Apocalyptic: adlai, here's the reasoning: If you just generate random (very likely composite) p and q's, it's not very practical to factor at scales (take for exemple the modulus I mentioned above, it still isn't fully factored and a great deal of CPU cycles were wasted already)

Apocalyptic: instead the enemy choses a prime from a know list, then multiplies by a random composite to get to the desired size

Apocalyptic: *known

Apocalyptic: this gives the appearence of random numbers should it be discovered, but gives people with access to the pre-defined prime list an easy and practical way to factor those, since the only have to factor the composite part which has many small factors

Apocalyptic: so I guess this matches your ring formulation above; note that pre-defined prime list can be quite large