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Quoting Wikipedia:

A password with, say, 42 bits of strength calculated in this way would be as strong as a string of 42 bits chosen randomly[.]

Assuming that interpretation is correct, combined with knowledge that cracking a random 256-bit number is infinitely expensive nay physically impossible, how can a passphrase with 256-bits of entropy practically be constructed & memorized?

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By writing a poem and using it as a passphrase? –  Philipp Jun 29 '14 at 18:27
A common word has 11 bits of entropy, so you need 24 words. Whether or not you can reliably memorize a 24 word poem you wrote yourself depends on you. –  Philipp Jun 29 '14 at 18:48
@Gracchus, it's not a question of belief. Try doing the arithmetic to see what it would cost to do exhaustive brute-force space against a128-bit key. I think you might be surprised. (The phrase "very reasonable" isn't one I would use.) Anyway, I'm sorry to hear you found my comment rude and presumptuous. That wasn't my intent, and I don't think it's rude to communicate expectations for questions on StackExchange sites. See, e.g., meta.stackoverflow.com/q/261592/781723. –  D.W. Jun 30 '14 at 5:05
Also note that your passphrase will never be stronger than the lenght of the password hash. So if the site uses a 128 bit hash to store your password anything above that will not help in preventing brute force against your password. - If they can brute force 128 bits of entropy they will not find your passphrase, but they will find a password that the system will accept as your password. –  Taemyr Jun 30 '14 at 7:40
Since you quoted an answer doing a thermodynamic analysis, knoweth: It is very unlikely that even a 128 bit key will be broken by brute force. Given an annual 10^20J available worldwide from uranium, and a total reserve of 10^20J each in natural gas and petroleum, using up all fossile fuel plus an entire year worth of uranium production on our planet would allow to do 10^42 elementary operations. Which equals counting to one 139 bit num, or 2048x to 128bits (decrypting 1 block is >1000 elem ops). Assuming there exist 3 keys more valuable than your key in the world, this isn't happening. –  Damon Jun 30 '14 at 15:00

2 Answers 2

up vote 12 down vote accepted

A common word of the English language has approximately 11 bits of entropy. That means a 256bit passphrase (passtext?) would require 24 words.

How could one make up a text of that length which is still easy to memorize? You could write a poem. The art of writing poetry and memorizing poems is not hard to learn. It doesn't even has to rhyme. In fact, not rhyming makes it harder to crack. Have fun!

However, we assumed that all words are completely random. In natural language, some words are more likely to appear after others. Poetry also usually conforms to certain rules and structures which further reduces the entropy. This answer on linguistics.stackexchange.com comes to the conclusion that the entropy per word in a poem goes down to about 5 bits per word, so you would need to write and memorize a poem of at least 52 words to get over 256 bits of entropy.

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Thank you! Do you happen to know how much grammar reduces entropy? –  user36556 Jun 29 '14 at 19:10
@Gracchus That's more of a question for a linguist. I am not sure, so I made a (I think pessimistic) guess of 99% in the answer. –  Philipp Jun 29 '14 at 19:12
Also, does it have to be a poem? In other words, does a poem have some special characteristic that contains more entropy than another set of text with an equal quantity of words? –  user36556 Jun 29 '14 at 21:00
You could make a long sentence, but a poem might be easier to remember –  GroundZero Jun 29 '14 at 21:01
@Gracchus: a poem (or any text that is easy to remember) contains less entropy than an equal number of words selected at random, and possibly less than prose although I wouldn't bet on it. Any internal structure or "rules" that make certain patterns more likely than others, is a reduction in entropy. This is more or less the same as the fact that a pronounceable series of letters has less entropy than a random sequence of any letters. However, 24 words is a very short poem, so you can compensate for quite a lot lower entropy if you just keep going. –  Steve Jessop Jun 30 '14 at 0:52

Such a question would prove that the person asking it has become serious about the strength of their passwords, at which point you simply should start using a password manager. Seriously strong passwords, without memorizing a bunch of poems.

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Yes, I definitely use a password manager for low value passwords. This question is for maximum value passwords where the risk from a single point of failure, online storage, or even local storage is unacceptable. –  user36556 Jun 29 '14 at 20:12
@Gracchus then get really good at memorization. :P youtube.com/watch?v=oNrWgjh9tnU –  Andrew Hoffman Jun 29 '14 at 20:42
I take it that one should use a strong password for one's password manager, so I don't think that the request to memorize "a passphrase" (that is, one) of this kind is dealt with by the password manager :-) –  Steve Jessop Jun 30 '14 at 1:19
@SteveJessop But password managers are more likely to allow multi-factor authentication, which should be more secure than just a password, and more practical too if you're trying to create, memorise and type something like long. –  Bob Jun 30 '14 at 6:18

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