Consider the following excerpt from Rueppels paper, titled When Shift Registers Clock Themselves:

Excerpt from Rueppel's paper

His example includes $C(D) = 1 + D + D^2 + D^3 + D^4 + D^5$ as a primitive connection polynomial. However, this polynomial is not irreducible, let alone primitive, since it can be factored as $(1+D)(1+D+D^2)^2$ over $\text{GF}(2)$.

What am I missing?

  • $\begingroup$ Is your question regarding the hardware or just the mathematics. You would be required to have 5 flip-flops, whether or not it is irreducible. The single vs. double clocking is interesting as it effectively behaves as adding some logic between blocks. $\endgroup$ – b degnan Oct 8 '17 at 0:29

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