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. 3. Each q has its recursive certificate following it. So if the smallest q is known to be prime, all the numbers are certified prime up the chain.
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