In this research article, I review the Cooley-Tukey FFT as a faster alternative than the naive approach for computing DFTs of lengths that are not a power of 2, but have highly composite factors.

In this research article, I review the barycentric interpolation method as a more efficient form for working with Lagrange bases especially when paired with nth roots of unity.

In this research article, I present a protocol for efficiently verifying the Ethereum Beacon chain's Casper FFG consensus proofs using a SNARK based scheme.

Polynomial commitment schemes are the foundational cryptographic primitive for things like computational integrity proofs (aka (ZK-)SNARKs) and verkle tries (a more efficient alternative to merkle-patricia tries). In this article i review their technical definition as well as their applications.

I explore the technical definition of consensus proofs, review their vulnerabilities, and examine the mitigations for these vulnerabilities. I then show how these mitigations enable consensus proofs to be safely used on-chain, allowing for the first-of-its-kind byzantine fault-tolerant cross-chain bridges.

State proofs are a critical primitive of the blockchain stack that enable things like trustless bridges, off-chain light clients that can access on-chain data in a permissionless and secure manner as well as modular blockchains architectures where the

*execution layer*can be decoupled from the*consensus layer*.In this research article, we propose an extension interface for

`ERC20`

, `ERC721`

and `ERC1155`

token contracts, in order for them to become native to multiple chains. This interface will become a necessity with with the upcoming Cambrian explosion of rollups on Ethereum with addition of EIP-4844. You can find the actual EIP proposal here.Merkle mountain ranges are an improvement over conventional merkle trees for growing, potentially unbounded lists. Where conventional merkle tree constructions over growing lists prove very inefficient to compute, as all nodes in the tree must be recomputed. Merkle mountain ranges amortise this cost by growing subtrees incrementally and merging subtrees at the same height, rather than growing the full tree.

Merkle multi proofs enable more efficient merkle proofs by re-using the intermediate nodes shared by the proof leaves during the recalculation of the root hash of the tree.

This technical specification assumes that you’re already aware of the sync committee protocol introduced in the altair, the first hard fork of the ethereum beacon chain. If not, tl;dr: The original attestation protocol unfortunately did not include succint BLS public key aggregation, which would’ve made it cheap to verify by light clients given that there are now almost 500k authorities actively validating blocks on the beacon chain.