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hiren_garai avatar Crypto

Extracting key bit for a stream cipher

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hiren_garai

My question is little bit general. A stream cipher uses a $5$-bit key $(k_0, k_1, k_2, k_3, k_4)$, $k_i \in \{0,1\}$ for $i = 0,1,2,3,4,5$. Now the design contains $3$ generators $G_1, G_2, G_3$ which generates sequences of $0$'s and $1$'s viz., $\{z^{G_1}_i\}$, $\{z^{G_2}_i\}$ and $\{z^{G_3}_i\}$ respectively. The final output is the keystream $z_i = z^{G_1}_i z^{G_2}_i \oplus z^{G_2}_iz^{G_3}_i \oplus k_{(i\mod 5)}$.

If all the $z_i$'s are known we have to explain how to find out $k_2$.

Now I do not seek for an exact solution, my understanding is as we have to find out $k_2$, $z_{2+5k}$'s, $k = 0,1,2 \dots $ are useful for us. We know all the $z_{2+5k}$'s as $z_i$'s are known to us.

Also we can separate $k_2$ from the given relation.

But after this I am getting stuck. I don't need an to the point answer. Having known all of these, can in any way we can find out $k_2$ ? Any idea will be helpful. Thank you.

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kelalaka avatar
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kelalaka
Hint: $$z^{G_1}_i z^{G_2}_i \oplus z^{G_2}_iz^{G_3}_i \oplus k_{(i\bmod 5)} = z^{G_2}_i (z^{G_1}_i \oplus z^{G_3}_i) \oplus k_{(i\bmod 5)} = $$ Apply a correlation attack?
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