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Is there a way to use a long (and possibly hashed) passphrase as a private key, so that all I need to recover my Bitcoins would be that password?
I was thinking of buying Bitcoins using a private key that's generated from a passphrase, and then deleting the Bitcoins software entirely. When I want to use my Bitcoins in the future, I will re-create the key using the password. This will eliminate the possibility of stealing my Bitcoins by hacking to my computer.
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The simplest and most common way of doing this is to do SHA256(passphrase) and to use the result as your private key. You can e.g. use the "Brain wallet" tab on www.bitaddress.org which is all javascript so that you can download it and do everything offline for increased security.
Be aware though that your pass phrase must be really long and truly unique since if someone would use the same pass phrase they would have access to your money. The comment from Pieter Wuille above is worth repeating:
Be aware that when you do this, you expose your wallet to the entire world to do a brute-force attack on your password. If many people start doing this, it may become very profitable to try to find their passwords. Software like Armory and Electrum have the ability to generate a determinsitic wallet from a 'seed', but this seed is still randomly generated, in order to be sure that it is hard to attack
To put some numbers on how long and unique it has to be, to be confident in its security, you should pick a passphrase such that an attacker would be expected to need to guess at least a billion billion passphrases before he'd come up with yours. For example, a passphrase that would probably be considered just barely strong enough would be firefly 80 m00se! – David Schwartz – 2012-08-23T16:23:02.840
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You ought to look into solutions such as Brainwallet. It hashes your passphrase into a Bitcoin address.
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Yes, you can. It's named Brain Wallet but... Both answears (@ThePiachu and @D.H.) lead you to catastrophy. Don't use external tools to create your brain wallet. Do it on your own on your PC disconnected from Intenet. There is a lot of tools to count SHA256 on your own. When you creating your passphrase use not only alphanumeric characters but also other.
6Be aware that when you do this, you expose your wallet to the entire world to do a brute-force attack on your password. If many people start doing this, it may become very profitable to try to find their passwords. Software like Armory and Electrum have the ability to generate a determinsitic wallet from a 'seed', but this seed is still randomly generated, in order to be sure that it is hard to attack. – Pieter Wuille – 2012-08-21T12:20:35.153