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Binius: Highly efficient proofs over binary fields
This post is primarily intended for readers roughly familiar with 2019-era cryptography, especially SNARKs and STARKs. If you are not, I recommend reading those articles first.
All SNARKs and STARKs are based on this idea of using a simple equation over polynomials (or sometimes vectors and matrices) to represent a large number of relationships between individual values. with Ethereum's EIP-4844, where the individual "chunks" of a blob have to be numbers modulo 52435875175126190479447740508185965837690552500527637822603658699938581184513, and so encoding binary data involves throwing away a bit of space and doing extra checks at the application layer to make sure that each element is storing a value less than \(2^{248}\). In practice, a major source of inefficiency comes from the fact that in real programs, most of the numbers we are working with are tiny: indices in for loops, True/False values, counters, and similar things.
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