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Two rational primes p, q are called dual elliptic if there is an el- liptic curve E mod p with q points. They were introduced as an interesting means for combining the strengths of the elliptic curve and cyclotomy primal- ity proving algorithms. By extending to elliptic curves some notions of galois theory of rings used in the cyclotomy primality tests, one obtains a new algo- rithm which has heuristic cubic run time and generates certificates that can be verified in quadratic time. After the break through of Agrawal, Kayal and Saxena has settled the complexity theoretical problem of primality testing, some interest remains for the practical aspect of state of the art implementable proving algorithms.