Web3 Daily Professor @Web3daily
🪶 #Crypto, and #Web3 Writer | Founder of Web3 Daily Agency | Amb: @BNBCHAIN / @Stake_Stone | Official #Binance Affiliate Partner 🔶 t.me/ThessAhnBio Onchain Joined April 2023-
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#Binius concept
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Back to building the # 1 restaking protocol Thank you for supporting the launch of $REZ #⃣ Eligible users: ~103.5k 🔥 Total $REZ allocation: 7% total supply 💰 $REZ distributed: ~$167m What’s Next 🚢 ezETH withdraws around the corner 🧑🌾 $REZ Season 2 incentives 🧺 $EIGEN &…
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Now, we get to Binius. The core idea in Binius is treating the same data as a hypercube, and as a grid. Here's an example (using regular integers):
I highly encourage people to understand and explore! Lots of innovations have been happening in ZK-proving recently. And I expect lots more to come soon.
This stuff is complicated! It took me a while to understand. You can find my implementation here: github.com/ethereum/resea… The good news is, binary fields are the only "advanced math" here. Nothing fancy like in elliptic curve pairings.
1. Put adjacent bits in each row into groups, and do Reed-Solomon extension on those groups. 2. Decompose each value in the row combination into bits, allowing the Reed-Solomon extension of that to be bit-wise.
Full Binius takes this idea, but restricts it so that the hypercube can only contain bits. To make the whole protocol operate over bits, Binius uses two clever tricks.
For the purposes of polynomial commitment, we look at the hypercube as a multilinear polynomial, and take its evaluations outside the cube. For the purposes of the proof, we take the rows of the grid, and use Reed-Solomon codes to extend them.
Here are addition and multiplication tables for 4-bit binary field elements, treating bitstrings as integers in little-endian representation. They follow weird rules: 2 + 2 = 0, 2 * 6 = 11, and a² + b² = (a+b)². But they satisfy all of the usual laws of arithmetic.
Binary fields are a fascinating mathematical structure that have many unique properties. The tower construction is a fascinating way of producing them, that adds even more advantages.