Cryptography is essential to Ethereum's ability to offer a trustless environment where funds are protected from theft, transactions are executed securely, and all participants in the system agree on the same history of transactions.
The method of how funds are stored and sent for example, is based on asymmetric cryptography, or the use of public and private key pairs. Each user has a private key, which funds are spent from, and a corresponding public key, which funds are spent to. By using these key pairs, users can securely store funds and authorise transactions without reliance on an outside party.
To ensure that transactions are spent by the rightful owner, Ethereum makes use of digital signatures. When a user sends ETH from their public key, they must use a corresponding private key to create a signature, which serves as a unique proof that the transaction was authorised by the correct owner.
In its transaction settlement process, Ethereum also uses cryptography to create a degree of randomness, fairness and unpredictability. For example, the RANDAO helps randomly select the validator who will propose the next block, making it challenging for bad actors to manipulate which transactions are being settled over time. This randomness contributes to Ethereum's neutrality and resistance to censorship.
The trust assumptions that Ethereum users make ultimately depend on these cryptographic tools remaining secure. By using these cryptographic tools — public and private keys, digital signatures, and randomness beacons — Ethereum secures transactions, prevents unauthorised access to funds, and upholds its reliability to eventually settle all user transactions.