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Example text

The process of gluing can be repeated, packing all initial equations into one MRHS equation. The resulting equation is nothing more than a system of linear equations, and can easily be solved. The solution we find will necessarily satisfy all the original initial MRHS equations, so this strategy will solve the system in question. The problem we face when applying the technique of gluing in practice, is that the number of right hand sides in glued equations tends to increase exponentially. Only when there are just a few equations remaining, with large A-matrices, will the restrictions on potential solutions be so limiting that the number of possible right hand sides rapidly decreases.

In [22], Yun et al. applied the techniques invented by Wang et al. in the cryptanalysis of MD5 and SHA-1 to the HAS-160 hash function. They show that a collision can be found for HAS-160 reduced to 45 steps with a complexity of about 212 . This attack was later extended by Cho et al. [3] to HAS-160 reduced to 53 steps. The attack has a complexity of about 255 53-step HAS-160 computations. Mendel and Rijmen [10] improved the attack and reduced the complexity to 235 and presented an actual colliding message pair for HAS-160 reduced to 53 steps.

Cw is one codeword of the linear code and therefore a differential characteristic. To construct the generator matrix for the linear code, we proceed as follows: 1. Compute cwj with the input difference ΔM = ej , where ej ∈ {0, 1}512 is the j-th unit vector and ΔM the difference of the message block in bit representation. 2. Repeat the computation for j = 1, . . , 512. The resulting generator matrix of the linear code representing linearized HAS160 is defined in the following way: ⎞ ⎛ cw1 ⎟ ⎜ G512×n·32 := ⎝ ...

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