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Suppose I had a gigantic sum of bitcoins and wanted to store them in my brain's memory instead of any physical form. How much data would I need to be able to remember? I understand I'd have to transfer it all to one address - would I need to remember both its private and public key, or can the public one be calculated based on the private key?

d33tah
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This is called a brain wallet. You need to remember only the generator for the private key, which can be anything you choose. You just have to remember something (anything) that is complicated enough that other people could not guess it, even if given trillions of guesses.

This is actually surprisingly complicated. If you write the passphrase down, it's not really a brain wallet anymore. If you don't, and you forget any bit of it and cannot precisely remember it, your funds are lost. If you make it simple enough to ensure you can remember it, the risk that someone else will find it is high. Lastly, the tools you use to convert the passphrase to a Bitcoin address can be flawed or sabotaged.

David Schwartz
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    I'd like to stress the importance of the risk that someone else can find it. As opposed to normal password-guesses, where you are rate-limited or shut out after some failed tries, here you are exposing yourself to an unlimited attack by the whole world, forever. If you didn't generate the passphrase by a computer or dice, it's probably not random enough (even though it is long, and looks non-random). – Pieter Wuille Oct 27 '13 at 00:35
  • What if I decide to just remember a random key? Wouldn't that theoretically be as secure as storing the key on the HDD? – d33tah Oct 28 '13 at 19:20
  • @d33tah So long as the key had at least 70 bits of equivalent entropy, you should be okay. But it's pretty hard to remember 13 random characters. Give it a try -- yXi2ano5SXJ8D. – David Schwartz Oct 28 '13 at 19:29
  • I think that if you store a million dollars there, you have quite a good incentive. – d33tah Oct 28 '13 at 22:52