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Let $E_k$ : {0,1}$^l$ be a block cipher encryption function with block-size $l$ and key-length $n$.

In class, we saw that a double encryption with two independent keys $E{}'_{k_1k_2}(x)$ = $E_{k_1}(E_{k_2}(x))$,

can be brute-forced in $O(2^n)$ time and $O((l + n)2^n)$ space using the meet-in-the-middle attack:

create a table with $2^n$ entries mapping keys of length $l$ to entries of length n by encrypting the known plain text $x$ with all possible keys $k_2$, then search the table by decrypting the corresponding cipher text $c$ using all possible keys $k_1$.

Suppose you only have $O((l + m)2^m)$ space (where $m < n$), Give an algorithm to break the double encryption in time $O(2^{2n-m})$.

I am stuck for days, can't figure out the trick that will allow such an efficient algorithm.

Any ideas? Tips? Hints?

This is homework of course.

ADDED:

Is it sufficient to show that if we choose smaller $n$ the encryption will be broken with non neglect-able probability? Or I must show an algorithm that ALWAYS brake the encryption?

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The reason for the homework tag not existing were discussed in Asking homework questions. –  Paŭlo Ebermann Jan 6 '13 at 12:50
    
@PaŭloEbermann , Do you think I should edit my question so it will not look like an homework assignment but more like a cryptographic problem? –  Yoni Hassin Jan 6 '13 at 13:07
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If you can't fit a full table, why not try using a smaller table? –  Antimony Jan 7 '13 at 7:29
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@YoniHassin, I think the info you added needs to be asked to the professor. –  mikeazo Jan 7 '13 at 14:24
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I encourage you to read the FAQ on homework questions. In particular, the FAQ asks you to show your work: show us what you've tried so far, where you've gotten stuck, etc. See also Paŭlo Ebermann♦'s answer. The problem is your question is too much like "Do my homework for me" and not enough like "Here is my task, I already did this part, and I have now this problem. Here is what I tried, but which didn't work." –  D.W. Jan 9 '13 at 7:22
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up vote 4 down vote accepted

Hint: suppose someone told you $n-m$ bits of $k_2$; how much time/space would a meet-in-the-middle attack take then?

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It will take $O(2^{2n-m})$ but with low (non neglectable) probability, while using $n$ entries will have very high probability (almost 1). –  Yoni Hassin Jan 7 '13 at 16:23
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Go through it again: if you know $n-m$ bits of the key, it'll take less than $O(2^{2n-m})$ time; how much exactly? How much space? And, once you have that, how do you extend that observation to the case where you weren't given the hint? –  poncho Jan 7 '13 at 16:27
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Again, do the exercise: if $k_1$ has $n$ unknown bits, and $k_2$ has $m$ unknown bits, how much time would it take to recover the unknown bits? How much space? Once you have answered that question, it should be simple to use that as a black-box to solve the case where both $k_1$ and $k_2$ has $n$ unknown bits. –  poncho Jan 7 '13 at 18:45
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