You can think of it as a magic trick. The magician is convincing you the he's computing a totally legit action X, all the while abusing your trust (or your cogn

You can think of it as a magic trick. The magician is convincing you the he's computing a totally legit action X, all the while abusing your trust (or your cognitive biases) to perform action Y instead right under your nose. In this case, the abuse is to have you believe the modulus---which defines the core mathematical structure you're working on---is a prime, when it is not. Once you manage to sell this, and clearly not everyone checks the primality of their moduli, the rest become implementation details.