━━━━━━━━━━┛ ╱ Constraint_0
... ... ╱
┏━━━━┳━━━━━━━━━━━━━━━━━┳━━━━━┳━━━━━━━━━━━━━━━━━━━━━━━━┓ │
┃ is │ wireId_nB ┃ V │ b_{0,wireId_nB} ┃ │
┗━━━━┻━━━━━━━━━━━━━━━━━┻━━━━━┻━━━━━━━━━━━━━━━━━━━━━━━━┛ │
┏━━━━┳━━━━━━━━━━━━━━━━━┓ │
┃ is │ nC ┃ │
┣━━━━╋━━━━━━━━━━━━━━━━━╋━━━━━┳━━━━━━━━━━━━━━━━━━━━━━━━┓ │
┃ is │ wireId_1 ┃ V │ c_{0,wireId_1} ┃ │
┣━━━━╋━━━━━━━━━━━━━━━━━╋━━━━━╋━━━━━━━━━━━━━━━━━━━━━━━━┫ │
┃ is │ wireId_2 ┃ V │ c_{0,wireId_2} ┃ │
┗━━━━┻━━━━━━━━━━━━━━━━━┻━━━━━┻━━━━━━━━━━━━━━━━━━━━━━━━┛ │
... ... │
┏━━━━┳━━━━━━━━━━━━━━━━━┳━━━━━┳━━━━━━━━━━━━━━━━━━━━━━━━┓ │
┃ is │ wireId_nC ┃ V │ c_{0,wireId_nC} ┃ ╱
┗━━━━┻━━━━━━━━━━━━━━━━━┻━━━━━┻━━━━━━━━━━━━━━━━━━━━━━━━┛ ╱
╱
┏━━━━┳━━━━━━━━━━━━━━━━━┓ ╲
┃ is │ nA ┃ ╲
┣━━━━╋━━━━━━━━━━━━━━━━━╋━━━━━┳━━━━━━━━━━━━━━━━━━━━━━━━┓ ╲
┃ is │ wireId_1 ┃ V │ a_{1,wireId_1} ┃ │
┣━━━━╋━━━━━━━━━━━━━━━━━╋━━━━━╋━━━━━━━━━━━━━━━━━━━━━━━━┫ │
┃ is │ wireId_2 ┃ V │ a_{1,wireId_2} ┃ │
┗━━━━┻━━━━━━━━━━━━━━━━━┻━━━━━┻━━━━━━━━━━━━━━━━━━━━━━━━┛ │
... ... │
┏━━━━┳━━━━━━━━━━━━━━━━━┳━━━━━┳━━━━━━━━━━━━━━━━━━━━━━━━┓ │
┃ is │ wireId_nA ┃ V │ a_{1,wireId_nA} ┃ │
┗━━━━┻━━━━━━━━━━━━━━━━━┻━━━━━┻━━━━━━━━━━━━━━━━━━━━━━━━┛ │
┏━━━━┳━━━━━━━━━━━━━━━━━┓ │
┃ is │ nB ┃ │
┣━━━━╋━━━━━━━━━━━━━━━━━╋━━━━━┳━━━━━━━━━━━━━━━━━━━━━━━━┓ │
┃ is │ wireId_1 ┃ V │ b_{1,wireId_1} ┃ │
┣━━━━╋━━━━━━━━━━━━━━━━━╋━━━━━╋━━━━━━━━━━━━━━━━━━━━━━━━┫ ╲
┃ is │ wireId_2 ┃ V │ b_{1,wireId_2} ┃ ╲
┗━━━━┻━━━━━━━━━━━━━━━━━┻━━━━━┻━━━━━━━━━━━━━━━━━━━━━━━━┛ ╱ Constraint_1
... ... ╱
┏━━━━┳━━━━━━━━━━━━━━━━━┳━━━━━┳━━━━━━━━━━━━━━━━━━━━━━━━┓ │
┃ is │ wireId_nB ┃ V │ b_{1,wireId_nB} ┃ │
┗━━━━┻━━━━━━━━━━━━━━━━━┻━━━━━┻━━━━━━━━━━━━━━━━━━━━━━━━┛ │
┏━━━━┳━━━━━━━━━━━━━━━━━┓ │
┃ is │ nC ┃ │
┣━━━━╋━━━━━━━━━━━━━━━━━╋━━━━━┳━━━━━━━━━━━━━━━━━━━━━━━━┓ │
┃ is │ wireId_1 ┃ V │ c_{1,wireId_1} ┃ │
┣━━━━╋━━━━━━━━━━━━━━━━━╋━━━━━╋━━━━━━━━━━━━━━━━━━━━━━━━┫ │
┃ is │ wireId_2 ┃ V │ c_{1,wireId_2} ┃ │
┗━━━━┻━━━━━━━━━━━━━━━━━┻━━━━━┻━━━━━━━━━━━━━━━━━━━━━━━━┛ │
... ... │
┏━━━━┳━━━━━━━━━━━━━━━━━┳━━━━━┳━━━━━━━━━━━━━━━━━━━━━━━━┓ │
┃ is │ wireId_nC ┃ V │ c_{1,wireId_nC} ┃ ╱
┗━━━━┻━━━━━━━━━━━━━━━━━┻━━━━━┻━━━━━━━━━━━━━━━━━━━━━━━━┛ ╱
╱
...
...
...
┏━━━━┳━━━━━━━━━━━━━━━━━┓ ╲
┃ is │ nA ┃ ╲
┣━━━━╋━━━━━━━━━━━━━━━━━╋━━━━━┳━━━━━━━━━━━━━━━━━━━━━━━━┓ ╲
┃ is │ wireId_1 ┃ V │ a_{m-1,wireId_1} ┃ │
┣━━━━╋━━━━━━━━━━━━━━━━━╋━━━━━╋━━━━━━━━━━━━━━━━━━━━━━━━┫ │
┃ is │ wireId_2 ┃ V │ a_{m-1,wireId_2} ┃ │
┗━━━━┻━━━━━━━━━━━━━━━━━┻━━━━━┻━━━━━━━━━━━━━━━━━━━━━━━━┛ │
... ... │
┏━━━━┳━━━━━━━━━━━━━━━━━┳━━━━━┳━━━━━━━━━━━━━━━━━━━━━━━━┓ │
┃ is │ wireId_nA ┃ V │ a_{m-1,wireId_nA} ┃ │
┗━━━━┻━━━━━━━━━━━━━━━━━┻━━━━━┻━━━━━━━━━━━━━━━━━━━━━━━━┛ │
┏━━━━┳━━━━━━━━━━━━━━━━━┓ │
┃ is │ nB ┃ │
┣━━━━╋━━━━━━━━━━━━━━━━━╋━━━━━┳━━━━━━━━━━━━━━━━━━━━━━━━┓ │
┃ is │ wireId_1 ┃ V │ b_{m-1,wireId_1} ┃ │
┣━━━━╋━━━━━━━━━━━━━━━━━╋━━━━━╋━━━━━━━━━━━━━━━━━━━━━━━━┫ ╲
┃ is │ wireId_2 ┃ V │ b_{m-1,wireId_2} ┃ ╲
┗━━━━┻━━━━━━━━━━━━━━━━━┻━━━━━┻━━━━━━━━━━━━━━━━━━━━━━━━┛ ╱ Constraint_{m-1}
... ... ╱
┏━━━━┳━━━━━━━━━━━━━━━━━┳━━━━━┳━━━━━━━━━━━━━━━━━━━━━━━━┓ │
┃ is │ wireId_nB ┃ V │ b_{m-1,wireId_nB} ┃ │
┗━━━━┻━━━━━━━━━━━━━━━━━┻━━━━━┻━━━━━━━━━━━━━━━━━━━━━━━━┛ │
┏━━━━┳━━━━━━━━━━━━━━━━━┓ │
┃ is │ nC ┃ │
┣━━━━╋━━━━━━━━━━━━━━━━━╋━━━━━┳━━━━━━━━━━━━━━━━━━━━━━━━┓ │
┃ is │ wireId_1 ┃ V │ c_{m-1,wireId_1} ┃ │
┣━━━━╋━━━━━━━━━━━━━━━━━╋━━━━━╋━━━━━━━━━━━━━━━━━━━━━━━━┫ │
┃ is │ wireId_2 ┃ V │ c_{m-1,wireId_2} ┃ │
┗━━━━┻━━━━━━━━━━━━━━━━━┻━━━━━┻━━━━━━━━━━━━━━━━━━━━━━━━┛ │
... ... │
┏━━━━┳━━━━━━━━━━━━━━━━━┳━━━━━┳━━━━━━━━━━━━━━━━━━━━━━━━┓ │
┃ is │ wireId_nC ┃ V │ c_{m-1,wireId_nC} ┃ ╱
┗━━━━┻━━━━━━━━━━━━━━━━━┻━━━━━┻━━━━━━━━━━━━━━━━━━━━━━━━┛ ╱
╱ ╱
````
#### Constraints
Each constraint contains 3 linear combinations A, B, C.
@@ -269,36 +391,35 @@ The constraint is such that:
A*B-C = 0
```
### Linear combination
#### Linear combination
Each linear combination is of the form:
$$
a_{0,0}s_0 + a_{0,1}s_1 + ... + a_{0,n-1}s_{n-1}
a_{j,0}w_0 + a_{j,1}w_1 + ... + a_{j,n}w_{n}
$$
### Number of nonZero Factors
#### Number of nonZero Factors
Size: nW bytes
Size: `ìs` bytes
Format: Little-Endian
Total number of non Zero factors in the linear compination.
The factors MUST be sorted in ascending order.
### Factor
#### Factor
For each factor we have the index of the factor and the value of the factor.
### Index of the factor
#### WireId of the factor
Size: nW bytes
Size: `is` bytes
Format: Little-Endian
Index of the nonZero Factor
WireId of the nonZero Factor
### Value of the factor
#### Value of the factor
The first byte indicate the length N in bytes of the number in the upcoming bytes.
@@ -307,7 +428,7 @@ The next N bytes represent the value in Little Endian format.
For example, to represent the linear combination:
$$
5s_4 +8s_5 + 260s_886
5w_4 +8w_5 + 260w_{886}
$$
The linear combination would be represented as:
@@ -323,6 +444,18 @@ The linear combination would be represented as:
┃ 76 03 00 00 ┃ 02 04 01 ┃
┗━━━━━━━━━━━━━━━━━┻━━━━━━━━━━━━━━━━━┛
````
### WireId2LabelId Map Section
Section Type: 0x03
````
┏━━┳━━━━━━━━━━━━━━━━━━━┳━━┳━━━━━━━━━━━━━━━━━━━┓ ┏━━┳━━━━━━━━━━━━━━━━━━━┓
┃is│ labelId of Wire_0 ┃is│ labelId of Wire_1 ┃ ... ┃is│ labelId of Wire_n ┃
┗━━┻━━━━━━━━━━━━━━━━━━━┻━━┻━━━━━━━━━━━━━━━━━━━┛ ┗━━┻━━━━━━━━━━━━━━━━━━━┛
````
## Rationale
Variable size for field elements allows to shrink the size of the file and allows to work with any field.
@@ -331,6 +464,8 @@ Version allows to update the format.
Have a very good comprasion ratio for sparse r1cs as it's the normal case.
The motivation of having a map between l and w is that this allows optimizers to calculate equivalent r1cs systems but keeping the original values geneated by the circuit.
## Backward Compatibility
@@ -344,37 +479,62 @@ Given this r1cs in a 256 bit Field:
$$
\left\{ \begin{array}{rclclcl}
(3s_5 + 8s_6) &\cdot& (2s_0 + 20s_2 + 12s_3) &-& (5s_0 + 7s_9) &=& 0 \\
(4s_1 + 8s_5 + 3s_9) &\cdot& (6s_6 + 44s_3) && &=& 0 \\
(4s_6) &\cdot& (6s_0 + 5s_3 + 11s_9) &-& (600s_700) &=& 0
(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:
````
┏━━━━━━━━━━━━━━┓
┃ 72 31 63 77 ┃ Magic
┏━━━━━━━━━━━━━━┓
┃ 72 31 63 77 ┃ Magic
┣━━━━━━━━━━━━━━┫
┃ 01 00 00 00 ┃ Version
┣━━━━━━━━━━━━━━┫
┃ 03 00 00 00 ┃ nSections
┗━━━━━━━━━━━━━━┛
┏━━━━━━━━━━━━━━┳━━━━━━━━━━━━━┓
┃ 01 00 00 00 ┃ 49 00 00 00 ┃ SectionType: Header
┗━━━━━━━━━━━━━━┻━━━━━━━━━━━━━┛
┏━━━━━━━━━━━━━━┳━━━━━━━━━━━━━┓
┃ 25 00 00 00 ┃ 10 00 00 00 ┃ FieldDefSize FieldDef
┣━━━━━━━━━━━━━━┻━━━━━━━━━━━━━┻━━━━━━━━━━━━━━━━━━━━━━━━━━━━━━━━━━━━━━━━━━━━━━┓
┃ 20 010000f0 93f5e143 9170b979 48e83328 5d588181 b64550b8 29a031e1 724e6430┃
┣━━━━━━━━━━━━━━┳━━━━━━━━━━━━━━━━━━━━━━━━━━━━━━━━━━━━━━━━━━━━━━━━━━━━━━━━━━━━┛
┃ 00 00 00 00 ┃ Big Int format
┣━━━━━━━━━━━━━━┫
┃ 01 00 00 00 ┃ Version
┃ 04 00 00 00 ┃ Id Size
┣━━━━━━━━━━━━━━┫
┃ 04 00 00 00 ┃ nW
┣━━━━━━━━━━━━━━┫
┃ 04 23 45 00 ┃ # of wires
┃ 07 00 00 00 ┃ # of wires
┣━━━━━━━━━━━━━━┫
┃ 01 00 00 00 ┃ # Public Outs
┣━━━━━━━━━━━━━━┫
┃ 02 00 00 00 ┃ # Public Ins
┃ 02 00 00 00 ┃ # Public Ins
┣━━━━━━━━━━━━━━┫
┃ 05 00 00 00 ┃ # Private Ins
┃ 03 00 00 00 ┃ # Private Ins
┣━━━━━━━━━━━━━━┫
┃ e8 03 00 00 ┃ # Labels
┣━━━━━━━━━━━━━━┫
┃ 03 00 00 00 ┃ # Constraints
┗━━━━━━━━━━━━━━┛
┏━━━━━━━━━━━━━━┓
┃ 03 00 00 00 ┃ # of constraints
┗━━━━━━━━━━━━━━┛
┏━━━━━━━━━━━━━━┓ Constraint 0: (3s_5 + 8s_6) * (2s_0 + 20s_2 + 12s_3) - (5s_0 + 7s_9) = 0
┏━━━━━━━━━━━━━━┳━━━━━━━━━━━━━┓
┃ 02 00 00 00 ┃ 8b 00 00 00 ┃ SectionType: Constraints
┗━━━━━━━━━━━━━━┻━━━━━━━━━━━━━┛
┏━━━━━━━━━━━━━━┓ Constraint 0: (3w_5 + 8w_6) * (2w_0 + 20w_2 + 12w_3) - (5w_0 + 7w_2) = 0
┃ 02 00 00 00 ┃
┣━━━━━━━━━━━━━━╋━━━━━━━━┓
┃ 05 00 00 00 ┃ 01 03 ┃
@@ -395,18 +555,17 @@ The format will be:
┣━━━━━━━━━━━━━━╋━━━━━━━━┓
┃ 00 00 00 00 ┃ 01 05 ┃
┣━━━━━━━━━━━━━━╋━━━━━━━━┫
┃ 09 00 00 00 ┃ 01 07 ┃
┃ 02 00 00 00 ┃ 01 07 ┃
┗━━━━━━━━━━━━━━┻━━━━━━━━┛
┏━━━━━━━━━━━━━━┓ Constraint 1: (4s_1 + 8s_5 + 3s_9) * (6s_6 + 44s_3) = 0
┏━━━━━━━━━━━━━━┓ Constraint 1: (4w_1 + 8w_4 + 3w_5) * (6w_6 + 44w_3) = 0
┃ 03 00 00 00 ┃
┣━━━━━━━━━━━━━━╋━━━━━━━━━┓
┃ 01 00 00 00 ┃ 01 04 ┃
┣━━━━━━━━━━━━━━╋━━━━━━━━━┫
┃ 05 00 00 00 ┃ 01 08 ┃
┃ 04 00 00 00 ┃ 01 08 ┃
┣━━━━━━━━━━━━━━╋━━━━━━━━━┫
┃ 09 00 00 00 ┃ 01 03 ┃
┃ 05 00 00 00 ┃ 01 03 ┃
┗━━━━━━━━━━━━━━┻━━━━━━━━━┛
┏━━━━━━━━━━━━━━┓
┃ 02 00 00 00 ┃
@@ -419,8 +578,7 @@ The format will be:
┃ 00 00 00 00 ┃
┗━━━━━━━━━━━━━━┛
┏━━━━━━━━━━━━━━┓ Constraint 2: (4s_6) * (6s_0 + 5s_3 + 11s_9) - (600s_700) = 0
┏━━━━━━━━━━━━━━┓ Constraint 2: (4w_6) * (6w_0 + 5w_3 + 11w_2) - (600w_6) = 0
┃ 01 00 00 00 ┃
┣━━━━━━━━━━━━━━╋━━━━━━━━━┓
┃ 06 00 00 00 ┃ 01 04 ┃
@@ -430,15 +588,34 @@ The format will be:
┣━━━━━━━━━━━━━━╋━━━━━━━━━┓
┃ 00 00 00 00 ┃ 01 06 ┃
┣━━━━━━━━━━━━━━╋━━━━━━━━━┫
┃ 03 00 00 00 ┃ 01 05 ┃
┃ 02 00 00 00 ┃ 01 0B ┃
┣━━━━━━━━━━━━━━╋━━━━━━━━━┫
┃ 09 00 00 00 ┃ 01 0B ┃
┃ 03 00 00 00 ┃ 01 05 ┃
┗━━━━━━━━━━━━━━┻━━━━━━━━━┛
┏━━━━━━━━━━━━━━┓
┃ 01 00 00 00 ┃
┣━━━━━━━━━━━━━━╋━━━━━━━━━━━━━┓
┃ BC 02 00 00 ┃ 02 58 02 ┃
┃ 06 00 00 00 ┃ 02 58 02 ┃
┗━━━━━━━━━━━━━━┻━━━━━━━━━━━━━┛
┏━━━━━━━━━━━━━━┳━━━━━━━━━━━━━┓
┃ 03 00 00 00 ┃ 1c 00 00 00 ┃ Wire to Label Map
┗━━━━━━━━━━━━━━┻━━━━━━━━━━━━━┛
┏━━━━━━━━━━━━━━┓
┃ 00 00 00 00 ┃
┣━━━━━━━━━━━━━━┫
┃ 03 00 00 00 ┃
┣━━━━━━━━━━━━━━┫
┃ 0a 00 00 00 ┃
┣━━━━━━━━━━━━━━┫
┃ 0b 00 00 00 ┃
┣━━━━━━━━━━━━━━┫
┃ 0c 00 00 00 ┃
┣━━━━━━━━━━━━━━┫
┃ 0f 00 00 00 ┃
┣━━━━━━━━━━━━━━┫
┃ 44 01 00 00 ┃
┗━━━━━━━━━━━━━━┛
````
And the binary representation in Hex:
@@ -446,38 +623,54 @@ And the binary representation in Hex:
````
72 31 63 77
01 00 00 00
03 00 00 00
01 00 00 00 49 00 00 00
25 00 00 00 10 00 00 00
20 010000f0 93f5e143 9170b979 48e83328 5d588181 b64550b8 29a031e1 724e6430
00 00 00 00
04 00 00 00
04 23 45 00
07 00 00 00
01 00 00 00
02 00 00 00
05 00 00 00
03 00 00 00
02 00 00 00
05 00 00 00 01 03
06 00 00 00 01 08
e8 03 00 00
03 00 00 00
00 00 00 00 01 02
02 00 00 00 01 14
03 00 00 00 01 0C
02 00 00 00 8b 00 00 00
02 00 00 00
00 00 00 00 01 05
09 00 00 00 01 07
05 00 00 00 01 03
06 00 00 00 01 08
03 00 00 00
01 00 00 00 01 04
05 00 00 00 01 08
09 00 00 00 01 03
00 00 00 00 01 02
02 00 00 00 01 14
03 00 00 00 01 0C
02 00 00 00
03 00 00 00 01 2C
06 00 00 00 01 06
00 00 00 00 01 05
02 00 00 00 01 07
03 00 00 00
01 00 00 00 01 04
04 00 00 00 01 08
05 00 00 00 01 03
02 00 00 00
03 00 00 00 01 2C
06 00 00 00 01 06
00 00 00 00
01 00 00 00
06 00 00 00 01 04
06 00 00 00 01 04
03 00 00 00
00 00 00 00 01 06
03 00 00 00 01 05
09 00 00 00 01 0B
00 00 00 00 01 06
02 00 00 00 01 0B
03 00 00 00 01 05
01 00 00 00
BC 02 00 00 02 58 02
06 00 00 00 02 58 02
03 00 00 00 1c 00 00 00
00 00 00 00
03 00 00 00
0a 00 00 00
0b 00 00 00
0c 00 00 00
0f 00 00 00
44 01 00 00
````
## Implementation
@@ -487,3 +680,4 @@ circom will output this format.
## Copyright
Copyright and related rights waived via [CC0](https://creativecommons.org/publicdomain/zero/1.0/).