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.
Talk to us
Join us for a 30 min session where you can share your feedback and ask us any queries you have