Factoring Safe Semiprimes with a Single Quantum Query

Shor's factoring algorithm (SFA), by its ability to efficiently factor large numbers, has the potential to undermine contemporary encryption. At its heart is a process called order finding, which quantum mechanics lets us perform efficiently. SFA thus consists of a \emph{quantum order finding algorithm} (QOFA), bookended by classical routines which, given the order, return the factors. But, with probability up to $1/2$, these classical routines fail, and QOFA must be rerun. We modify these routines using elementary results in number theory, improving the likelihood that they return the factors. The resulting quantum factoring algorithm is better than SFA at factoring safe semiprimes, an important class of numbers used in cryptography. With just one call to QOFA, our algorithm almost always factors safe semiprimes. As well as a speed-up, improving efficiency gives our algorithm other, practical advantages: unlike SFA, it does not need a randomly picked input, making it simpler to construct in the lab; and in the (unlikely) case of failure, the same circuit can be rerun, without modification. We consider generalizing this result to other cases, although we do not find a simple extension, and conclude that SFA is still the best algorithm for general numbers (non safe semiprimes, in other words). Even so, we present some simple number theoretic tricks for improving SFA in this case.Comment: v2 : Typo correction and rewriting for improved clarity v3 : Slight expansion, for improved clarit

Automated Termination Proofs for Logic Programs by Term Rewriting

There are two kinds of approaches for termination analysis of logic programs: "transformational" and "direct" ones. Direct approaches prove termination directly on the basis of the logic program. Transformational approaches transform a logic program into a term rewrite system (TRS) and then analyze termination of the resulting TRS instead. Thus, transformational approaches make all methods previously developed for TRSs available for logic programs as well. However, the applicability of most existing transformations is quite restricted, as they can only be used for certain subclasses of logic programs. (Most of them are restricted to well-moded programs.) In this paper we improve these transformations such that they become applicable for any definite logic program. To simulate the behavior of logic programs by TRSs, we slightly modify the notion of rewriting by permitting infinite terms. We show that our transformation results in TRSs which are indeed suitable for automated termination analysis. In contrast to most other methods for termination of logic programs, our technique is also sound for logic programming without occur check, which is typically used in practice. We implemented our approach in the termination prover AProVE and successfully evaluated it on a large collection of examples.Comment: 49 page

Techniques for the Synthesis of Reversible Toffoli Networks

This paper presents novel techniques for the synthesis of reversible networks of Toffoli gates, as well as improvements to previous methods. Gate count and technology oriented cost metrics are used. Our synthesis techniques are independent of the cost metrics. Two new iterative synthesis procedure employing Reed-Muller spectra are introduced and shown to complement earlier synthesis approaches. The template simplification suggested in earlier work is enhanced through introduction of a faster and more efficient template application algorithm, updated (shorter) classification of the templates, and presentation of the new templates of sizes 7 and 9. A novel ``resynthesis'' approach is introduced wherein a sequence of gates is chosen from a network, and the reversible specification it realizes is resynthesized as an independent problem in hopes of reducing the network cost. Empirical results are presented to show that the methods are effective both in terms of the realization of all 3x3 reversible functions and larger reversible benchmark specifications.Comment: 20 pages, 5 figure

A rewriting grammar for heat exchanger network structure evolution with stream splitting

The design of cost optimal heat exchanger networks is a difficult optimisation problem due both to the nonlinear models required and also the combinatorial size of the search space. When stream splitting is considered, the combinatorial aspects make the problem even harder. This paper describes the implementation of a two level evolutionary algorithm based on a string rewriting grammar for the evolution of the heat exchanger network structure. A biological analogue of genotypes and phenotypes is used to describe structures and specific solutions respectively. The top level algorithm evolves structures while the lower level optimises specific structures. The result is a hybrid optimisation procedure which can identify the best structures including stream splitting. Case studies from the literature are presented to demonstrate the capabilities of the novel procedure

Minimization Strategies for Maximally Parallel Multiset Rewriting Systems

Maximally parallel multiset rewriting systems (MPMRS) give a convenient way to express relations between unstructured objects. The functioning of various computational devices may be expressed in terms of MPMRS (e.g., register machines and many variants of P systems). In particular, this means that MPMRS are computationally complete; however, a direct translation leads to quite a big number of rules. Like for other classes of computationally complete devices, there is a challenge to find a universal system having the smallest number of rules. In this article we present different rule minimization strategies for MPMRS based on encodings and structural transformations. We apply these strategies to the translation of a small universal register machine (Korec, 1996) and we show that there exists a universal MPMRS with 23 rules. Since MPMRS are identical to a restricted variant of P systems with antiport rules, the results we obtained improve previously known results on the number of rules for those systems.Comment: This article is an improved version of [1

Structure Pattern Analysis Using Term Rewriting and Clustering Algorithm

Biological data is accumulated at a fast pace. However, raw data are generally difficult to understand and not useful unless we unlock the information hidden in the data. Knowledge/information can be extracted as the patterns or features buried within the data. Thus data mining, aims at uncovering underlying rules, relationships, and patterns in data, has emerged as one of the most exciting fields in computational science. In this dissertation, we develop efficient approaches to the structure pattern analysis of RNA and protein three dimensional structures. The major techniques used in this work include term rewriting and clustering algorithms. Firstly, a new approach is designed to study the interaction of RNA secondary structures motifs using the concept of term rewriting. Secondly, an improved K-means clustering algorithm is proposed to estimate the number of clusters in data. A new distance descriptor is introduced for the appropriate representation of three dimensional structure segments of RNA and protein three dimensional structures. The experimental results show the improvements in the determination of the number of clusters in data, evaluation of RNA structure similarity, RNA structure database search, and better understanding of the protein sequence-structure correspondence

Stable normal forms for polynomial system solving

This paper describes and analyzes a method for computing border bases of a zero-dimensional ideal $I$. The criterion used in the computation involves specific commutation polynomials and leads to an algorithm and an implementation extending the one provided in [MT'05]. This general border basis algorithm weakens the monomial ordering requirement for \grob bases computations. It is up to date the most general setting for representing quotient algebras, embedding into a single formalism Gr\"obner bases, Macaulay bases and new representation that do not fit into the previous categories. With this formalism we show how the syzygies of the border basis are generated by commutation relations. We also show that our construction of normal form is stable under small perturbations of the ideal, if the number of solutions remains constant. This new feature for a symbolic algorithm has a huge impact on the practical efficiency as it is illustrated by the experiments on classical benchmark polynomial systems, at the end of the paper

Automated Code Generation for Lattice Quantum Chromodynamics and beyond

We present here our ongoing work on a Domain Specific Language which aims to simplify Monte-Carlo simulations and measurements in the domain of Lattice Quantum Chromodynamics. The tool-chain, called Qiral, is used to produce high-performance OpenMP C code from LaTeX sources. We discuss conceptual issues and details of implementation and optimization. The comparison of the performance of the generated code to the well-established simulation software is also made