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What Is the Difference between an On-Chain and an Off-Chain Cryptographic Proof?

An on-chain cryptographic proof is a signature or mathematical proof that is verified directly by the smart contract on the blockchain. This verification consumes gas but provides the highest level of security.

An off-chain cryptographic proof is verified by a separate system or node off the blockchain. While faster and cheaper, it requires the consumer to trust the off-chain verifier.

Decentralized oracles typically generate the proof off-chain but verify the final aggregated signature on-chain.

What Is the Trade-off in Computational Complexity between the Two Rollup Types?
How Do Arbitrageurs Profit by Interacting with an AMM Pool Experiencing Impermanent Loss?
How Do Formal Verification Methods Aim to Address Smart Contract Security Vulnerabilities?
How Do Oracles Ensure the Data They Provide Is Accurate and Tamper-Proof?