In a series of papers in the 90s, Don Coppersmith shocked the cryptography world by proving that poorly designed RSA cryptosystems can be broken by a clever use of lattice reduction algorithms. Despite being well studied, it has since been an open question as to whether Coppersmith's techniques can succeed past his original proven bounds.

In this talk I will give an overview of Coppersmith's methods and explain how tools from capacity theory, a subfield of arithmetic geometry having its roots in physics and approximation theory, can settle the optimality of Coppersmith's theorems. This is based on joint work with T. Chinburg, B. Hemenway and N. Heninger.