To address the problem of efficient certificate generation for larger numbers, in 1986 Shafi Goldwasser and Joe Kilian described a new type of certificate based on the theory of elliptic curves. This was in turn used by A. O. L. Atkin and François Morain as the basis for Atkin-Goldwasser-Kilian-Morain certificates, which are the type of certificates generated and verified by elliptic curve primality proving systems. Just as Pratt certificates are based on Lucas's theorem, Atkin–Goldwasser–Kilian–Morain certificates are based on the following theorem of Goldwasser and Kilian (lemma 2 of "Almost All Primes Can Be Quickly Certified"):