The Good Tech Companies - How Binius Reduces Computational Complexity with Towers of Binary Fields

Episode Date: September 13, 2024

This story was originally published on HackerNoon at: https://hackernoon.com/how-binius-reduces-computational-complexity-with-towers-of-binary-fields. Discover how Binius leverages Towers of Binary Fields to enhance computational efficiency in Zero-Knowledge SNARKs. Check more stories related to web3 at: https://hackernoon.com/c/web3. You can also check exclusive content about #zero-knowledge-proofs, #sin7y, #binius, #blockchain-technology, #binary-fields, #zk-cryptography, #towers-of-binary-fields, #good-company, and more. This story was written by: @sin7y. Learn more about this writer by checking @sin7y's about page, and for more stories, please visit hackernoon.com. The Ulvetanna (IRREDUCIBLE) team addressed this question in their research paper titled Succinct Arguments over Towers of Binary Fields. They implemented it in Rust with their project, Binius: a Hardware-Optimized SNARK. In Binius, Binary Fields are constructed using towers of field extensions.

This audio is presented by Hacker Noon, where anyone can learn anything about any technology. How Binneas Reduces Computational Complexity with Towers of Binary Fields Bison 7y Reducing computational complexity has always been one of the primary goals of blockchain technology. One effective approach to achieving this is by reducing the bit-width of the computation field. For example, SNARKs based on elliptic curves perform arithmetic operations in fields with bit widths of 256 or higher, while STARKs have evolved from using the 64-bit Goldilocks field to the 31-bit Mersenne 31 and Baby Bear fields.

Beyond the efficiency of the prime numbers during modular operations, the significant reduction in bit width has led to Planonky 2 being hundreds of times faster than its predecessor, Plonky. Following this trajectory, one might wonder, is it possible to set the field width to 1, specifically $mathbbf underscore 2 dollar? The Ulvitana, Irreducible, team addressed this question in their research paper titled Succinct Arguments Over Towers of Binary Fields, 1, and implemented it in Rust with their project, Binneas. A hardware-optimized SNARK, 2, 3, since its release, Binneas has garnered significant attention in the ZK zero-knowledge community. The Lambda-class team has provided several technical analyzes,

4, 5, 6, and Vitalik Buterin offered a more accessible explanation, 7. In this article, we will explore the foundations of Binneas, focusing on the towers of binary fields, from both a technical and implementation perspective. Binary Fields The implementation of Binneas is based on binary fields. In Binneas, binary fields are constructed using towers of field extensions. The simplest binary field is, which contains only two elements, with operations performed model O2. Addition corresponds to bitwise XOR, and multiplication corresponds to bitwise N. By choosing an irreducible polynomial, we can form the field, where the elements are

remainders of polynomials of degree at most 1. While one method to extend fields involves taking remainders using irreducible polynomials, Binneas employs a more efficient approach. The use of multilinear Lagrange polynomials as a basis for tower extensions. This method allows for recursive field extensions, where each extension field is nested within the previous one. The specific implementation of tower extensions is as follows. First, then, next, from the construction process of the field extensions, it is clear that the extensions satisfy the following relationship. For, the field extension can also be expressed in the direct form of a ring as. Based on this implementation, different extensions can be

obtained as follows. From the elements contained in the extended field it is evident that for annealment derived from, it can be decomposed into the sum of two parts. For example, by iteratively decomposing, we can finally express. Additionally, for, since, addition and multiplication can be efficiently implemented in the binary extended field. Implementation of binary fields. Irreducible provides the open-source Rust implementation of Binus 3. The source code includes complete implementations and operations for the tower of binary fields 8, 9, 10, in 8. As shown in figure 1, the implementation includes the complete definition of operations for binary fields and the construction of the tower of binary fields. The tower of binary fields,

supporting up to a 128-bit binary field, is defined as follows. Additionally, 8. provides test and verification code for the tower of binary fields and related operations. References 1. HTTPS://eprint.iacr.org/.2023/.1784 Opening Square Bracket 2 Closing Square Bracket https://www.irreducible.com/.posts/.binius-hardware-optimized-snark-opening-square-bracket-3-closing-square-bracket-https-colon-slash-slash-gitlab.com-slash-irreducibloss-slash-binneas-opening-square-bracket-4-closing-square-bracket-snarks-on-binary-fields binneas part 1 lambda class dot com 5

snarks on binary fields binneas part 2 lambda class dot com 6 how binneas. Part 2. LambdaClass.com. 6. How Binneas is Helping Move the ZK Industry Forward. LambdaClass.com. 7. HTTPS colon slash slash vitalik dot eth dot limo slash general slash 2024 slash 04 slash 29 slash binneas dot html Opening square bracket 8 Closing square bracket bineus slash crate slash field slash src slash binary underscore field dot rs at main irreducible oss bineus github 9. bineus slash crate slash field slash src slash binary underscore field

underscore arithmetic dot rs at main irreducible oss bineus github 10. bineus slash crate slash Thank you for listening to this Hackernoon story, read by Artificial Intelligence. Visit hackernoon.com to read, write, learn and publish.