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I'm trying to calculate how many possible combinations of files there can be using a signed 64-bit file length, but can't seem to find a formula (or I'm using the wrong keywords). For example, the number of unique files that have a length of 256 bytes is $2^{256*8}$ = $2^{2048}$. I assume the total number of combinations across all file lengths will be some kind of geometric sequence?

The reason I want to calculate it is because I'm curious how it relates to the number of possible SHA-256 hashes ($2^{256}$). Also, if I calculate the SHA-256 hash of the SHA-256 hash of a file's contents, how much have I reduced the entropy (if that's the correct term)?

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It looks like you did not work on that one hard enough. Some hints follow. You will find the restrictions on the input of SHA-256 in the standard. That does not match the question and title, any way I read length. Worse, there's a clash in the usual meaning of the term. You need to define what length is in your context. –  fgrieu Dec 6 '11 at 12:50
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2 Answers 2

For a given file length L bytes the combinations for that length is 256^L. The total combination is the sum of combinations for all file lengths L from zero to (2^63)-1. Just substitute parameters in the formula for summing the n first terms of a geometric series. The rest is left as an exercise to the reader :-)

If all you need is a rough order-of-magnitude estimate it is really sufficient to just look at the highest order term, which provides a quick lower-bound estimate and which will already tells you the ratio is a very large number (very large for cryptographic purposes), which means the factor you have "reduced entropy" is a number very close to zero (once again, for cryptographic purposes).

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For example if the max file size was 8 bits, we get 2$^{8}$+2$^{7}$+...+2$^{0}$ = 2$^{9}$-1. In your example for a signed 64-bit integer the answer is 2$^{64}$-1 or 18,446,744,073,709,551,615, and for SHA-256 the answer is 2$^{65}$-1 or 36,893,488,147,419,103,231. For SHA-512 the answer is 2$^{129}$-1, which is around 680 trillion trillion trillion. –  Richie Frame Dec 22 '13 at 0:31
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Update:

The theory

If you have a file system like NTFS, this uses 64 bit for the file size, so you can have 2^64 Bytes = 16 EB in one file.

Imagine you have only one 16 EB large file. There are 256^18446744073709551616 different file contents possible. In total there are even more, because you need to sum up all file sizes.

From AES you get 2^256 different hashes. Compared to the possible combinations of file content, that's off by a huge factor and it seems that there must be many collisions.

The reality

The universe has approximately 10^82 = 2^272 atoms, Earth has about 10^50 = 2^166 atoms.

So even if you could store a whole file on every atom on Earth, there is still the possibility for a unique AES hash for every file.

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Thanks for downvoting. I updated my answer. –  Thomas W. Dec 21 '13 at 22:26
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