A recursive primality certificate for a prime p. The certificate consists of a list of 1. A point on an elliptic curve C y^2 = x^3 + g_2 x + g_3 (mod p) for some numbers g_2 and g_3. 2. A prime q with q>(p^(1/4) + 1)^2, such that for some other number k and m = k q with k!=1, m C(x, y, g_2, g_3, p) is the identity on the curve, but k C(x, y, g_2, g_3, p) is not the identity. This guarantees primality of p by a theorem of Goldwasser and Kilian.

elliptic curve primality proving | elliptic pseudoprime | Pratt certificate | primality certificate | witness