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A beginner’s intro to coding zero-knowledge proofs

Zero-knowledge proofs have gotten more and more well-liked, however it may be a tough area to navigate and discover the sources to get began as a developer. After spending a while learning the subject, I am compiling right here what I realized in hope it helps others. Observe from the title that this can be a newbie’s information, as in its writer (me!) being a newbie on ZKP programming, so please attain out should you discover something off on this article!

This text will cowl what a zero-knowledge proof is, how it’s used from a developer’s perspective, and undergo a number of languages for writing them. It’ll not undergo the underlying math: we’ll deal with ZKPs as a cryptographic black field.

What’s a zero-knowledge proof?

A zero-knowledge proof (ZKP for brief) is a cryptographic proof a few reality that doesn’t reveal any data about mentioned reality.

A basic instance to introduce ZKPs is that I (the prover) can show to you (the verifier) that I do know a valid coloring for a graph with out revealing it. To do it, we play a sport the place I colour the graph utilizing a random set of colours, then cowl it, and also you select an edge at random. If my coloring was legitimate, then each nodes related by that edge should have totally different colours. Then we repeat, with a special set of colours every time, till you might be happy. Since I re-paint the graph earlier than each spherical, you do not study something in regards to the coloring, however you’ll be able to confirm that it was legitimate each time.

One other well-known instance of what could possibly be thought-about a ZKP is a digital signature: I can produce a proof that I maintain a secret string (my personal key) that corresponds to a public one (my public key) by signing a message, and that signature doesn’t reveal any bits from my personal key.

What’s a zkSNARK?

SNARK stands for Succinct Non-interactive ARgument of Data. The vital bits listed below are the primary two phrases: succinct and non-interactive.

A proof is succinct if it is, properly, small. Which means, whatever the complexity of the truth that we’re proving, we will maintain proofs small and fast to confirm. That is what makes ZKPs enticing for building rollups: you’ll be able to generate a succinct proof that each one operations that occurred on the L2 chain have been legitimate, succinct sufficient which you could confirm it in a contract on L1. Protocols like Mina take this to the acute with recursive proofs, which permit them to show the validity of all the historical past of the chain with a constant-size proof.

As for non-interactivity, simply suppose how cumbersome it will be to cope with signatures should you wanted the signer to be on-line each single time somebody wanted to confirm it. Within the examples above, the graph coloring proof was interactive, requiring an internet sport between the prover and the verifier. Happily, zkSNARKs don’t have this requirement.

ZKPs flavors

Seems that zkSNARKs should not the one sort of attention-grabbing ZKPs. There are additionally STARKs (pioneered by Starwkare and used to energy Starknet) and Bulletproofs (utilized in Monero). These differ within the operating time for producing and verifying a proof, in addition to the ensuing proof dimension, or within the want for trusted setups (extra on this within the subsequent part). MatterLabs has an ideal comparability between them here.

Even inside SNARKs, you will discover that there are a number of flavors. Groth16 is among the hottest. The PLONK household, which incorporates TurboPLONK and UltraPLONK, can take away the necessity for a circuit-specific trusted setup on the expense of proof era and verification time, or add assist for brand spanking new varieties of operations.

Moreover, there are totally different elliptic curves to choose from, or totally different polynomial dedication schemes that can be utilized. Consensys Gnark offers an ideal explainer on SNARK proving schemes and curves here.

Often, choosing a ZKP language to work with will decide which ZK backend you find yourself utilizing, although there are some like Circom or Gnark that support more than a single proving system.

Trusted setup

Some flavors of proving techniques require what’s referred to as a trusted setup. A trusted setup is a one-time ceremony, run by a number of contributors, which produces a set of random values that energy the ZK proving system.

These ceremonies are important as a result of if all of its contributors collude they’ll break the proving system. Right here, breaking the proving system means with the ability to generate a proof for an invalid declare that will get accepted by the verifier. Happily, having only one sincere participant does the trick: that is why you will discover calls for as many community members as possible to take part in such a ceremony.

Sure flavors, like Groth16, require a trusted setup for every circuit: because of this, for each program you code, you have to run a brand new ceremony. Others, like PLONK, want simply one common trusted setup that may be reused on any program constructed with that taste. And others, like STARKs, do not want any trusted setup in any respect.


The primary factor you have to learn about ZKPs internals is that the proof operates on a mathematical constraint system. In different phrases, prover engines don’t cope with code like x := y * z, however with constraints of the shape x - y * z = 0. Relying on the language you are working with and the prover backend, these might affect the way you construction your code.

For example, Circom solely permits you to write quadratic expressions in your variables, whereas Halo2 helps you to write polynomial constraints of any degree. As we’ll see later, one of many outputs of compiling a program in a ZKP language would be the set of constraints.

This has the implication that sure operations are simpler to signify in a ZKP than others: addition may be represented in a single constraint, whereas bit manipulation might require a whole bunch, as each bit might must be represented as a separate variable. That is the explanation why particular cryptographic primitives are sometimes chosen when engaged on ZKPs. For instance, Pedersen hashes are hash features that may be represented in an arithmetic circuit way more effectively than keccak, so they’re steadily present in ZKPs.

Finite fields

One other vital bit you have to know is that constraints function in a finite subject of a dimension decided by the underlying elliptic curve. What this implies is that each one arithmetic operations are computed modulo a sure worth, that means they may wrap-around that worth. Don’t fret although, this worth is often massive sufficient. For example, in Circom it’s:

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This additionally implies that, if you wish to outline particular thresholds to your integer variables, you will want so as to add further constraints for it. Some languages like Noir will do it for you, whereas others would require you to do it manually.

Why all the excitement?

ZKPs have drawn quite a lot of consideration currently. The primary motive is that they can be utilized as a method to cheaply confirm any computation, successfully turning into a general-purpose cryptographic constructing block.

Gubhseep describes ZKPs as programmable cryptography, within the sense that you need to use them to program cryptographic proofs for normal issues. For example, you might show set membership utilizing ring signatures that are particularly constructed to sort out that downside, or simply coding a zkSNARK circuit for it.

Within the Devcon6 panel on Zero Knowledge, there’s an ideal level made about their energy in enabling interoperability throughout totally different techniques: upon getting a system that may confirm a ZKP, then you’ll be able to confirm that another system with any computational atmosphere behaved as anticipated, so long as you’ll be able to specific its computation as a ZKP. That is what powers zk-rollups: should you can write the foundations of your chain as a ZKP, then you’ll be able to push that proof to Ethereum and confirm it on the EVM – even when your rollup does not use the EVM as an execution atmosphere!

What does a circuit seem like?

Let’s now look into how all this works. Step one is to write down a program in your ZK language or framework of selection, resembling Circom, Halo2, or Noir. These packages are also known as circuits, which specific constraints amongst your set of variables, or alerts.

For instance, we will construct a circuit in Circom that lets us show that we all know two numbers, a and b, such that c = (a*b)^2, with out revealing them:

template MultiplierSq() {
  sign enter a;
  sign enter b;
  sign ab;
  sign output c;
  ab <== a * b;
  c <== ab * ab;
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Right here we’re defining two personal enter alerts, an intermediate variable, and an output worth. Observe that, since Circom doesn’t permit us to write down something extra complicated than quadratic constraints, we can’t immediately write c <== (a*b)^2, that is why we’d like the intermediate ab sign.

Creating and verifying a proof

The workflow for utilizing a ZKP circuit isn’t the identical as a daily program, since we’ve got two totally different brokers: the prover and the verifier. And even inside the prover, there are a number of steps concerned to generate the proof. Let’s undergo them:

  1. We first write and compile the circuit. This outputs a number of artifacts, such because the set of constraints outlined by the circuit, and a script or binary for use within the step beneath.
  2. Relying on the SNARK taste getting used, it could be essential to run a trusted setup ceremony to generate the proving and verifying keys, that are cryptographic materials required for the proving and verifying course of, as their names indicate.
  3. When interacting with our zk-app, the person enters the personal and public inputs for the circuit. Step one is to “execute” the circuit as if it have been a program, utilizing the script or binary generated throughout compilation. This calculates the values for all intermediate and output variables, given the enter. Within the instance above, given a = 2 and b = 3, it’s going to calculate that ab = 6 and c = 36. This task of each sign to its concrete worth known as a witness or hint.
  4. Given the hint from step 3 and the proving key from step 2, the prover can now generate a zero-knowledge proof that each one the constraints outlined within the circuit maintain, and solely reveals the output worth c. This proof can now be despatched to the verifier.
  5. The verifier, utilizing the submitted proof and the verifying key, can now confirm that the proof is appropriate for its public output. In our examples, the verifier can cryptographically examine that the prover is aware of three values (a, b, ab) such that a * b = ab and ab * ab = c, the place c = 36.

Remember the fact that the verification step is reasonable, so the verifier may be run as an EVM sensible contract. This permits to trustlessly confirm any off-chain computation that may be expressed as a ZKP.

If you wish to check out the workflow above your self, I strongly reccomend Circom’s getting started guide that may take you thru the required steps. In Circom, compiling a circuit will output a WASM or a C++ program that computes the witness, in addition to a R1CS representation that, along with the trusted setup ceremony, outputs the proving and verifying keys. Then, in an effort to generate the proof, you provide the calculated witness and the proving key. To confirm it, simply use the generated proof together with the verifying key. You can too autogenerate a Solidity contract for operating the verification on-chain.

Execution vs constraints

Within the workflow above, you’ll have famous that the circuit is used twice by the prover: first to generate the witness, after which to derive the constraints which are coated by the proof. These two runs are fully totally different beasts, and understanding their distinction is among the keys to understanding how ZKPs work.

Let’s return to our Circom instance from earlier than, the place we had two directions:

ab <== a * b;
c <== ab * ab;
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In Circom, a fat arrow is just syntax sugar for two different instructions, task and constrain, so the above may be expanded to:

ab <-- a*b
ab === a*b
c <-- ab * ab
c === ab * ab
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The a <-- a*b directions means through the execution section, assign a*b to ab, and will get ignored when compiling the constraints. Alternatively, ab === a*b means add a constraint that forces ab to be equal to a*b, which will get ignored throughout execution. In different phrases, when writing a circuit you are writing two totally different packages, that belong to 2 totally different programming paradigms, in a single one.

Whereas you’ll often write assignments and constraints which are equal, generally you have to cut up them up. instance of that is the IsZero circuit.

The IsZero circuit

Writing a circuit that outputs 1 if an enter sign is zero and 0 in any other case is surprisingly troublesome if you find yourself constrained (pun supposed) to only quadratic expressions. In case you are pondering of utilizing an expression like x === 0, observe that that won’t return a boolean on whether or not x is zero or not, however as an alternative it’s going to constrain the sign x to be zero on this circuit.

Happily, Circom has a library of helpful constructing blocks that features an IsZero circuit that returns 0 or 1 relying on whether or not the enter is zero, and appears like this:

template IsZero() {
  sign enter in;
  sign output out;
  sign inv;

  inv <-- in!=0 ? 1/in : 0;
  out <-- -in*inv +1;
  out === -in*inv +1;
  in*out === 0;
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This can be a good instance of decoupling execution and constraints in a circuit. Let’s start analyzing the constraints. The constraint in * out = 0 tells us that in or out is zero. If in is zero, then out = -in * inv + 1 = 1, so we’re good. And if out is zero, then it follows that in * inv = 1, which implies that in can’t be zero. Nice!

However observe that the precise worth of inv by no means popped up in our reasoning, since we didn’t want it. That is legitimate for writing constraints, however relating to populating the hint, it falls quick. We have to inform Circom calculate the worth for inv within the witness era section: that is the place the inv <-- in!=0 ? 1/in : 0 expression is available in. Cairo appropriately calls this type of expressions hints.

Observe that we’re utilizing operations way more complicated than a quadratic expression right here: we’ve got a conditional, a comparability, and a division. It’s because the execution section is unrelated to the constraints, and as such, is not certain by the restrictions of producing a ZKP. So we’re again to just about regular programming when working within the execution section, gaining access to extra operators, management stream statements, and even auxiliary functions.

Underconstrained computation bugs

Nevertheless, this distinction between execution and constraint era phases can open the door to a really nasty sort of bugs. Going back to the IsZero example, what would have occurred if we forgot to write down one of many constraints, say the in*out = 0 one?

Let’s comply with the workflow we all know, assuming in=3 as enter:

  1. The circuit would have compiled positive, since there are not any errors.
  2. Witness era would have adopted the execution directions, and accurately calculated the values inv=3^-1, out=0.
  3. The prover would have generated a proof primarily based on the witness and the general public output, because the constraint out = -in*inv +1 holds.
  4. The verifier would have accepted it as legitimate, trusting that the prover has a non-public non-zero worth.

To this point so good. However what occurs if a malicious prover desires to persuade us that they’ve a zero worth, once they even have in=3?

  1. The malicious person can manually craft a witness the place in=3, inv=0, out=1.
  2. For the reason that witness satisfies the constraint out = -in*inv +1, the prover can generate a proof for it.
  3. The verifier fortunately accepts it, and believes that the person has a zero worth once they truly do not.

This type of points, dubbed under-constrained computation, are significantly arduous to catch. Any unit take a look at that treats the circuit as a black field would by no means set off them, as they use the circuit’s execution logic to construct the witness, which is able to move the constraints. The one technique to reproduce them is to manually assemble a witness with custom logic, which requires that you just precisely know the assault you might be in search of.

Alternatively, there are efforts for verifying the soundness of a circuit you have written with regard to the operate you need to encode, resembling Ecne, whcih I nonetheless need to check out.

The underside line right here is that try to be significantly cautious once you write execution code separate out of your constraint era code, because it opens the door to underconstrained ZK packages.

An actual-world instance

Tornado Cash is a superb studying useful resource on construct a real-world zk-app with Circom circuits. I strongly advocate going by the 3-page-long white paper, in addition to Gubsheep’s breaking down Tornado Cash session from the 0xparc course, plus Porter’s overview of the white paper.

The gist of Twister, in case you aren’t aware of it, is which you could deposit ETH in discrete quantities (there are notes of 0.1, 1, 10, and 100 ETH) right into a contract, and later anonymously withdraw every observe into different addresses. This has the impact of blending your notes with these from different individuals, granting anonimity so long as there’s sufficient utilization.

How does it work?

For the sake of the reason, we’ll undergo a barely simplified model of the protocol, and comply with Gubsheep’s version as an alternative of the one truly applied.

At any time when a person desires to deposit a observe, they generate a secret worth s, and submit their 1 ETH observe together with a hash of the key H(s) to the contract. The contract collects all submitted secret hashes right into a merkle tree, and shops its root R.

Now, when a person desires to withdraw their observe, they should show that they’ve the key that corresponds to H(s). To do that, the person submits a ZKP that exhibits that they know a non-public worth s such that H(s) is within the merkle tree of root R. It’s not revealed which of all of the leaves of the tree is the person’ s H(s), because the ZKP solely public output is R, so this preserves anonimity. The merkle proof is checked by a circuit within the ZKP.

We additionally want a method to make sure that a person can’t withdraw the identical observe twice. Right here is the place the idea of nullifier is available in. A nullifier is a price, deterministically generated from the unique secret, that’s tracked to stop double-spends.

So, along with proving membership to the merkle tree of root R, the ZKP additionally outputs a price G(s) for use as a nullifier, the place G is one other hash operate totally different to H. Then, at any time when a observe is withdrawn, its nullifier is added to a nullifier set within the sensible contract, and the withdrawn is rejeted if it already exists. Observe that G must be totally different to H, in any other case the ZKP would reveal H(s), which may be traced again to the unique deposit and break anonimity.

In different phrases, at any time when a person desires to withdraw, they submit a ZKP that states that they know a secret worth that is in a merkle tree of root R, and exposes the nullifier of that secret worth. Then the contract can examine that the merkle root matches and the nullifier hasn’t been used it.

Should you went by Twister’s white paper, you will observe that the nullifier idea is the place their implementation diverges from what we described right here. Twister truly makes use of two totally different secret values, referred to as randomness r and nullifier okay, and the general public worth they retailer known as the nullifier hash. Twister additionally makes use of a single hash operate (Pedersen hashes) as an alternative of two totally different ones, and so they compute the merkle tree leaf as H(r || okay) and the nullifier hash as H(okay). The vital factor is that the nullifier can’t be linked again to the merkle tree leaf, and that it’s deterministically generated from the key. You’ll be able to try Twister’s full circuits in Circom here.

Languages for ZKPs

There are a number of languages and frameworks which you could select from for writing ZK circuits, both normal objective (like Circom) or particular to a sure platform (like Cairo). I’ve tried out three totally different languages, Circom, Halo2, and Noir, and implemented the same circuit to compare them. The circuit proves that the person is aware of the outcomes of a non-public set of rock-paper-scissors matches, such that the whole rating matches a public output. The rating is calculated as specified within the definition of advent of code 2022 day 2, which I used for inspiration for the issue.

Your whole rating is the sum of your scores for every spherical. The rating for a single spherical is the rating for the form you chose (1 for Rock, 2 for Paper, and three for Scissors) plus the rating for the end result of the spherical (0 should you misplaced, 3 if the spherical was a draw, and 6 should you received).

What follows is an summary of every language, my private impression of it, and the way the rock-paper-scissors circuit seems like in it. Bear in mind I am a newbie in all of them, so there could also be errors. Please attain out should you discover any so I can repair this text.

Different languages and frameworks I nonetheless need to check out are Zokrates, Cairo, SnarkyJS, Gnark, and Leo. I hope to replace this publish as I am going by them!


I discovered Circom to be the very best language to begin studying ZKPs, as steered by CyberPorter. To me, Circom has the perfect stage of abstraction for studying: not too excessive stage that it abstracts away a few of the quirks of ZKP, and never too low stage that you just get slowed down in annoying particulars. It additionally has been round for fairly a while (modulo how new all the things ZK is!), so you will discover extra documentation and samples, in addition to the awesome Circom workshops from 0xparc. Final however not least, there’s circomlib, a library of frequent constructing blocks for Circom.

Circom ships with a rust-based compiler CLI, plus an npm package for trusted setups and proof era and verification. The advantage of having an npm bundle is which you could generate proofs within the browser, opening the door to constructing zk-apps. You can too autogenerate Solidity contracts for verifying proofs on-chain. Circom additionally helps you to to decide on your proving engine between Groth16 and PLONK.

As talked about earlier than, in Circom you’re employed assembling circuits, the place every circuit has a set of enter, inside, and output signals. Circuits are outlined as templates, which may be parameterized with values identified at compile-time, that are just like C++ function templates. Circom additionally has the idea of functions as a reusable constructing block within the execution section.

The primary challenges are Circom is understanding if you find yourself writing constraint-generation code, and if you find yourself writing witness-generation code, and monitoring what values are known at compile-time versus solely identified at runtime. Many language constructs are solely obtainable for witness era, like boolean or comparison operators, or can solely depend upon values identified at compile-time, resembling control flow statements. Additionally, preserving in thoughts the distinction between constraint and witness era occasions can guard you towards underconstrained computation bugs.

Rock-paper-scissors in Circom

The circuits for the rock-paper-scissors problem have been enjoyable to write down in Circom. Since it isn’t doable to write down conditional expressions in constraints, you have to resort to math-based tips. As a easy instance, the primary circuit, which validates the enter of a play by checking that it’s 0, 1, or 2, seems like this:

template AssertIsRPS() {
  sign enter x;
  sign isRP <== (x-0) * (x-1);
  isRP * (x-2) === 0;
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The code for calculating the score of a single round makes use of related constructs, together with the IsEqual circuit from CircomLib:

// Returns the rating for a single spherical, given the performs by x and y
template Spherical() {
  sign enter x, y;
  sign output out;

  // make sure that every enter is inside 0,1,2

  // examine if match was a draw
  sign isDraw <== IsEqual()([x, y]);

  sign diffYX <== (y+3)-x;

  // y wins if y-x = 1 mod 3
  sign yWins1 <== (diffYX-1) * (diffYX-4);
  sign yWins <== IsZero()(yWins1);

  // x wins if y-x = 2 mod 3
  sign xWins1 <== (diffYX-2) * (diffYX-5);
  sign xWins <== IsZero()(xWins1);

  // examine that precisely considered one of xWins, yWins, isDraw is true
  // we in all probability can do with out these constraints
  sign xOrYWins <== (xWins - 1) * (yWins - 1);
  xOrYWins * (isDraw - 1) === 0;
  xWins + yWins + isDraw === 1;

  // rating is 6 if y wins, 3 if draw, 0 if x wins
  // plus 1, 2, 3 relying on the selection of RPS
  out <== yWins * 6 + isDraw * 3 + y + 1;
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Lastly, the outermost circuit loops by a parameterizable variety of rounds n, and aggregates all scores. Observe {that a} loop can be utilized right here because it will depend on n, which is a template parameter identified at compile-time.

template Recreation(n) {
  sign enter xs[n];
  sign enter ys[n];
  sign scores[n];
  sign output out;

  var rating = 0;
  for (var i = 0; i < n; i++) {
    scores[i] <== Spherical()(xs[i], ys[i]);
    rating += scores[i];

  out <== rating;
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Halo2 isn’t a language however a Rust-based framework, maintained by the ZCash workforce. Halo2 is restricted to PLONKish, providing you with very direct management of how circuits are represented in the arithmetization. This makes Halo2 very low-level, however preferrred for writing extremely optimized circuits.

To me, Halo2 had the steepest studying curve by far. Not simply due to having to know how PLONKish arithmetization works to construct the circuits, however largely as a result of I discovered Halo2’s API fairly complicated and its documentation arduous to search out. There are additionally few sources for studying Halo2: the very best I discovered have been the 0xparc course which offers a number of really valuable code samples, in addition to the examples in the main repo. You might also need to try awesome-halo2 for up to date sources.

When beginning with Halo2, understand that there are two flavors of the framework on the market: the vanilla one maintained by ZCash, and the fork by Ethereum’s Privacy and Scaling Explorations team which makes use of a special polynomial dedication (KZG as an alternative of IPA) that is extra pleasant for Ethereum.

PLONKish arithmetization

The important thing to constructing circuits in Halo2 is to start by designing how the underlying PLONKish matrix will seem like. In PLONKish, circuits are outlined over a matrix that you just immediately outline. This matrix consists by a number of columns, the place a column might signify a public output (referred to as occasion), a non-public witness worth (referred to as recommendation), a relentless worth (referred to as mounted), a flag on whether or not a constraint is enabled (referred to as selector, extra on this beneath), or a lookup value (used for lookup tables, a complicated characteristic).

After getting the matrix arrange, it is time to outline the polynomial constraints utilizing gates. A gate in Halo2 defines a number of constraints to be utilized over a set of cells within the matrix. Every gate is managed by way of a selector column: if the selector is enabled at a row, then the constraints imposed by the gate relative to that row can be enforced.

A gate can outline constraints throughout a number of rows. For instance, this is an instance of an L-shaped multiplication gate straight out of the Halo2 book:

meta.create_gate("mul", |meta| 
    // );
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Within the code pattern above, a0 and a1 are recommendation solumns, and s_mul is the selector that defines whether or not the mul multiplication gate can be enforced. Whether it is, then the worth for a0 within the subsequent row can be constrained to be equal to the product of a0 and a1 on the present row.

As well as, Halo2 permits you to outline equality constraints, the place you require {that a} particular cell in your matrix should be equal to a different. That is helpful for “copying” values throughout your matrix, or for constraining a public occasion cell to equal a selected personal recommendation cell in an effort to expose a price.

As one other instance, you might outline a circuit for proving the N-th Fibonacci number, the place row i of the matrix has the values x[i-2], x[i-1], x[i]. This implies you’d want three recommendation columns to start with, let’s name them a, b, c. Then, you’d constrain c to be equal to a + b in the identical row utilizing a gate, which requires including a selector column for controlling it. And also you’d need to set up equality constraints in order that a and b are equal to b and c from the earlier row. Final, in an effort to expose the N-th number as a public value, you’d want an instance column, constrained to equal the value of c at the last row.

Chips, devices, and areas

The primary constructing blocks for circuits in Halo2 are gadgets and chips. Chips are the bottom stage unit. A chip will sometimes expose a technique for configuring its gates, in addition to a number of strategies for assigning values to its cells throughout synthesis (extra on this section beneath). You can too construct chips composed out of different chips. Gadgets, however, function at the next stage of abstraction, hiding the underlying chips’ implementation particulars, although you’ll be able to construct your circuits immediately out of chips and skip gadgets altogether.

To advertise reusability, you will discover that chips at all times function over relative offsets. This permits a number of chips to be grouped into regions in a circuit. As soon as all areas and their shapes have been outlined, it’s the accountability of a floor planner to rearrange these areas on the matrix, so that you need not immediately outline the place every chip is definitely positioned. Nevertheless, relying on the way you construction your circuits, it’s completely doable to pack all the things right into a single area and never delegate placement to the planner.

Halo2 Rust API

In Halo2, just like Circom, the primary factor to remember is that your code can be invoked a number of occasions in numerous circumstances: whether or not it’s to configure the matrix, generate the constraints, create a proof, or calculate a witness.

Your circuits must implement a specific Circuit trait that defines strategies that can be referred to as all through this lifecycle, both with concrete or unknown Values:

pub trait Circuit<F: Subject> {
    sort Config: Clone;
    sort FloorPlanner: FloorPlanner;

    fn without_witnesses(&self) -> Self;
    fn configure(meta: &mut ConstraintSystem<F>) -> Self::Config;
    fn synthesize(
        config: Self::Config, 
        layouter: impl Layouter<F>
    ) -> Outcome<(), Error>;
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The vital bits listed below are configure and synthesize. The TLDR is that configure will arrange the matrix form and its gates, and synthesize will calculte the witness and populate the matrix with the corresponding values. Nevertheless, synthesize can also be invoked with unknown values throughout different levels, so you have to at all times work with wrapped Values. For example, and whereas it could appear anti-intuitive, polynomial constraints in gates are outlined throughout configuration, however equality constraints are arrange at synthesis.

The example within the Halo2 e-book has a superb step-by-step on implement a easy circuit with a single chip in case you want a scaffold to comply with.

Rock-paper-scissors in Halo2

The RPS implementation in Halo2 is way more verbose than in Circom, so we’ll solely reproduce some components right here. To start with, the matrix is ready up with the next columns:

  • An recommendation column for every of the 2 gamers, xs and ys, the place the worth in row i represents what transfer the participant selected in spherical i.
  • A 3rd recommendation column accum that tracks the accrued whole rating, so row i incorporates the sum of the scores of all rounds as much as i.
  • A public occasion column, the place the final cell is constrained to be equal to the worth of accum, so solely the whole rating is revealed and never the intermediate ones.
  • A selector for enabling the one gate that validates inputs and calculates the rating of every spherical.

The main chip defines a customized gate that packs all constraints for every spherical: validating that inputs are in vary, and calculating the rating for the spherical and including it to the accrued whole rating. This chip makes use of the values of xs, ys, and accum in a row as “inputs”, and “outputs” the brand new rating within the accum column on the subsequent row.

meta.create_gate("spherical", |meta| {
    let s = meta.query_selector(selector);

    let x = meta.query_advice(col_x, Rotation::cur());
    let y = meta.query_advice(col_y, Rotation::cur());
    let accum = meta.query_advice(col_accum, Rotation::cur());

    // We retailer the output within the accum column within the subsequent row
    let out = meta.query_advice(col_accum, Rotation::subsequent());

    // Constraints for every spherical
        // out = y_wins * 6 + is_draw * 3 + y + 1 + accum
        s.clone() * (out - (y_wins.expr() * F::from(6) + is_draw.expr() * F::from(3) + y.clone() + const_val(1) + accum)),
        // x in (0,1,2)
        s.clone() * x.clone() * (x.clone() - const_val(1)) * (x.clone() - const_val(2)),
        // y in (0,1,2)
        s.clone() * y.clone() * (y.clone() - const_val(1)) * (y.clone() - const_val(2)),
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The y_wins and is_draw above are IsZero chips outlined like the next. Observe that we will reuse the identical selector column for all constraints, since there is no such thing as a row during which we’d need to allow some however disable others.

// yWins <==> (y+2-x) * (y-1-x) == 0;
let y_wins = IsZeroChip::configure(
    |meta| meta.query_selector(selector), 
    |meta| {
        let x = meta.query_advice(col_x, Rotation::cur());
        let y = meta.query_advice(col_y, Rotation::cur());
        (y.clone() + const_val(2) - x.clone()) * (y - const_val(1) - x)
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When synthesizing the circuit, we loop over the inputs for every spherical, calculate the accrued rating, and assign the calculated values to our matrix. Observe that, for the “execution” mode, we will immediately use conditional expressions to calculate the rating.

// Assign one row per spherical
for row in 0..N {
  let [x, y] = [xs[row], ys[row]]; 

  // Allow the selector for the spherical gate
  self.config.selector.allow(&mut area, row)?;

  // That is requiring us so as to add a relentless column to the chip config simply with zeros
  if row == 0 

  // Set x and y recommendation columns to the enter values
  area.assign_advice(|| format!("x[{}]", row),col_x,row,|| x)?;
  area.assign_advice(|| format!("y[{}]", row),col_y,row,|| y)?;

  // Assign the is_zero chips to the identical expressions outlined within the gates
  // yWins <==> (y+2-x) * (y-1-x) == 0;
  let y_wins_chip = IsZeroChip::assemble(y_wins.clone());
  let y_wins_value = (y + Worth::identified(F::from(2)) - x) * (y - Worth::identified(F::ONE) - x);
  let y_wins = y_wins_chip.assign(&mut area, row, y_wins_value)?;

  // isDraw <==> y-x == 0;
  let is_draw_chip = IsZeroChip::assemble(is_draw.clone());
  let is_draw_value = y - x;
  let is_draw = is_draw_chip.assign(&mut area, row, is_draw_value)?;

  // Calculate the rating of this spherical
  let round_score =|(y_wins, is_draw)| {
      let partial_score = if y_wins { 6 } else if is_draw { 3 } else { 0 };
      Worth::identified(F::from(partial_score)) + y + Worth::identified(F::ONE)

  // Assign the col_accum *within the subsequent row* to the brand new rating
  accum_value = accum_value + round_score;
  out_cell = area.assign_advice(
      || format!("out[{}]", row),
      row + 1,
      || accum_value
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The final bit is exposing the whole rating as a public output of the circuit, by constraining the instance column to match the accum one within the final row of the matrix:

layouter.constrain_instance(out_cell.cell(), self.config.occasion, N-1)
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Noir, constructed by Aztec, is the latest language of the pack and is beneath lively growth. It’s distributed as nargo, a rust-based CLI, and as a set of npm packages. The npm packages releases appear to lag barely behind the crate, which has the bleeding edge options and solely obtained its first stable release a number of days in the past (early February). Each the rust crate and the npm bundle can be utilized to compile Noir packages, generate and confirm proofs, and create a verifier Solidity contract.

Noir is closely impressed in Rust, as in it shares virtually the identical syntax. It helps integer, booleans, tuples, structs, and arrays, in addition to a fundamental field type which represents a component within the native subject of the proving backend (suppose an unsigned int capped to an enormous prime of about 254 bits), and is extra performant than common integers.

Not like Circom, Noir doesn’t mean you can outline code used just for witness era (aka hints). Noir helps management stream statements and operators out of the field, and converts them to equal constraints as wanted. This helps forestall any unconstrained computation bugs, on the expense of efficiency. That mentioned, you can outline constraints unrelated to witness era, utilizing the particular constrain keyword.

Rock-paper-scissors in Noir

Noir feels a bit like dishonest in comparison with the opposite languages, because it abstracts away most complexities associated to ZKPs, and looks like writing common Rust code. The whole thing of this system suits in 30 traces, and reads like a daily program:

world N: Subject = 3;

fn len<T>(_input : [T]) -> comptime Subject {}

fn spherical(x: Subject, y: Subject) -> Subject {
    constrain (x as u8) <= 2;
    constrain (y as u8) <= 2;

    let mut rating = 0;
    let diffYX = (y + 3 - x);

    if x == y {
        rating = 3;
    if (diffYX == 4) | (diffYX == 1) {
        rating = 6;

    rating + y + 1

fn fundamental(xs: [Field; N], ys: [Field; N]) -> pub Subject {
    let mut outcome = 0;

    for i in 0..len(xs) {
        outcome = outcome + spherical(xs[i], ys[i]);
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It is value mentioning that I did not run any evaluation relating to the variety of constraints generated by every language, however it’s anticipated that Noir, given its greater abstraction stage, produces the next variety of constraints, which affect proving time. However, because the compiler matures, extra optimizations ought to be added that get routinely integrated to Noir-written packages.


Should you obtained up to now and need to continue to learn, listed below are a few of the sources I used for getting up to the mark with ZKPs. I encourage you to undergo them if you wish to study immediately from the masters!

  • CyberPorter produced a set of 6 classes that present an ideal overview of ZKPs, together with the motivation, a mapping of the area and who’s engaged on it, a deep dive on TornadoCash, an summary of Circom, Cairo, SnarkyJS, and Noir, and a walkthrough of the mathematics behind Groth16.
  • 0xPARC has two full programs on-line, one on Circom and one other on Halo2. I strongly advocate beginning with the Circom one, led largely by the good Gubsheep, the man behind the Dark Forest game. These programs embody hands-on workshops that take you step-by-step on construct a number of circuits. If you wish to take a stab at Halo2, I would say they’re a must-see, since there should not many different resoruces on it.
  • Vitalik has a number of weblog posts on just about each cryptographic idea you want for understanding SNARKs and STARKs, in addition to attention-grabbing uses for the tech. If you wish to go deeper into the workings of ZKPs, search on his weblog first.
  • The Scroll team, who’re constructing a ZK-EVM, run a wonderful instructional weblog relating to ZKPs. Particularly, the article on proof generation offers an ideal step-by-step on the entire workflow concerned when making a ZKP.
  • Awesome lists are superior, and Matter Labs, the workforce behind zkSync, retains an awesome-zero-knowledge-proofs with hyperlinks to all sort of sources.

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