Lower bounds for decision problems in imaginary, norm-Euclidean quadratic integer rings

Authors:
J. Busch
Affiliations:
The Archer School for Girls, Mathematics Department, 11725 Sunset Boulevard, Los Angeles, CA 90049, USA
Venue:
Journal of Symbolic Computation
Year:
2009

Citing 11
Cited 0

Discrete logarithms in GF(p)

Algorithmica
A lower bound for integer greatest common divisor computations

Journal of the ACM (JACM)
Computation of discrete logarithms in prime fields

Designs, Codes and Cryptography
Lower bounds for arithmetic problems

Information Processing Letters
Lower bounds for computations with the floor operation

SIAM Journal on Computing
(1+i)-ary GCD computation in Z[i] as an analogue to the binary GCD algorithm

Journal of Symbolic Computation
On the Optimality of the Binary Algorithm for the Jacobi Symbol

Fundamenta Informaticae
Arithmetic complexity

ACM Transactions on Computational Logic (TOCL)
Lower bounds in arithmetic complexity via asymmetric embeddings

Lower bounds in arithmetic complexity via asymmetric embeddings
Efficient algorithms for the gcd and cubic residuosity in the ring of Eisenstein integers

Journal of Symbolic Computation
Recursion and complexity

CiE'05 Proceedings of the First international conference on Computability in Europe: new Computational Paradigms

Quantified Score

Hi-index	0.00

Visualization

Abstract

We prove lower bounds for the complexity of deciding several relations in imaginary, norm-Euclidean quadratic integer rings, where computations are assumed to be relative to a basis of piecewise-linear operations. In particular, we establish lower bounds for deciding coprimality in these rings, which yield lower bounds for gcd computations. In each imaginary, norm-Euclidean quadratic integer ring, a known binary-like gcd algorithm has complexity that is quadratic in our lower bound.