26 research outputs found
On Taking Square Roots without Quadratic Nonresidues over Finite Fields
We present a novel idea to compute square roots over finite fields, without
being given any quadratic nonresidue, and without assuming any unproven
hypothesis. The algorithm is deterministic and the proof is elementary. In some
cases, the square root algorithm runs in bit operations
over finite fields with elements. As an application, we construct a
deterministic primality proving algorithm, which runs in
for some integers .Comment: 14 page
Structure computation and discrete logarithms in finite abelian p-groups
We present a generic algorithm for computing discrete logarithms in a finite
abelian p-group H, improving the Pohlig-Hellman algorithm and its
generalization to noncyclic groups by Teske. We then give a direct method to
compute a basis for H without using a relation matrix. The problem of computing
a basis for some or all of the Sylow p-subgroups of an arbitrary finite abelian
group G is addressed, yielding a Monte Carlo algorithm to compute the structure
of G using O(|G|^0.5) group operations. These results also improve generic
algorithms for extracting pth roots in G.Comment: 23 pages, minor edit
Identifying supersingular elliptic curves
Given an elliptic curve E over a field of positive characteristic p, we
consider how to efficiently determine whether E is ordinary or supersingular.
We analyze the complexity of several existing algorithms and then present a new
approach that exploits structural differences between ordinary and
supersingular isogeny graphs. This yields a simple algorithm that, given E and
a suitable non-residue in F_p^2, determines the supersingularity of E in O(n^3
log^2 n) time and O(n) space, where n=O(log p). Both these complexity bounds
are significant improvements over existing methods, as we demonstrate with some
practical computations.Comment: corrected a typo, 10 page