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Two rational primes p, q are called dual elliptic if there is an ellipticcurve E mod p with q points. They were introduced as an interesting means forcombining the strengths of the elliptic curve and cyclotomy primality provingalgorithms. By extending to elliptic curves some notions of galois theory ofrings used in the cyclotomy primality tests, one obtains a new algorithm whichhas heuristic cubic run time and generates certificates that can be verified inquadratic time.  After the break through of Agrawal, Kayal and Saxena has settled thecomplexity theoretical problem of primality testing, some interest remains forthe practical aspect of state of the art implementable proving algorithms.

Dual Elliptic Primes and Applications to Cyclotomy Primality Proving