Is it possible to modify the account storage trie for non-contract accounts? For example, if I want to prove that I have control of an account without having to send any ETH, could I generate a transaction that would put a given value in a given key of my account storage trie? If so, how would you do this in the Geth console and show that it's done afterwards?

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No, this is not possible, as storage is only modified by the SSTORE opcode.

However, you can prove ownership of an account by signing a message using the private keys of that account. You can do this in geth using the eth.sign method. Web3js probably has web3.eth.sign.

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