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 $E\times E$ and $E$, respectively, with respect to a given projective embedding of $E$ in $\mathbb{P}^r$. 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 $2$-torsion or $3$-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