An M3 public-key encryption scheme
Lecture notes in computer sciences; 218 on Advances in cryptology---CRYPTO 85
A binary algorithm for the Jacobi symbol
ACM SIGSAM Bulletin
A public-key cryptosystem utilizing cyclotomic fields
Designs, Codes and Cryptography
Efficient algorithms for computing the Jacobi symbol
Journal of Symbolic Computation
(1+i)-ary GCD computation in Z[i] as an analogue to the binary GCD algorithm
Journal of Symbolic Computation
A fast Euclidean algorithm for Gaussian integers
Journal of Symbolic Computation
Asymptotically Fast GCD Computation in Z[i]
ANTS-IV Proceedings of the 4th International Symposium on Algorithmic Number Theory
Lower bounds for decision problems in imaginary, norm-Euclidean quadratic integer rings
Journal of Symbolic Computation
On the l-ary GCD-algorithm in rings of integers
ICALP'05 Proceedings of the 32nd international conference on Automata, Languages and Programming
A new GCD algorithm for quadratic number rings with unique factorization
LATIN'06 Proceedings of the 7th Latin American conference on Theoretical Informatics
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We present simple and efficient algorithms for computing the gcd and cubic residuosity in the ring of Eisenstein integers, Z[@z], i.e. the integers extended with @z, 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.