$ETH Ethereum (ETH), a groundbreaking decentralized platform, employs the Ethereum Virtual Machine (EVM) to execute smart contracts. EVM's operation relies on Turing completeness, ensuring it can simulate any computation. Cryptographic security is ensured through elliptic curve cryptography (ECC), specifically the secp256k1 curve, defined by

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. The transition from Proof of Work (PoW) to Proof of Stake (PoS) involves cryptographic puzzles and probability theory. PoW employs the SHA-3 hash function, requiring miners to solve

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. PoS utilizes the Byzantine Fault Tolerance (BFT) algorithm, enhancing security and scalability through mathematical consensus. ETH integrates mathematics at every layer.