# Binary format for R1CS --- eip: title: r1cs binary format author: Jordi Baylina discussions-to: status: draft type: Standards Track category: ERC created: 2019-09-24 requires: --- ## 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 bytes The file start with a constant 4 bytes (magic number) "r1cs" ``` 0x72 0x31 0x63 0x73 ``` #### Version Size: 4 bytes Format: Little-Endian For this standard it's fixed to ``` 0x01 0x00 0x00 0x00 ``` #### Number of Sections Size: 4 bytes Format: Little-Endian Number of sections contained in the file #### SectionType 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: ``` 0x01 0x00 0x00 0x00 ``` #### SectionSize 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 Size: `is` bytes Format: Little-Endian Total Number of wires public output wires. They should be starting at idx 1 #### Number of public inputs 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 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 (m) Size: `ìs` bytes Format: Little-Endian Total Number of 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. 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: `ìs` bytes Format: 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: `is` bytes Format: Little-Endian WireId of the nonZero Factor #### Value of the factor The first byte indicate the length N in bytes of the number in the upcoming bytes. The next N bytes represent the value in Little Endian format. 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 ┃ 01 05 ┃ ┣━━━━━━━━━━━━━━━━━╋━━━━━━━━━━━━━━━━━┫ ┃ 05 00 00 00 ┃ 01 08 ┃ ┣━━━━━━━━━━━━━━━━━╋━━━━━━━━━━━━━━━━━┫ ┃ 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. 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: ```` ┏━━━━━━━━━━━━━━┓ ┃ 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 ┣━━━━━━━━━━━━━━┫ ┃ 04 00 00 00 ┃ Id Size ┣━━━━━━━━━━━━━━┫ ┃ 07 00 00 00 ┃ # of wires ┣━━━━━━━━━━━━━━┫ ┃ 01 00 00 00 ┃ # Public Outs ┣━━━━━━━━━━━━━━┫ ┃ 02 00 00 00 ┃ # Public Ins ┣━━━━━━━━━━━━━━┫ ┃ 03 00 00 00 ┃ # Private Ins ┣━━━━━━━━━━━━━━┫ ┃ e8 03 00 00 ┃ # Labels ┣━━━━━━━━━━━━━━┫ ┃ 03 00 00 00 ┃ # Constraints ┗━━━━━━━━━━━━━━┛ ┏━━━━━━━━━━━━━━┳━━━━━━━━━━━━━┓ ┃ 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 ┃ ┣━━━━━━━━━━━━━━╋━━━━━━━━┫ ┃ 06 00 00 00 ┃ 01 08 ┃ ┗━━━━━━━━━━━━━━┻━━━━━━━━┛ ┏━━━━━━━━━━━━━━┓ ┃ 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 ┃ ┣━━━━━━━━━━━━━━╋━━━━━━━━┓ ┃ 00 00 00 00 ┃ 01 05 ┃ ┣━━━━━━━━━━━━━━╋━━━━━━━━┫ ┃ 02 00 00 00 ┃ 01 07 ┃ ┗━━━━━━━━━━━━━━┻━━━━━━━━┛ ┏━━━━━━━━━━━━━━┓ Constraint 1: (4w_1 + 8w_4 + 3w_5) * (6w_6 + 44w_3) = 0 ┃ 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 ┃ ┗━━━━━━━━━━━━━━┛ ┏━━━━━━━━━━━━━━┓ Constraint 2: (4w_6) * (6w_0 + 5w_3 + 11w_2) - (600w_6) = 0 ┃ 01 00 00 00 ┃ ┣━━━━━━━━━━━━━━╋━━━━━━━━━┓ ┃ 06 00 00 00 ┃ 01 04 ┃ ┗━━━━━━━━━━━━━━┻━━━━━━━━━┛ ┏━━━━━━━━━━━━━━┓ ┃ 03 00 00 00 ┃ ┣━━━━━━━━━━━━━━╋━━━━━━━━━┓ ┃ 00 00 00 00 ┃ 01 06 ┃ ┣━━━━━━━━━━━━━━╋━━━━━━━━━┫ ┃ 02 00 00 00 ┃ 01 0B ┃ ┣━━━━━━━━━━━━━━╋━━━━━━━━━┫ ┃ 03 00 00 00 ┃ 01 05 ┃ ┗━━━━━━━━━━━━━━┻━━━━━━━━━┛ ┏━━━━━━━━━━━━━━┓ ┃ 01 00 00 00 ┃ ┣━━━━━━━━━━━━━━╋━━━━━━━━━━━━━┓ ┃ 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: ```` 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 07 00 00 00 01 00 00 00 02 00 00 00 03 00 00 00 e8 03 00 00 03 00 00 00 02 00 00 00 8b 00 00 00 02 00 00 00 05 00 00 00 01 03 06 00 00 00 01 08 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 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 03 00 00 00 00 00 00 00 01 06 02 00 00 00 01 0B 03 00 00 00 01 05 01 00 00 00 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 circom will output this format. ## Copyright Copyright and related rights waived via [CC0](https://creativecommons.org/publicdomain/zero/1.0/).