Efficient algorithms for the gcd and cubic residuosity in the ring of Eisenstein integers

Abstract

We present simple and efficient algorithms for computing the gcd and cubic residuosity in the ring of Eisenstein integers, Z [ ζ ] , i.e. the integers extended with ζ , a complex primitive third root of unity. The algorithms are similar and may be seen as generalisations of the binary integer gcd and derived Jacobi symbol algorithms. Our algorithms take time O ( n 2 ) for n -bit input. For the cubic residuosity problem this is an improvement from the known results based on the Euclidean algorithm, and taking time O ( n ⋅ M ( n ) ) , where M ( n ) denotes the complexity of multiplying n -bit integers. For the gcd problem our algorithm is simpler and faster than an earlier algorithm of complexity O ( n 2 ) . The new algorithms have applications in practical primality tests and the implementation of cryptographic protocols.