18,613 research outputs found
Improved quantum algorithms for the ordered search problem via semidefinite programming
One of the most basic computational problems is the task of finding a desired
item in an ordered list of N items. While the best classical algorithm for this
problem uses log_2 N queries to the list, a quantum computer can solve the
problem using a constant factor fewer queries. However, the precise value of
this constant is unknown. By characterizing a class of quantum query algorithms
for ordered search in terms of a semidefinite program, we find new quantum
algorithms for small instances of the ordered search problem. Extending these
algorithms to arbitrarily large instances using recursion, we show that there
is an exact quantum ordered search algorithm using 4 log_{605} N \approx 0.433
log_2 N queries, which improves upon the previously best known exact algorithm.Comment: 8 pages, 4 figure
The geometry of efficient arithmetic on elliptic curves
The arithmetic of elliptic curves, namely polynomial addition and scalar
multiplication, can be described in terms of global sections of line bundles on
and , respectively, with respect to a given projective embedding
of in . By means of a study of the finite dimensional vector
spaces of global sections, we reduce the problem of constructing and finding
efficiently computable polynomial maps defining the addition morphism or
isogenies to linear algebra. We demonstrate the effectiveness of the method by
improving the best known complexity for doubling and tripling, by considering
families of elliptic curves admiting a -torsion or -torsion point
A Formalization of Polytime Functions
We present a deep embedding of Bellantoni and Cook's syntactic
characterization of polytime functions. We prove formally that it is correct
and complete with respect to the original characterization by Cobham that
required a bound to be proved manually. Compared to the paper proof by
Bellantoni and Cook, we have been careful in making our proof fully contructive
so that we obtain more precise bounding polynomials and more efficient
translations between the two characterizations. Another difference is that we
consider functions on bitstrings instead of functions on positive integers.
This latter change is motivated by the application of our formalization in the
context of formal security proofs in cryptography. Based on our core
formalization, we have started developing a library of polytime functions that
can be reused to build more complex ones.Comment: 13 page
Scalable parallel computation of the translation operator in three dimensions
We propose a novel algorithm for the parallel, distributed-memory computation of the translation operator in the three-dimensional multilevel fast multipole algorithm (MLFMA). Sequential algorithms can compute the translation operator with L multipoles and O(L-2) sampling points in O(L-2) time. State-of-the-art hierarchical parallelization schemes of the MLFMA rely on the distribution of radiation patterns and associated translation operators among P = O(L-2) parallel processes, necessitating the development of distributed-memory algorithms for the computation of the translation operator. Whereas a baseline parallel algorithm computes this translation operator in O(L) time, we propose an algorithm that achieves this in only O(log L) time. For large translation operators and a high number of parallel processes, our algorithm proves to be roughly ten times faster than the baseline algorithm
Isogenies of Elliptic Curves: A Computational Approach
Isogenies, the mappings of elliptic curves, have become a useful tool in
cryptology. These mathematical objects have been proposed for use in computing
pairings, constructing hash functions and random number generators, and
analyzing the reducibility of the elliptic curve discrete logarithm problem.
With such diverse uses, understanding these objects is important for anyone
interested in the field of elliptic curve cryptography. This paper, targeted at
an audience with a knowledge of the basic theory of elliptic curves, provides
an introduction to the necessary theoretical background for understanding what
isogenies are and their basic properties. This theoretical background is used
to explain some of the basic computational tasks associated with isogenies.
Herein, algorithms for computing isogenies are collected and presented with
proofs of correctness and complexity analyses. As opposed to the complex
analytic approach provided in most texts on the subject, the proofs in this
paper are primarily algebraic in nature. This provides alternate explanations
that some with a more concrete or computational bias may find more clear.Comment: Submitted as a Masters Thesis in the Mathematics department of the
University of Washingto
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