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I have a Truecrypt container which I want to synchronise between computers (i.e. different people, that I want to share the data with).

If I used Dropbox for synchronisation and someone downloaded all the different versions of a publicly accessible Truecrypt container over time, whenever it gets changed, is it then easier for them to find the password than if they just had one version?

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The truecrypt folks seem to have done a pretty good job with their crypto, so I'm going to say the answer is no, it would not get easier for them to find the password for any practical sense of the word "easier". They even have a write up on Sharing over Network which seems pertinent.

Typically in a cryptanalytical attack, however, the more information the analyst has, the easier their job. So I guess in that sense, theoretically the answer would be yes, their job is easier since they have extra information. But I personally wouldn't be worried about it as to gain any real-world advantage the analyst would probably need petabytes (if not more) of information for any attack to be practical. By the time they gather that much information, they probably could have guessed your password.

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Your guess is correct. There are all kinds of things an active attacker (one who can induce you to use modified containers) might be able to do if they can watch the container over time. (For the most obvious attack, they can replace your container with an older version.) But no information expected to be useful for finding the password is leaked. – David Schwartz Sep 14 '11 at 13:49

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