Fast reduction and composition of binary quadratic forms
ISSAC '91 Proceedings of the 1991 international symposium on Symbolic and algebraic computation
(1+i)-ary GCD computation in Z[i] as an analogue to the binary GCD algorithm
Journal of Symbolic Computation
Asymptotically Fast GCD Computation in Z[i]
ANTS-IV Proceedings of the 4th International Symposium on Algorithmic Number Theory
A computational introduction to number theory and algebra
A computational introduction to number theory and algebra
Efficient algorithms for the gcd and cubic residuosity in the ring of Eisenstein integers
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
| Hi-index | 0.00 |
We present an algorithm to compute a greatest common divisor of two integers in a quadratic number ring that is a unique factorization domain. The algorithm uses $O(n {\rm log}^{2} n {\rm log log} n + \Delta^ {\raisebox{0.8mm}{\scriptsize 1}{\scriptsize /}\raisebox{-0.5mm}{\scriptsize 2}} +^{\epsilon})$ bit operations in a ring of discriminant Δ. This appears to be the first gcd algorithm of complexity o(n2) for any fixed non-Euclidean number ring. The main idea behind the algorithm is a well known relationship between quadratic forms and ideals in quadratic rings. We also give a simpler version of the algorithm that has complexity O(n2) in a fixed ring. It uses a new binary algorithm for reducing quadratic forms that may be of independent interest.