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[Isn't this] just as hard as the RSA problem? Oddly enough, no. If we were given a random $p$, and were asked to see if there's a large prime factor $q$ of $p-1$, yes, that would be, on average, a hard problem. However, that's not what we actually do. Instead, we get to pick $p$, and so that gives us a lot of flexibility. One way is to pick $q$ ...

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