My understanding is that this particular proof works by constructing a curve over a finite field built of the number to be tested with a "sufficiently large" pr

My understanding is that this particular proof works by constructing a curve over a finite field built of the number to be tested with a "sufficiently large" prime order, you prove its order by just multiplying a generator by the order. The curve won't work right if the field isn't a field (or in other words unless our candidate is actually prime). The proof has to be applied recursively to prove the order of the test group is itself prime. Take your pick of redundant WP articles for a more technical explanation. https://en.wikipedia.org/wiki/Primality_certificate#Atkin%E2%80%93Goldwasser%E2%80%93Kilian%E2%80%93Morain_certificates https://en.wikipedia.org/wiki/Primality_certificate#Atkin%E2... https://en.wikipedia.org/wiki/Elliptic_curve_primality https://en.wikipedia.org/wiki/Elliptic_curve_primality