[karl-j]

I think I'm almost as uninformed as you, but I believe it comes down to the difference between perfect solutions and close enough solutions. Consider the classic NP problem of the traveling salesman problem.

"[Modern heuristic and approximation algorithms] can find solutions for extremely large problems (millions of cities) within a reasonable time which are with a high probability just 2–3% away from the optimal solution." [0]

When close enough is enough, NP problems can often be solved in P time, and I suspect this is one of those cases. For crypto however, close enough is not enough.

[0] https://en.wikipedia.org/wiki/Travelling_salesman_problem#He...