zkDaily - 2026-04-12

零知识证明 zkDaily

ZKP Frontier Tracker 🎯

Sun

04.12

2026

Boolean Arithmetic over $\mathbb{F}_2$ from Group Commutators

Paper

https://eprint.iacr.org/2026/701

Marc Joye Boolean Group

Marc Joye proposes methods for binary field arithmetic using nonabelian group operations in the paper, implementing Boolean computation via commutators with explicit realizations in alternating groups A5 and A6, achieving state-of-the-art efficiency.

Notes

Implements binary field arithmetic using only group operations (multiplication and inversion) in nonabelian groups
Uses commutators for Boolean computation within group structure with two complementary approaches
Realizes universal Boolean gate (NAND) and direct XOR/AND operations
Applies to finite nonabelian simple groups like alternating groups A5 and A6
Achieves state-of-the-art efficiency in number of group operations for A5
Potential applications in fully homomorphic encodings and cryptography

Collected by @icerdesign

零知识证明 zkDaily

Q&A Deep Dive 💬

Sun

04.12

2026

What is a commutator?

A commutator is a basic group-theoretic operation, usually written as [a,b] = a⁻¹b⁻¹ab, measuring whether two elements commute. The paper uses it to encode NAND, XOR, and AND.

What two construction approaches does the paper propose?

The first approach realizes the universal Boolean gate NAND. The second directly realizes the field operations XOR and AND over 𝔽₂. Both rely only on intrinsic group operations.

Why is using only multiplication and inversion significant?

It means all computation is internal to the group structure, without requiring extra arithmetic circuits, reducing implementation complexity and improving elegance and composability.

Prompted by @icerdesign