PlonKathon

PlonKathon is part of the program for [MIT IAP 2023] Modern Zero Knowledge Cryptography. Over the course of this weekend, we will get into the weeds of the PlonK protocol through a series of exercises and extensions. This repository contains a simple python implementation of PlonK adapted from py_plonk, and targeted to be close to compatible with the implementation at https://zkrepl.dev.

Cheatsheet

Following are notes that I made while working on the exercises. Don't look at it if you haven't made an attempt.

Variables Structure & Examples

## Round 1

Program(["e public", "c <== a * b", "e <== c * d"], 8)

witness: {None: 0, 'a': 3, 'b': 4, 'c': 12, 'd': 5, 'e': 60}

self.group_order = 8

program.constraints = 3

program.wires(): [Wire(L='e', R=None, O=None),
                  Wire(L='a', R='b', O='c'),
                  Wire(L='c', R='d', O='e')]

A_values: [60, 3, 12]
B_values: [None, 4, 5]
C_values: [None, 12, 60]

self.pk.QL = [1, 0, 0, 0, 0, 0, 0, 0]
self.pk.QR = [0, 0, 0, 0, 0, 0, 0, 0]
self.pk.QM = [0, 21888242871839275222246405745257275088548364400416034343698204186575808495616, 21888242871839275222246405745257275088548364400416034343698204186575808495616, 0, 0, 0, 0, 0]
self.pk.QO = [0, 1, 1, 0, 0, 0, 0, 0]
self.PI = [21888242871839275222246405745257275088548364400416034343698204186575808495557, 0, 0, 0, 0, 0, 0, 0]
self.pk.QC = [0, 0, 0, 0, 0, 0, 0, 0]

Variables Mapping

Notations

$[ \ ]_1$: commitment to Group 1

Paper	Codebase	In Proof / Round Outputs	Function	Appears	Description
$n$	`group_order`			PR1
$n$	`program.constraints`			PR1
$\omega_iL_i(X)$	`list_a`		`A_values + padding`	PR1	Scalar values of witness wire LEFT values
$\omega_{n+i}L_i(X)$	`list_b`		`B_values + padding`	PR1	Scalar values of witness wire RIGHT values
$\omega_{2n+i}L_i(X)$	`list_c`		`C_values + padding`	PR1	Scalar values of witness wire OUTPUT values
$a(X)$	`self.A`		`Polynomial(list_a, Basis.LAGRANGE)`	PR1	LEFT wire polynomials in Lagrange Basis
$b(X)$	`self.B`		`Polynomial(list_b, Basis.LAGRANGE)`	PR1	RIGHT wire polynomials in Lagrange Basis
$c(X)$	`self.C`		`Polynomial(list_c, Basis.LAGRANGE)`	PR1	OUTPUT wire polynomials in Lagrange Basis
$[a]_1$	`a_1`	Y	`setup.commit(self.A)`	PR1	Commitment of $a$
$[b]_1$	`b_1`	Y	`setup.commit(self.A)`	PR1	Commitment of $b$
$[c]_1$	`c_1`	Y	`setup.commit(self.A)`	PR1	Commitment of $c$
$q_M(X)$	`QM`			PR1, PR3	multiplication selector polynomial
$q_L(X)$	`QL`			PR1, PR3	left selector polynomial
$q_R(X)$	`QR`			PR1, PR3	right selector polynomial
$q_O(X)$	`QO`			PR1, PR3	output selector polynomial
$q_C(X)$	`QC`			PR1, PR3	constants selector polynomial
$S_{\sigma1}(X)$	`S1`			PR1, PR3	1st permutation polynomial
$S_{\sigma2}(X)$	`S2`			PR1, PR3	2nd permutation polynomial
$S_{\sigma3}(X)$	`S3`			PR1, PR3	3rd permutation polynomial
$[z]_1$	`z_1`	Y	`setup.commit(self.Z)`	PR2	Commitment of $z$
$z(X)$	`self.Z`		`Polynomial(Z_values, Basis.LAGRANGE)`	PR2	Permutation Grand Product polynomial in Lagrange Basis
$\omega^j$	`roots_of_unity`		`Scalar.roots_of_unity(group_order)`	PR2
$k_1\omega^j$	`2 * roots_of_unity`			PR2
$k_2\omega^j$	`3 * roots_of_unity`			PR2
	`roots_of_unity_by_4`		`Scalar.roots_of_unity(4 * group_order)`	PR3
$X$	`X`		`roots_of_unity_by_4_poly * self.fft_cofactor`	PR3
$k_1X$	`two_X`		`roots_of_unity_by_4_poly * self.fft_cofactor * 2`	PR3
$k_2X$	`three_X`		`roots_of_unity_by_4_poly * self.fft_cofactor * 3`	PR3
$a(X)$	`A_big`		`self.fft_expand(self.A)`	PR3	A in coset extended Lagrange basis
$b(X)$	`B_big`		`self.fft_expand(self.B)`	PR3	B in coset extended Lagrange basis
$c(X)$	`C_big`		`self.fft_expand(self.C)`	PR3	C in coset extended Lagrange basis
$PI(X)$	`PI_big`		`self.fft_expand(self.PI)`	PR3	Public Inputs in coset extended Lagrange basis
$QL(X), ...$	`QL_big`		`self.fft_expand(self.pk.QL)`	PR3	Selector polynomials QL, QR, QM, QO, QC in coset extended Lagrange basis
$z(X)$	`Z_big`		`self.fft_expand(self.Z)`	PR3	Permutation Grand Product polynomial in coset extended Lagrange basis
$Z(X\omega)$	`Z_shifted_big`		`Z_big.shift(4)`	PR3	Shifted Permutation Grand Product polynomial in coset extended Lagrange basis
$S_{\sigma1}(X)$	`S1_big`		`self.fft_expand(self.pk.S1)`	PR3	1st permutation polynomial in coset extended Lagrange Basis
$S_{\sigma2}(X)$	`S2_big`		`self.fft_expand(self.pk.S2)`	PR3	2nd permutation polynomial in coset extended Lagrange Basis
$S_{\sigma3}(X)$	`S3_big`		`self.fft_expand(self.pk.S3)`	PR3	3rd permutation polynomial in coset extended Lagrange Basis
$Z_H(X)$	`Z_H`		Polynomial( [((self.fft_cofactor * x) ** group_order - 1) for x in roots_of_unity_by_4], Basis.LAGRANGE)	PR3	$Z_H = X^N - 1$ in evaluation form in coset extended Lagrange basis
$L_1(X)$	`L0`		`Polynomial([1] + [0] * (group_order - 1), Basis.LAGRANGE)`	PR3	Lagrange basis polynomial: $L_1(x) = 1$ for $x=1$ $L_1(x) = 0$ for any other $x$
	`QUOT_big_coeffs`		`self.expanded_evals_to_coeffs(QUOT_big)`	PR3
$t_{lo}(X)$	`self.T1`		`Polynomial(QUOT_big_coeffs.values[:group_order], Basis.MONOMIAL).fft()`	PR3	$t(X)$ where: degree < n
$t_{mid}(X)$	`self.T2`		`Polynomial(QUOT_big_coeffs.values[group_order : 2 * group_order], Basis.MONOMIAL).fft()`	PR3	$t(X)$ where: n <= degree < 2n
$t_{high}(X)$	`self.T3`		`Polynomial(QUOT_big_coeffs.values[2 * group_order : 3 * group_order], Basis.MONOMIAL).fft()`	PR3	$t(X)$ where: 2n <= degree < 3n
$[t_{lo}]_1$	`t_lo_1`	Y	`setup.commit(self.T1)`	PR3	Commitment of $t_{lo}(X)$
$[t_{mid}]_1$	`t_mid_1`	Y	`setup.commit(self.T2)`	PR3	Commitment of $t_{mid}(X)$
$[t_{hi}]_1$	`t_hi_1`	Y	`setup.commit(self.T3)`	PR3	Commitment of $t_{hi}(X)$
$\bar{a} = a(\zeta)$	`a_eval`	Y	`self.A.barycentric_eval(self.zeta)`	PR4	A evaluated at $zeta$
$\bar{b} = a(\zeta)$	`b_eval`	Y	`self.B.barycentric_eval(self.zeta)`	PR4	B evaluated at $zeta$
$\bar{c} = a(\zeta)$	`c_eval`	Y	`self.C.barycentric_eval(self.zeta)`	PR4	C evaluated at $zeta$
$\bar{s}{\sigma1} = S{\sigma1}(\zeta)$	`s1_eval`	Y	`self.S1.barycentric_eval(self.zeta)`	PR4	S1 evaluated at $zeta$
$\bar{s}{\sigma2} = S{\sigma1}(\zeta)$	`s2_eval`	Y	`self.S2.barycentric_eval(self.zeta)`	PR4	S2 evaluated at $zeta$
$\omega$	`root_of_unity`		`Scalar.root_of_unity(self.group_order)`	PR4	1st root of unity
$\bar{z}{\omega} = z(\zeta\omega)$	`z_shifted_eval`	Y	`self.Z.barycentric_eval(self.zeta * root_of_unity)`	PR4	S2 evaluated at $zeta$
$Z_H(\zeta) = \zeta^n - 1$	`Z_H_eval`		`zeta ** group_order - 1`	PR5	Zero polynomial evaluation at Zeta
$L_1(\zeta)$	`L_1_eval`		`Z_H_eval / (group_order * (zeta - 1))`	PR5	Lagrange polynomial evaluation at Zeta
$W_\zeta(X)$	`W_z`		`Polynomial(W_z_coeffs[:group_order], Basis.MONOMIAL).fft()`	PR5
$W_{\zeta\omega}(X)$	`W_zw`		`Polynomial(W_zw_coeffs[:group_order], Basis.MONOMIAL).fft()`	PR5	Proof of
$[W_{\zeta}]_1$	`W_z_1`	Y	`setup.commit(W_z)`	PR5	Commitment of $W_{\zeta}(X)$
$[W_{\zeta\omega}]_1$	`W_zw_1`	Y	`setup.commit(W_zw)`	PR5	Commitment of $W_{\zeta\omega}(X)$

Exercises

Each step of the exercise is accompanied by tests in test.py to check your progress.

Step 1: Implement setup.py

Implement Setup.commit and Setup.verification_key.

Step 2: Implement prover.py

Implement Round 1 of the PlonK prover
Implement Round 2 of the PlonK prover
Implement Round 3 of the PlonK prover
Implement Round 4 of the PlonK prover
Implement Round 5 of the PlonK prover

Step 3: Implement verifier.py

Implement VerificationKey.verify_proof_unoptimized and VerificationKey.verify_proof. See the comments for the differences.

Step 4: Pass all the tests!

Pass a number of miscellaneous tests that test your implementation end-to-end.

Extensions

Add support for custom gates. TurboPlonK introduced support for custom constraints, beyond the addition and multiplication gates supported here. Try to generalise this implementation to allow circuit writers to define custom constraints.
Add zero-knowledge. The parts of PlonK that are responsible for ensuring strong privacy are left out of this implementation. See if you can identify them in the original paper and add them here.
Add support for lookups. A lookup argument allows us to prove that a certain element can be found in a public lookup table. PlonKup introduces lookup arguments to PlonK. Try to understand the construction in the paper and implement it here.
Implement Merlin transcript. Currently, this implementation uses the merlin transcript package. Learn about the Merlin transcript construction and the STROBE framework which Merlin is based upon, and then implement the transcript class MerlinTranscript yourself!

Getting started

To get started, you'll need to have a Python version >= 3.8 and poetry installed: curl -sSL https://install.python-poetry.org | python3 -.

Then, run poetry install in the root of the repository. This will install all the dependencies in a virtualenv.

Then, to see the proof system in action, run poetry run python test.py from the root of the repository. This will take you through the workflow of setup, proof generation, and verification for several example programs.

The main branch contains code stubbed out with comments to guide you through the tests. The hardcore branch removes the comments for the more adventurous amongst you. The reference branch contains a completed implementation.

For linting and types, the repo also provides poetry run black . and poetry run mypy .

Compiler

Program

We specify our program logic in a high-level language involving constraints and variable assignments. Here is a program that lets you prove that you know two small numbers that multiply to a given number (in our example we'll use 91) without revealing what those numbers are:

n public
pb0 === pb0 * pb0
pb1 === pb1 * pb1
pb2 === pb2 * pb2
pb3 === pb3 * pb3
qb0 === qb0 * qb0
qb1 === qb1 * qb1
qb2 === qb2 * qb2
qb3 === qb3 * qb3
pb01 <== pb0 + 2 * pb1
pb012 <== pb01 + 4 * pb2
p <== pb012 + 8 * pb3
qb01 <== qb0 + 2 * qb1
qb012 <== qb01 + 4 * qb2
q <== qb012 + 8 * qb3
n <== p * q

Examples of valid program constraints:

a === 9
b <== a * c
d <== a * c - 45 * a + 987

Examples of invalid program constraints:

7 === 7 (can't assign to non-variable)
a <== b * * c (two multiplications in a row)
e <== a + b * c * d (multiplicative degree > 2)

Given a Program, we can derive the CommonPreprocessedInput, which are the polynomials representing the fixed constraints of the program. The prover later uses these polynomials to construct the quotient polynomial, and to compute their evaluations at a given challenge point.

@dataclass
class CommonPreprocessedInput:
    """Common preprocessed input"""

    group_order: int
    # q_M(X) multiplication selector polynomial
    QM: list[Scalar]
    # q_L(X) left selector polynomial
    QL: list[Scalar]
    # q_R(X) right selector polynomial
    QR: list[Scalar]
    # q_O(X) output selector polynomial
    QO: list[Scalar]
    # q_C(X) constants selector polynomial
    QC: list[Scalar]
    # S_σ1(X) first permutation polynomial S_σ1(X)
    S1: list[Scalar]
    # S_σ2(X) second permutation polynomial S_σ2(X)
    S2: list[Scalar]
    # S_σ3(X) third permutation polynomial S_σ3(X)
    S3: list[Scalar]

Assembly

Our "assembly" language consists of AssemblyEqns:

class AssemblyEqn:
    """Assembly equation mapping wires to coefficients."""
    wires: GateWires
    coeffs: dict[Optional[str], int]

where:

@dataclass
class GateWires:
    """Variable names for Left, Right, and Output wires."""
    L: Optional[str]
    R: Optional[str]
    O: Optional[str]

Examples of valid program constraints, and corresponding assembly:

program constraint	assembly
a === 9	([None, None, 'a'], {'': 9})
b <== a * c	(['a', 'c', 'b'], {'a*c': 1})
d <== a _ c - 45 _ a + 987	(['a', 'c', 'd'], {'a*c': 1, 'a': -45, '': 987})

Setup

Let $\mathbb{G}_1$ and $\mathbb{G}_2$ be two elliptic curves with a pairing $e : \mathbb{G}_1 \times \mathbb{G}_2 \rightarrow \mathbb{G}_T$. Let $p$ be the order of $\mathbb{G}_1$ and $\mathbb{G}_2$, and $G$ and $H$ be generators of $\mathbb{G}_1$ and $\mathbb{G}_2$. We will use the shorthand notation

$$[x]_1 = xG \in \mathbb{G}_1 \text{ and } [x]_2 = xH \in \mathbb{G}_2$$

for any $x \in \mathbb{F}_p$.

The trusted setup is a preprocessing step that produces a structured reference string: $$\mathsf{srs} = ([1]_1, [x]_1, \cdots, [x^{d-1}]_1, [x]_2),$$ where:

$x \in \mathbb{F}$ is a randomly chosen, secret evaluation point; and
$d$ is the size of the trusted setup, corresponding to the maximum degree polynomial that it can support.

@dataclass
class Setup(object):
    #   ([1]₁, [x]₁, ..., [x^{d-1}]₁)
    # = ( G,    xG,  ...,  x^{d-1}G ), where G is a generator of G_2
    powers_of_x: list[G1Point]
    # [x]₂ = xH, where H is a generator of G_2
    X2: G2Point

In this repository, we are using the pairing-friendly BN254 curve, where:

p = 21888242871839275222246405745257275088696311157297823662689037894645226208583
$\mathbb{G}_1$ is the curve $y^2 = x^3 + 3$ over $\mathbb{F}_p$;
$\mathbb{G}2$ is the twisted curve $y^2 = x^3 + 3/(9+u)$ over $\mathbb{F}{p^2}$; and
$\mathbb{G}_T = {\mu}r \subset \mathbb{F}{p^{12}}^{\times}$.

We are using an existing setup for $d = 2^{11}$, from this ceremony. You can find out more about trusted setup ceremonies here.

Prover

The prover creates a proof of knowledge of some satisfying witness to a program.

@dataclass
class Prover:
    group_order: int
    setup: Setup
    program: Program
    pk: CommonPreprocessedInput

The prover progresses in five rounds, and produces a message at the end of each. After each round, the message is hashed into the Transcript.

The Proof consists of all the round messages (Message1, Message2, Message3, Message4, Message5).

Round 1

def round_1(
    self,
    witness: dict[Optional[str], int],
) -> Message1

@dataclass
class Message1:
    # - [a(x)]₁ (commitment to left wire polynomial)
    a_1: G1Point
    # - [b(x)]₁ (commitment to right wire polynomial)
    b_1: G1Point
    # - [c(x)]₁ (commitment to output wire polynomial)
    c_1: G1Point

Round 2

def round_2(self) -> Message2

@dataclass
class Message2:
    # [z(x)]₁ (commitment to permutation polynomial)
    z_1: G1Point

Round 3

def round_3(self) -> Message3

@dataclass
class Message3:
    # [t_lo(x)]₁ (commitment to t_lo(X), the low chunk of the quotient polynomial t(X))
    t_lo_1: G1Point
    # [t_mid(x)]₁ (commitment to t_mid(X), the middle chunk of the quotient polynomial t(X))
    t_mid_1: G1Point
    # [t_hi(x)]₁ (commitment to t_hi(X), the high chunk of the quotient polynomial t(X))
    t_hi_1: G1Point

Round 4

def round_4(self) -> Message4

@dataclass
class Message4:
    # Evaluation of a(X) at evaluation challenge ζ
    a_eval: Scalar
    # Evaluation of b(X) at evaluation challenge ζ
    b_eval: Scalar
    # Evaluation of c(X) at evaluation challenge ζ
    c_eval: Scalar
    # Evaluation of the first permutation polynomial S_σ1(X) at evaluation challenge ζ
    s1_eval: Scalar
    # Evaluation of the second permutation polynomial S_σ2(X) at evaluation challenge ζ
    s2_eval: Scalar
    # Evaluation of the shifted permutation polynomial z(X) at the shifted evaluation challenge ζω
    z_shifted_eval: Scalar

Round 5

def round_5(self) -> Message5

@dataclass
class Message5:
    # [W_ζ(X)]₁ (commitment to the opening proof polynomial)
    W_z_1: G1Point
    # [W_ζω(X)]₁ (commitment to the opening proof polynomial)
    W_zw_1: G1Point

Verifier

Given a Setup and a Program, we can generate a verification key for the program:

def verification_key(self, pk: CommonPreprocessedInput) -> VerificationKey

The VerificationKey contains:

verification key element	remark
$[q_M(x)]_1$	commitment to multiplication selector polynomial
$[q_L(x)]_1$	commitment to left selector polynomial
$[q_R(x)]_1$	commitment to right selector polynomial
$[q_O(x)]_1$	commitment to output selector polynomial
$[q_C(x)]_1$	commitment to constants selector polynomial
$[S_{\sigma1}(x)]_1$	commitment to the first permutation polynomial $S_{\sigma1}(X)$
$[S_{\sigma2}(x)]_1$	commitment to the second permutation polynomial $S_{\sigma2}(X)$
$[S_{\sigma3}(x)]_1$	commitment to the third permutation polynomial $S_{\sigma3}(X)$
$[x]_2 = xH$	(from the $\mathsf{srs}$)
$\omega$	an $n$-th root of unity, where $n$ is the program's group order.

Name		Name	Last commit message	Last commit date
Latest commit History 629 Commits
.vscode		.vscode
compiler		compiler
test		test
.envrc		.envrc
.gitignore		.gitignore
README.md		README.md
TESTING_verifier_DO_NOT_OPEN.py		TESTING_verifier_DO_NOT_OPEN.py
__init__.py		__init__.py
curve.py		curve.py
poetry.lock		poetry.lock
poly.py		poly.py
prover.py		prover.py
pyproject.toml		pyproject.toml
setup.py		setup.py
test.py		test.py
transcript.py		transcript.py
utils.py		utils.py
verifier.py		verifier.py

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PlonKathon

Cheatsheet

Variables Structure & Examples

Variables Mapping

Notations

Exercises

Step 1: Implement setup.py

Step 2: Implement prover.py

Step 3: Implement verifier.py

Step 4: Pass all the tests!

Extensions

Getting started

Compiler

Program

Assembly

Setup

Prover

Round 1

Round 2

Round 3

Round 4

Round 5

Verifier

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PlonKathon

Cheatsheet

Variables Structure & Examples

Variables Mapping

Notations

Exercises

Step 1: Implement setup.py

Step 2: Implement prover.py

Step 3: Implement verifier.py

Step 4: Pass all the tests!

Extensions

Getting started

Compiler

Program

Assembly

Setup

Prover

Round 1

Round 2

Round 3

Round 4

Round 5

Verifier

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