zkDaily - 2026-04-27

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ZKP Frontier Tracker 🎯

Mon

04.27

2026

The Sum-Check Protocol over the Monomial Basis, and Other Optimizations

Paper

https://eprint.iacr.org/2026/762

Ari Biswas Sum-Check Monomial Basis

Biswas et al. proposed a projective variant of the Sum-Check protocol in their paper, changing the interpolating set from Boolean hypercube to infinity hypercube to optimize prover performance, with additional optimizations for prime field multiplication.

Notes

Proposed a projective Sum-Check variant using infinity hypercube instead of Boolean hypercube
Achieves end-to-end prover speedup on BN254 and 128-bit pseudo-Mersenne prime fields
Eliminates all field subtractions when binding multilinear polynomials
Provides evaluation procedures with fewer field operations for structured polynomials
Monomial-coefficient form aligns naturally with polynomial commitment schemes like WHIR
Optimized prime field multiplication by sampling challenges from specific subsets

Collected by @icerdesign

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Q&A Deep Dive 💬

Mon

04.27

2026

On what domain does standard sum-check typically operate?

Standard sum-check typically runs over the Boolean hypercube, where each variable is 0 or 1. This fits naturally with multilinear polynomials.

What is the key idea of the monomial basis optimization?

The paper replaces the Boolean hypercube with an “infinity hypercube,” allowing evaluations at certain points to directly extract monomial coefficients, changing the representation used in sum-check.

Why does the monomial coefficient form align better with some polynomial commitments?

Some commitment schemes like WHIR naturally use monomial coefficients, while standard sum-check uses multilinear basis. This mismatch requires conversions, which the new approach avoids.

Prompted by @icerdesign