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Private keys of Ethereum accounts are usually represented as 64-character hex strings. Will any such string work as a private key? Are there limitations on what the key could be? If not, I can just make a random string and use a privateKey -> address function and I have generated an account, as secure as any wallet generation library. Is this correct?

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You can make a random string yourself. 256 bit with two restrictions. Can't be zero and must be less than the order of the curve. Read more about it here How to validate a private key?.

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