How to implement a modular form

AbstractWe present a model for Fourier expansions of arbitrary modular forms. This model takes precisions and symmetries of such Fourier expansions into account. The value of this approach is illustrated by studying a series of examples. An implementation of these ideas is provided by the author. We discuss the technical background of this implementation, and we explain how to implement arbitrary Fourier expansions and modular forms. The framework allows us to focus on the considerations of a mathematical nature during this procedure. We conclude with a list of currently available implementations and a discussion of possible computational research

How to implement the modular building principle of the electronic textbook based on enlargement of didactic units

The article presents a way of how to build the e-textbook by a modular principle, which is proposed to implement in respect of a problematic rather than a thematic ground of educational material. The problematic approach of the educational content in e-textbooks is proved by the need to perform a complete didactic cycle of training and to organize the new material in the form of integrated didactic unitsВ статье излагается один из способов построения электронного учебника - модульный, который предлагается реализовать не по тематическому, а по проблемному принципу предъявления учебного материала. Проблемный подход к подобному представлению образовательного контента в электронных учебниках обосновывается необходимостью реализации полного дидактического цикла обучения и организации нового материала в виде укрупнённых дидактических едини

The design of a framework for compilers development

DOLPHIN framework is a solution conceived to support the development of modular compilers. Its supplies a large set of components, like: front-end’s, back-end’s, code analysis, code optimizations and measure components that can be combined to build new compilers. All these components work over a single form of intermediate code representation, the DOLPHIN Internal code Representation. The main principle that guides the conception of DOLPHIN framework was to build a user-friendly solution to develop high quality compilers. Such solution was achieved based on three main concepts: components, components reuse and data consistency. This paper, that aims to present the architectural design of DOLPHIN framework, demonstrates: how the concepts presented above influence the framework architecture; how they were ”implemented” on the framework, namely shows the interfaces defined for the components and for the code representation; how the components are related; how to use the components to implement concrete compilers; and how to evolve the components and the framework to support new features.info:eu-repo/semantics/publishedVersio

Quantum resource estimates for computing elliptic curve discrete logarithms

We give precise quantum resource estimates for Shor's algorithm to compute discrete logarithms on elliptic curves over prime fields. The estimates are derived from a simulation of a Toffoli gate network for controlled elliptic curve point addition, implemented within the framework of the quantum computing software tool suite LIQ$Ui|\rangle$. We determine circuit implementations for reversible modular arithmetic, including modular addition, multiplication and inversion, as well as reversible elliptic curve point addition. We conclude that elliptic curve discrete logarithms on an elliptic curve defined over an $n$-bit prime field can be computed on a quantum computer with at most $9n + 2\lceil\log_2(n)\rceil+10$ qubits using a quantum circuit of at most $448 n^3 \log_2(n) + 4090 n^3$ Toffoli gates. We are able to classically simulate the Toffoli networks corresponding to the controlled elliptic curve point addition as the core piece of Shor's algorithm for the NIST standard curves P-192, P-224, P-256, P-384 and P-521. Our approach allows gate-level comparisons to recent resource estimates for Shor's factoring algorithm. The results also support estimates given earlier by Proos and Zalka and indicate that, for current parameters at comparable classical security levels, the number of qubits required to tackle elliptic curves is less than for attacking RSA, suggesting that indeed ECC is an easier target than RSA.Comment: 24 pages, 2 tables, 11 figures. v2: typos fixed and reference added. ASIACRYPT 201

Arithmetic and computational aspects of modular forms over global fields

This thesis consists of two parts. In the first part, we present a positive characteristic analogue of Shimura's theorem on the special values of modular forms at CM points. More precisely, we show using Hayes' theory of Drinfeld modules that the special value at a CM point of an arithmetic Drinfeld modular form of arbitrary rank lies in the Hilbert class field of the CM field up to a period, independent of the chosen modular form.This is achieved via Pink's realization of Drinfeld modular forms as sections of a sheaf over the compactified Drinfeld modular curve. In the second part of the thesis, we present various computational and algorithmic aspects both for the classical theory (over C) and function field theory. First, we implement the rings of quasimodular forms in SageMath and give some applications such as the symbolic calculation of the derivative of a classical modular form. Second, we explain how to compute objects associated with a Drinfeld modules such as the exponential, the logarithm, and Potemine's set of basic J-invariants. Lastly, we present a SageMath package for computing with Drinfeld modular forms and their expansion at infinity using the nonstandard A-expansion theory of López and Petrov