Binius: highly efficient proofs over binary fields

@Sakura please summarize this article, thanks uwu.

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Reading https://vitalik.eth.limo/general/2024/04/29/binius.html

:cherry_blossom: TLDR: Binius is a new cryptographic proof system that operates directly over binary fields (zeroes and ones) to achieve highly efficient proofs, surpassing previous SNARK and STARK technologies.

Key Points:

  • :mag: Binius uses a unique approach of representing data as a “hypercube” of binary evaluations, rather than traditional univariate polynomials.
  • :rocket: By working directly with bits, Binius can perform arithmetic operations like addition and multiplication much more efficiently than previous systems.
  • :lock: Binius introduces novel techniques like “tensor product” and “row/column combinations” to enable secure proofs over the binary hypercube.
  • :brain: The Binius protocol is complex, but builds upon the foundations of finite fields, arithmetization, and polynomial commitments used in SNARKs and STARKs.
  • :mag: Binius aims to push the boundaries of what’s possible with cryptographic proofs, making them faster and more efficient than ever before.

In-depth Summary:

Binius is a new cryptographic proof system that takes a radically different approach compared to previous SNARK and STARK technologies. While those systems operate over large prime fields or extension fields, Binius works directly with binary fields - zeroes and ones.

The key innovation in Binius is the way it represents the computational data. Instead of a traditional univariate polynomial, Binius uses a “hypercube” of binary evaluations. Each dimension of the hypercube corresponds to a single bit, and the full cube represents the entire computational trace.

This binary representation allows Binius to perform arithmetic operations like addition and multiplication much more efficiently than previous systems. Addition becomes a simple XOR, and multiplication can be parallelized in clever ways. The Binius team has developed specialized techniques like “tensor products” and “row/column combinations” to enable secure proofs over this binary hypercube structure.

The Binius protocol is quite complex, but it builds upon the foundational concepts of finite fields, arithmetization, and polynomial commitments that underlie SNARKs and STARKs. By working directly with bits, Binius aims to push the boundaries of what’s possible with cryptographic proofs, making them faster and more efficient than ever before.

ELI5:

Binius is a new way to prove that a computer program did what it was supposed to do, but it’s way faster and more efficient than older methods. Instead of using big numbers, Binius just uses 0s and 1s, which computers are really good at working with. It does this by breaking the program down into a bunch of little cubes, each one representing a single bit of information. This lets Binius do all the math really quickly, without having to deal with huge numbers that slow things down.

Writer’s Main Point:

The primary goal of Binius is to achieve highly efficient cryptographic proofs by operating directly over binary fields (zeroes and ones), rather than the larger prime or extension fields used in previous SNARK and STARK technologies. By representing data as a “hypercube” of binary evaluations and leveraging specialized techniques, Binius aims to push the boundaries of what’s possible with cryptographic proofs, making them faster and more practical than ever before.