[SHA, Right] What’s Joux Talkin’ Bout?
April 29th, 2008
"Differential Collisions in SHA-0"
Chabaud & Joux begin by attacking a weakened SHA-0 variation. The first version, which
they call SHI1, is equivalent to SHA-0 with the addition operations in A_{i+1} replaced
with XORs and the f_i functions replaced by XOR. This effectively removes the
non-linearity introduced by the f_i function and the addition operations.
The analysis of SHI1 begins by examing the effect of perturbations of the vector
W_{i} (0 <= i < 80) directly (instead of trying to study the perturbations of the
message block, M or W_i (0 <= i < 16)). In other words, suppose we directly supply the
vector W, instead of using the expansion function E. Since we are mounting a
differential attack, we will start with some vector W and study how permutations of this
vector effect the final result.
The paper gives the following differential path to correct a flip of W^{i}_{1}:
W^{i+1}_{6}
W^{i+2}_{1}
W^{i+3}_{31}
W^{i+4}_{31}
W^{i+5}_{31}
This differential path gives the bits we need to flip to "undo" a given bit flip (bit
1 of W^{i} or W^{i}_{1} for this example).
TODO: Give the output of the python test script -- showing the correction?
TODO: Add image from the Joux paper (Figure 2)
Performing the above flips, we can produce a second vector, W', which will produce the
same result -- a collision!
This shows we can produce a collision if we can supply both W and W' in their
entirety, but SHI1 contains an expansion function producing W from the message block
M. The last 64 words of W are derived from a recurrence primed with the first 16
words (which is M). Some values we could choose for W (or W') would be impossible
to generate using any M. This leads to our next step; we must ensure that both W
and W' can result from the expansion process.
Let's start by representing the collisions as an error vector, m_{0}. m_{0} is
80 bits long and will contain a 1 in position i if we intend to negate W^{i}_{1}
(for now, we will only concern ourselves with correcting flips in bit 1 -- see
Note 1 in the paper). m_{0} must be 0 for the i >= 75 "since a perturbation in
round i is never corrected before round i + 6, and since all perturbations must be
corrected by round 80."
The intuitive explanation of the process
----------------------------------------
Important observation:
* The expansion process does not interleave bits! This turns it into a function
from 16 bits to 80 bits over each bit in the word.
1. Find valid perturbations -- these are deduced by ensuring they fit the expansion
recurrence relation. It is important to see that since the compression functions
starts primed with some A - E, the recurrence must actually start at the 11th word
(5 steps [A-E] behind the 16th that the recurrence is defined at).
The search is brute force with a search space of 2^16. It is simple.
We will call the chosen error vector e_{0}
TODO: Include the functions taken from sha_exp_rev.py that compute valid error
vectors.
2. Now, derive the global differential mask (which is M in the paper -- M is also
the message block... bad naming). The global differential mask is derived by fixing
the flips found in the previous step with the differential path described in the
prebvious section. Since the SHI1 defines all combination function in the
compression function as XOR, we can XOR the differential paths for all the bits
flipped in e_{0} to compute up with the global mask. We will call the global mask
G. We only need the first 16 words of this mask since those will define the rest of
it (via the expansion function).
NOTE: Maybe worth pointing out that this will generate a valid W' because e_{0}
satisfies (9).
3. Given the global mask M and *any* input message M, SHI1(M) == SHI1(M \xor G).
Collision! Hooray! Wait. That's just SHI1. It's all linear. We just solved an
algebra equation. Oh.
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áëàãîäàðñòâóþ!!…
February 10th, 2015 at 11:21 pm
earthmens@aparicio.mounted” rel=”nofollow”>.…
thank you….
February 11th, 2015 at 5:36 am
modified@diversified.catching” rel=”nofollow”>.…
ñýíêñ çà èíôó….
February 13th, 2015 at 4:24 pm
thatched@comprise.toscaninis” rel=”nofollow”>.…
ñïñ!!…
February 13th, 2015 at 5:39 pm
suvorovs@correlations.occupancy” rel=”nofollow”>.…
ñýíêñ çà èíôó….
February 14th, 2015 at 6:50 am
presidential@narebs.voulez” rel=”nofollow”>.…
thanks for information!…