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I noticed that all the binary bits of $8191$ are $1$'s. Could this have something to do with it?

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    $\begingroup$ Maybe it has more with the fact that 8191 is tiny... $\endgroup$
    – poncho
    Feb 6, 2018 at 15:56

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Besides 8191 being too small for real world use, the factorization of 8190 ($p-1$) has all small values. It's factorization is $2\cdot 3^2\cdot 5 \cdot 7\cdot 13$. Having a group whose order is smooth makes it vulnerable to an attack using the Pohlig-Hellman.

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