Hello, Hypercube!

Motivation

This is a project as part of ZK Hack Istanbul 2023. zk-SNARKs have always been black boxs to most ZK developers, and our motivation is to deconstruct the black box into its base level primitives, showcasing the various techniques that can be used to transform a much complex constraints problem into simpler univariate polynomials that are much easier and efficient to dealt with.

The goal is to implement cryptographic primitives to reduce sumcheck-based zk-SNARKS proving system into a univariate polynomial eventually. We call it "Hello, Hypercube!" because a multilinear polynomial can be represented as points on a boolean hypercube and reduced through evaluations on it partial sum (using sumcheck).

Such reduction can be used in:

• Sumcheck-based zk-SNARKS
• Nova-like folding schemes
• Lasso/Jolt zkVM systems
• GKR-based protocols

Overview

The above diagram shows the outline of this project. Starting from the top level, we have a ZK proving system (a simplified version of Spartan is implemented here). The proving system can be represented as a multilinear polynomial. We can think of the multilinear polynomial as a Boolean hypercube. For example, if the polynomial is of form $f(x_1, x_2, x_3)$, then the hypercube is of dimension 3. We can think of f as following:

f(0, 0, 0) = v_0
f(0, 0, 1) = v_1
f(0, 1, 0) = v_2
f(0, 1, 1) = v_3
f(1, 0, 0) = v_4
f(1, 0, 1) = v_5
f(1, 1, 0) = v_6
f(1, 1, 1) = v_7

where $v_i$ is the value of the polynomial at the point $(x_1, x_2, x_3)$ = $(i_1, i_2, i_3)$. Point $v_i$ is each a vertex of the hypercube. We can then evaluate over the hypercube to get the partial sums from the sumcheck protocol. The purpose of evaluating the polynomial on a Boolean hypercube is that multilinear sumcheck is efficient for bitwise operations.

Thinking of zk-SNARKs construction over hypercubes is very intuitive. For example, R1CS follows the general form of $A\vec{z}\circ B\vec{z} = C\vec{z}$, which we can decompose it into multiple polynomials as shown in the diagram below. Each polynomial is a hypercube, and to evalutate their relationships we just need to evaluate by folding the vertices of the hypercubes.

After the sumcheck protocol, we have reduced the polynomial to a simpler form where it is a polynomial dependent on the verifier's randomness during sumcheck. This is represented as $f(\vec{r})$ and is still a multilinear polynomial, though a much smaller one. We want to further reduce this to a univariate polynomial. Three ways that have been proposed in literature are LogUp+, Gemini and ZeroMorph. They each use different techniques, which are:

1. LogUp+: reduction of multivariate polynomial in evaluation form to univariate in evaluation form
2. Gemini: reduction of multivariate polynomial in coefficient form to univariate in coefficient form
3. ZeroMorph: reduction of multivariate polynomial in evaluation form to univariate in coefficient form

It's much easier to deal with univariate polynomial using commitment schemes such as KZG10. Especially when a degree-N polynomial has really large degrees (i.e. N), the evaluation proofs of such polynomials become prohibitively expensive, which motivates the entire project.

Running the code

cd univarization
cargo test

Code Organization

The main files showcasing our work are as follows (in the univarization/src folder):

• mle/*.rs: representing polynomials using multilinear extension (MLE)
• unipoly.rs: representing univariate polynomials
• sumcheck.rs: sumcheck protocol on multilinear polynomials
• bcho_pcs.rs: implementation of the Gemini univarization technique as mentioned above
• ph23_pcs.rs: implementation of the LogUp+ technique as mentioned above
• zeromorph.rs: implementation of the Zeromorph technique as mentioned above
• unipoly.rs: implementation of univariate polynomials
• snark.rs: implementation of a Spartan like proving system

We use ark_bn254, ark_std and ark_ff in our implementations.

File	blank	comment	code
unipoly.rs	353	497	1023
mle/mod.rs	31	186	125
mle/coeffs_sparse.rs	132	157	709
mle/evals_sparse.rs	66	18	349
mle/evals.rs	58	36	272
ph23_pcs.rs	175	296	558
zeromorph.rs	194	235	530
bcho_pcs.rs	127	141	453
sumcheck.rs	165	86	418
snark.rs	93	67	388
kzg10.rs	97	163	293
lib.rs	45	9	140
unisumcheck.rs	37	12	130
transcript.rs	23	2	107
bits.rs	26	3	102
groupsim.rs	18	0	72
SUM:	1640	1908	5669

Future Work

• We look to benchmark the three approaches in the future.
• Adding shifting (to the next).
• Adding zero knowledge construction.

Credits

Our work is based on the following papers and libraries:

• LogUp+
• Gemini
• ZeroMorph
• Arkworks