AirAssembly

This library contains specifications and JavaScript runtime for AirAssembly - a language for encoding Algebraic Intermediate Representation (AIR) of computations. AIR is a representation used in zk-STARKs to construct succinct proofs of computational integrity.

AirAssembly is a low-level language and is intended to be a compilation target for other, higher-level, languages (e.g. AirScript). It uses a simple s-expression-based syntax to specify:

1. Inputs required by the computations.
2. Logic for generating execution traces for the computations.
3. Logic for evaluating transition constraints for the computations.
4. Metadata needed to compose the computations with other computations.

Full specifications for AirAssembly can be found here.

Usage

This library is not intended for standalone use, but is rather meant to be a component used in STARK provers/verifiers (e.g. genSTARK). Nevertheless, you can install it separately like so:

$ npm install @guildofweavers/air-assembly --save

Once installed, you can use the library to compile AirAssembly source code into AirModules, and then use these modules to generate execution trace tables and constraint evaluation tables for computations.

The code below illustrates how to do this on the example of a MiMC computation. Other examples can be found here.

import { compile, instantiate } from '@guildofweavers/air-assembly';

const source = `
(module
    (field prime 4194304001)
    (const $alpha scalar 3)
    (function $mimcRound
        (result vector 1)
        (param $state vector 1) (param $roundKey scalar)
        (add 
            (exp (load.param $state) (load.const $alpha))
            (load.param $roundKey)))
    (export mimc
        (registers 1) (constraints 1) (steps 32)
        (static
            (cycle (prng sha256 0x4d694d43 32)))
        (init
            (param $seed vector 1)
            (load.param $seed))
        (transition
            (call $mimcRound (load.trace 0) (get (load.static 0) 0)))
        (evaluation
            (sub
                (load.trace 1)
                (call $mimcRound (load.trace 0) (get (load.static 0) 0))))))`;

// instantiate AirModule object
const schema = compile(Buffer.from(source));
const air = instantiate(schema, 'mimc');

// generate trace table
const context = air.initProvingContext([], [3n]);
const trace = context.generateExecutionTrace();

// generate constraint evaluation table
const tracePolys = air.field.interpolateRoots(context.executionDomain, trace);
const constraintEvaluations = context.evaluateTransitionConstraints(tracePolys);

API

Complete API definitions can be found in air-assembly.d.ts. Here is a quick overview of the provided functionality.

Top-level functions

The library exposes a small set of functions that can be used to compile AirAssembly source code, instantiate AirModules, and perform basic analysis of the underlying AIR. These functions are:

• compile(source: Buffer | string, limits?: StarkLimits): AirSchema
  Parses and compiles AirAssembly source code into an AirSchema object. If source parameter is a Buffer, it is expected to contain AirAssembly code. If source is a string, it is expected to be a path to a file containing AirAssembly code. If limits parameter is provided, generated AirSchema will be validated against these limits.

• instantiate(schema: AirSchema, component: string, options?: ModuleOptions): AirModule
  Creates an AirModule object for the specified component within the provided schema. The AirModule can then be used to generate execution trace tables and evaluate transition constraints. The optional options parameter can be used to control instantiation of the AirModule.

• instantiate(schema: AirSchema, options?: ModuleOptions): AirModule
  An overloaded version of the instantiate() function with component name assumed to equal "default". If the provided schema does not have a default component export, an error will be thrown.

• analyze(schema: AirSchema, component: name): SchemaAnalysisResult
  Performs basic analysis of the component within the provided schema to infer such things as degree of transition constraints, number of additions and multiplications needed to evaluate transition function etc.

Air module options

When instantiating an AirModule object, an AirModuleOptions object can be provided to specify any of the following parameters for the module:

Property	Description
limits	Limits to be imposed on the instantiated module. If not provided, default limit values will be used.
wasmOptions	Options for finite fields which can take advantage of WebAssembly optimization. This property can also be set to a boolean value to turn the optimization on or off.
extensionFactor	Number by which the execution trace is to be "stretched." Must be a power of 2 at least 2x of the highest constraint degree. This property is optional, the default is the smallest power of 2 that is greater than 2x of the highest constraint degree.

Stark limits

StarkLimits object defines limits against which AirSchema and AirModule objects are validated. StarkLimits objects can contain any of the following properties:

Property	Description
maxTraceLength	Maximum number of steps in an execution trace; defaults to 220
maxTraceRegisters	Maximum number of state registers; defaults to 64
maxStaticRegisters	Maximum number of static registers; defaults to 64
maxConstraintCount	Maximum number of transition constraints; defaults to 1024
maxConstraintDegree	Highest allowed degree of transition constraints; defaults to 16

Generating execution trace

To generate an execution trace for a computation defined by AirAssembly source code, the following steps should be executed:

1. Compile AirAssembly source code into AirSchema using the top-level compile() function.
2. Pass the resulting AirSchema to the top-level instantiate() function to create an AirModule. You'll also need to specify the name of the AirComponent exported from the AirSchema because AirModules are instantiated for a specific component.
3. Create a ProvingContext by invoking AirModule.initProvingContext() method. If the computation contains input registers, then input values for these registers must be passed to the initProvingContext() method. This instantiates the ProvingContext for a specific set of inputs.
4. Generate the execution trace by invoking ProvingContext.generateExecutionTrace() method.

The code block bellow illustrates these steps:

const schema = compile(Buffer.from(source));
const air = instantiate(schema, `mimc`, { extensionFactor: 16 });
const context = air.initProvingContext([], [3n]);
const trace = context.generateExecutionTrace();

In the above:

• source is a string variable containing AirAssembly source code similar to the one shown in the usage section.
• The AirModule is instantiated for the exported mimc component using default limits but setting extension factor to 16.
• An empty inputs array is passed as the first parameter to the initProvingContext() method since the computation shown in the usage section does not define any input registers.
• A seed array with value 3 is passed as the second parameter to the initProvingContext() method since component initializer expects a vector parameter to initialize the first row of the execution trace.
• After the code is executed, the trace variable will be a matrix with 1 row and 32 columns (corresponding to 1 register and 32 steps).

The execution trace for the single register will look like so:

[
             3, 1539309651, 3863242857, 3506640509, 1371547896, 215222094,  220283781,  2120321425,
    2290167095, 3044083866, 3673976270, 2694057310,  995327947, 2470701222, 798926004,  2416031839, 
    4124930959,  680273881,  115120944, 2405022753,  963841868, 327198005,  34356700,   1065113318,
    2951801258,  791752781, 1878966595, 2503692690, 1792666246, 3884924604, 3800788053, 2681237718
]

Evaluating transition constraints

Transition constraints described in AirAssembly source code can be evaluated either as a prover or as a verifier. Both methods are described below.

Evaluating transition constraints as a prover

When generating a STARK proof, transition constraints need to be evaluated at all points of the evaluation domain. This can be done efficiently by invoking ProvingContext.evaluateTransitionConstraints() method. To evaluate the constraints, the following steps should be executed:

1. Create a proving context and generate an execution trace as described in the previous section.
2. Generate a set of trace polynomials by interpolating execution trace columns over the execution domain.
3. Evaluate the constraints by invoking ProvingContext.evaluateTransitionConstraints() method and passing trace polynomials to it.

The code block bellow illustrates steps 2 and 3:

const tracePolys = air.field.interpolateRoots(context.executionDomain, trace);
const constraintEvaluations = context.evaluateTransitionConstraints(tracePolys);

In the above:

• AirModule's field object is used to interpolate the execution trace. The interpolateRoots() method takes a vector of x values as the first parameter, and a matrix of y values as the second parameter (the matrix is expected to have a distinct set of y values in each row). The output is a matrix where each row contains an interpolated polynomial.
• evaluateTransitionConstraints() method returns a matrix where each row corresponds to a transition constraint evaluated over the composition domain.

For example, evaluating constraints for AirAssembly code from the usage section, will produce a matrix with a single row containing the following values:

[
    0, 1888826267,  934997684,  522697873,         0, 3636300716,  301925789,  369141145,
    0,  767283131,  270628806, 1668446351,         0, 1739694248, 3247199818, 2569615536,
    0,   44729160, 4039819553, 3564072931,         0, 1616917451, 1151293301, 3209868277,
    0, 3410907990, 4004509077, 4190379432,         0, 3101507817, 3553581961, 2793433224,
    0,  330772896, 4060647779, 2512435701,         0, 3403188821,  235591542, 3772363484,
    0, 2256420389, 2357121513,   61957993,         0, 3272390069,  197242509, 2878395132,
    0,  155740407,  298885317, 3310802262,         0,   19161130,  691333255, 1102311751,
    0, 1751005830, 2349558192, 3473961491,         0, 4006336837,  565227775, 4021023132,
    0, 3315940573,  989407555, 2088778801,         0,  898450568, 3610287112, 3576441219,
    0,  326707597, 2532917782, 3330991749,         0, 4162556873, 1554019377, 4171366685,
    0,  984976271, 2011763604,  728626530,         0, 3611841258, 2245193661, 2605704194,
    0, 2583926003, 3992303847, 2748879594,         0, 2379703446,  430289311, 3052280185,
    0,  179547660, 1215051408, 2628504587,         0, 2862551083, 2740849758,  925951430,
    0, 4000243259,  913649599, 1118200600,         0, 1484209861, 1897468182,  190582872,
    0, 4135707956, 1007284323, 2027805646,         0, 1310083809, 2946378676,  350300836,
    0, 3019962854, 1468795609, 1874742277, 803208359, 4116321517, 3116095172,   77399359
]

Note: The constraints are evaluated over the composition domain (not evaluation domain). The evaluations can be extended to the full evaluation domain by interpolating them over the composition domain, and then evaluating the resulting polynomials over the evaluation domain. But in practice, to improve performance, the evaluations are first merged into a single vector via random linear combination, and then this single vector is extended to the full evaluation domain.

Evaluating transition constraints as a verifier

When verifying a STARK proof, transition constraints need to be evaluated at a small subset of points. This can be done by invoking VerificationContext.evaluateConstraintsAt() method which evaluates constraints at a single point of evaluation domain. To evaluate constraints at a single point, the following steps should be executed:

1. Compile AirAssembly source code into AirSchema using the top-level compile() function.
2. Pass the resulting AirSchema together with exported component name to the top-level instantiate() function to create an AirModule.
3. Create a VerificationContext by invoking AirModule.initVerificationContext() method. If the computation contains input registers, input shapes for these registers must be passed to the initVerificationContext() method. If any of the input registers are public, input values for these registers must also be passed to the method.
4. Evaluate constraints by invoking VerificationContext.evaluateConstraintsAt() method and passing to it an x-coordinate of the desired point from the evaluation domain, as well as corresponding values of register traces.

The code block bellow illustrates these steps:

const schema = compile(Buffer.from(source));
const air = instantiate(schema, 'mimc', { extensionFactor: 16 });
const context = air.initVerificationContext();
const x = air.field.exp(context.rootOfUnity, 16n);
const rValues = [1539309651n], nValues = [3863242857n];
const evaluations = context.evaluateConstraintsAt(x, rValues, nValues, []);

In the above:

• source is a string variable containing AirAssembly source code similar to the one shown in the usage section.
• The AirModule for mimc component is instantiated using default limits but setting extension factor to 16.
• No input shapes or inputs are passed to the initVerificationContext() method since the computation shown in the usage section does not define any input registers.
• x is set to the 17th value in of the execution domain, while rValues and nValues contain register traces for 2nd and 3rd steps of the computation. This is because when the extension factor is 16, the 2nd value of the execution trace aligns with the 17th value of the evaluation domain.
• After the code is executed, the evaluations variable will be an array with a single value 0. This is because applying transition function to value 1539309651 (current state of the execution trace) results in 3863242857, which is the next state of the execution trace.

Air Schema

An AirSchema object contains a semantic representation of AirAssembly source code. This representation makes it easy to analyze the source code, and serves as the basis for generating AirModules. An AirSchema object can be created by compiling AirAssembly code with the top-level compile() function.

AirSchema has the following properties:

Property	Description
field	A finite field object instantiated for the field specified for the computations contained withing schema.
constants	An array of Constant objects describing module constants defined for the computations.
functions	an array of AirFunction objects describing module functions defined for the computations.
components	A map of component export, where the key is the name of the component, and the value is an AirComponent object.

Note: definitions for Constant and AirFunction objects mentioned above can be found in air-assembly.d.ts file.

Air Component

An AirComponent object is a semantic representation of a specific computation contained within AirSchema. That is, a single AirSchema object can contain many AirComponents each describing a distinct computation. This allows packaging AIR of many computations into a single physical file.

AirComponent has the following properties:

Property	Description
name	String value containing name of the exported component.
staticRegisters	An array of StaticRegister objects describing static registers defined for the computation.
secretInputCount	An integer value specifying number of secret input registers defined for the computation.
traceInitializer	An AirProcedure object describing execution trace initializer expression defined for the computation.
transitionFunction	An AirProcedure object describing transition function expression defined for the computation.
constraintEvaluator	An AirProcedure object describing transition constraint evaluator expression defined for the computation.
constraints	An array of ConstraintDescriptor objects containing metadata for each of the defined transition constraints (e.g. constraint degree).
maxConstraintDegree	An integer value specifying the highest degree of transition constraints defined for the computation.

Note: definitions for StaticRegister, AirProcedure, and ConstraintDescriptor objects mentioned above can be found in air-assembly.d.ts file.

Air Module

An AirModule object contains JavaScript code needed to create ProvingContext and VerificationContext objects. These objects can then be used to generate execution trace and evaluate transition constraints for a computation. An AirModule can be instantiated for a specific component of an AirSchema by using the top-level instantiate() function.

AirModule has the following properties:

Property	Description
field	A finite field object used for all arithmetic operations of the computation.
traceRegisterCount	Number of state registers in the execution trace.
staticRegisterCount	Number of static registers in the execution trace.
inputDescriptors	An array of input descriptor objects describing inputs required by the computation.
secretInputCount	An integer value specifying number of secret input registers defined for the computation.
constraints	An array of ConstraintDescriptor objects containing metadata for each of the defined transition constraints (e.g. constraint degree).
maxConstraintDegree	An integer value specifying the highest degree of transition constraints defined for the computation.
extensionFactor	An integer value specifying how much the execution trace is to be "stretched."

AirModule exposes the following methods:

• initProvingContext(inputs?: any[], seed?: bigint[]): ProvingContext
  Instantiates a ProvingContext object for a specific instance of the computation. This context can then be used to generate execution trace table and constraint evaluation table for the computation.

  • inputs parameter must be provided only if the computation contains input registers. In such a case, the shape of input objects must be in line with the shapes specified by the computation's input descriptors.
  • seed parameter must be provided only if the trace initializer for the computation expects a vector parameter.

• initVerificationContext(inputShapes?: InputShape[], publicInputs?: any[]): VerificationContext
  Instantiates a VerificationContext object for a specific instance of the computation. This context can then be used to evaluate transition constraints at a given point of the evaluation domain of the computation. If the computation contains input registers, inputShapes parameter must be provided to specify the shapes of consumed inputs. If any of the input registers are public, publicInputs parameter must also be provided to specify the actual values of all public inputs consumed by the computation.

Proving context

A ProvingContext object contains properties and methods needed to help generate a proof for a specific instance of a computation. Specifically, a ProvingContext can be used to generate an execution trace for a specific set of inputs, and to evaluate transition constraints derived form this trace. To create a ProvingContext, use initProvingContext() method of AirModule object.

ProvingContext has the following properties:

Property	Description
field	Reference to the finite field object of the AirModule which describes the computation.
rootOfUnit	Primitive root of unity of the evaluation domain for the instance of the computation.
traceLength	Length of the execution trace for the instance of the computation.
extensionFactor	Extension factor of the execution trace.
constraints	An array of constraint descriptors with metadata for the defined transition constraints.
inputShapes	Shapes of all input registers for the instance of the computation.
executionDomain	A vector defining domain of the execution trace.
evaluationDomain	A vector defining domain of the low-degree extended execution trace. The length of the evaluation domain is equal to traceLength * extensionFactor.
compositionDomain	A vector defining domain of the low-degree extended composition polynomial. The length of the composition domain is equal to traceLength * compositionFactor, where compositionFactor is set to be the smallest power of 2 greater than or equal to the highest constraint degree of the computation. For example, if highest constraint degree is 3, the compositionFactor will be set to 2.
secretRegisterTraces	Values of secret input registers evaluated over the evaluation domain.

ProvingContext exposes the following methods:

• generateExecutionTrace(): Matrix
  Generates an execution trace for a computation. The return value is a matrix where each row corresponds to a dynamic register, and every column corresponds to a step in a computation (i.e. the number of columns will be equal to the length of the execution trace).

• evaluateTransitionConstraints(tracePolys: Matrix): Matrix
  Evaluates transition constraints for a computation. The tracePolys parameter is a matrix where each row represents a polynomial interpolated from a corresponding register of the execution trace. The return value is a matrix where each row represents a transition constraint evaluated over the composition domain.

Verification context

A VerificationContext object contains properties and methods needed to help verify a proof of an instance of a computation (i.e. instance of a computation for a specific set of inputs). Specifically, a VerificationContext can be used to evaluate transition constraints at a specific point of an evaluation domain. To create a VerificationContext, use initVerificationContext() method of AirModule object.

VerificationContext has the following properties:

Property	Description
field	Reference to the finite field object of the AirModule which describes the computation.
rootOfUnit	Primitive root of unity of the evaluation domain for the instance of the computation.
traceLength	Length of the execution trace for the instance of the computation.
extensionFactor	Extension factor of the execution trace.
constraints	An array of constraint descriptors with metadata for the defined transition constraints.
inputShapes	Shapes of all input registers for the instance of the computation.

VerificationContext exposes the following methods:

• evaluateConstraintsAt(x: bigint, rValues: bigint[], nValues: bigint[], sValues: bigint[]): bigint[]
  Returns an array of values resulting from evaluating transition constraints at point x. For example, if the computation is defined by a single transition constraint, an array with one value will be returned. The meaning of the parameters is as follows:
  • x is the point of the evaluation domain corresponding to the current step of the computation.
  • rValues is an array of dynamic register values at the current step of the computation.
  • nValues is an array of dynamic register values at the next step of the computation.
  • sValues is an array of secret register values at the current step of the computation.

Input descriptor

An InputDescriptor object contains information about an input register defined for the computation.

InputDescriptor has the following properties:

Property	Description
rank	An integer value indicating the position of the register in the input dependency tree. For example, rank of a register without parents is 1, rank of a register with a single ancestor is 2, rank of register with 2 ancestors is 3 etc.
secret	A boolean value indicating wither the inputs for the register are public or secret.
binary	A boolean value indicating whether the register can accept only binary values (ones and zeros).
offset	A signed integer value specifying the number of steps by which an input value is to be shifted in the execution trace.
parent	An integer value specifying an index of the parent input register. If the register has no parents, this property will be undefined.
steps	An integer value specifying the number of steps by which a register trace is to be expanded for each input value. For non-leaf registers, this property will be undefined.

License

MIT © 2019 Guild of Weavers