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Let talk about Implementation and performance analysis of Barkan, Biham and Keller’s attack on A5/2. In this paper, mentioned that we need to brute force attack on register 4 ($R_4$) to get keystream from below:

$$HP^{-1}C=HP^{-1}K. \tag{1}$$ Where $HP^{-1}$ is parity check and $C$ is ciphertext and $K$ is keystream.

On the other hand, the equations on the key stream must be adapted into equations on the variables of the $LFSRs$. So we have the following linear system:

$$Sr=K. \tag{2}$$

where $S$ is multiplication matrix of size $1368*656$ and $K$ is the concatenation of $3$ unknown key streams $k_1$, $k_2$, and $k_3$, and $r$ is the vector of unknowns representing the state of the $LFSRs$.

  1. Where is $R_4$ influence in the first equation?

  2. How we can construct the second equation(Please with details)?

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The authors state in 4.3 that the design of the cipher has the following clever feature:

The dependencies between the LFSRs and the keystream vary greatly with the initial value of $R_4$ and since the variables of $R_4$ play no role in the value of the keystream, so it is not possible to find them.

So, later on they suggest the values of $R_4$ must be bruteforced, as a precomputation.

As for your question 2, you need to plug things in, as in the equations at the end of section 4.3.

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  • $\begingroup$ Thanks, but it is not clear for me how multiplication matrix($S$) created? If it is scalar, how we can build it? $\endgroup$ – R. Jalaei Salahi Dec 29 '19 at 12:59

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