The complexity of class polynomial computation via floating point approximations

We analyse the complexity of computing class polynomials, that are an important ingredient for CM constructions of elliptic curves, via complex floating point approximations of their roots. The heart of the algorithm is the evaluation of modular functions in several arguments. The fastest one of the presented approaches uses a technique devised by Dupont to evaluate modular functions by Newton iterations on an expression involving the arithmetic-geometric mean. It runs in time $O (|D| \log^5 |D| \log \log |D|) = O (|D|^{1 + \epsilon}) = O (h^{2 + \epsilon})$ for any $\epsilon > 0$, where $D$ is the CM discriminant and $h$ is the degree of the class polynomial. Another fast algorithm uses multipoint evaluation techniques known from symbolic computation; its asymptotic complexity is worse by a factor of $\log |D|$. Up to logarithmic factors, this running time matches the size of the constructed polynomials. The estimate also relies on a new result concerning the complexity of enumerating the class group of an imaginary-quadratic order and on a rigorously proven upper bound for the height of class polynomials

Efficient implementation of the Hardy-Ramanujan-Rademacher formula

We describe how the Hardy-Ramanujan-Rademacher formula can be implemented to allow the partition function $p(n)$ to be computed with softly optimal complexity $O(n^{1/2+o(1)})$ and very little overhead. A new implementation based on these techniques achieves speedups in excess of a factor 500 over previously published software and has been used by the author to calculate $p(10^{19})$, an exponent twice as large as in previously reported computations. We also investigate performance for multi-evaluation of $p(n)$, where our implementation of the Hardy-Ramanujan-Rademacher formula becomes superior to power series methods on far denser sets of indices than previous implementations. As an application, we determine over 22 billion new congruences for the partition function, extending Weaver's tabulation of 76,065 congruences.Comment: updated version containing an unconditional complexity proof; accepted for publication in LMS Journal of Computation and Mathematic

A comparison and a combination of SST and AGM algorithms for counting points of elliptic curves in characteristic 2

International audienceSince the first use of a p-adic method for counting points of elliptic curves, by Satoh in 1999, several variants of his algorithm have been proposed. In the current state, the AGM algorithm, proposed by Mestre is thought to be the fastest in practice, and the algorithm by Satoh­-Skjernaa­-Taguchi has the best asymptotic complexity but requires precomputations. We present an amelioration of the SST algorithm, borrowing ideas from the AGM. We make a precise comparison between this modified SST algorithm and the AGM, thus demonstrating that the former is faster by a significant factor, even for small cryptographic sizes

Computing Jacobi's θ\theta in quasi-linear time

Jacobi's $\theta$ function has numerous applications in mathematics and computer science; a naive algorithm allows the computation of $\theta(z,\tau)$, for $z, \tau$ verifying certain conditions, with precision $P$ in $O(\mathcal{M}(P) \sqrt{P})$ bit operations, where $\mathcal{M}(P)$ denotes the number of operations needed to multiply two complex $P$-bit numbers. We generalize an algorithm which computes specific values of the $\theta$ function (the \textit{theta-constants}) in asymptotically faster time; this gives us an algorithm to compute $\theta(z, \tau)$ with precision $P$ in $O(\mathcal{M}(P) \log P)$ bit operations, for any $\tau \in \mathcal{F}$ and $z$ reduced using the quasi-periodicity of $\theta$

The complex AGM, periods of elliptic curves over C and complex elliptic logarithms

We give an account of the complex Arithmetic-Geometric Mean (AGM), as first studied by Gauss, together with details of its relationship with the theory of elliptic curves over \C, their period lattices and complex parametrisation. As an application, we present efficient methods for computing bases for the period lattices and elliptic logarithms of points, for arbitrary elliptic curves defined over \C. Earlier authors have only treated the case of elliptic curves defined over the real numbers; here, the multi-valued nature of the complex AGM plays an important role. Our method, which we have implemented in both \Magma\ and \Sage, is illustrated with several examples using elliptic curves defined over number fields with real and complex embeddings.Comment: The addional file elog_ex.sage contains a Sage script for the examples in the last section of the paper, and the file elog_ex.out contains the result of running that script with Sage version 5.

Operation of meshed high voltage direct current (HVDC) overlay grids: from operational planning to real time operation

Energy turnaround from conventional to renewable energy generation needs bulk power long distance transmission. This new transmission objective can be meet with an HVDC overlay grid spanning the existing AC transmission system. This thesis proposes an operation management strategy for future HVDC overlay grids subdivided in tertiary, secondary and primary control instances. The tertiary control ensures coordination among HVDC converters and with the AC system. It determines converter reference values on a regular basis. It is proposed for the case of having multiple as well as a single system operator responsible for the overlay HVDC grid. The secondary control instance locally adapts tertiary control’s converter references to the actual grid requirements (e.g. after disturbances). The primary control ensures DC energy balance. Therefore, a continuous p-v-characteristic is proposed as well as two appropriate parameterization methods. One emulates piecewise linear p-v-characteristics and the other performs an automatic parameterization according to available balancing power provision capabilities on related AC point of common coupling. All control methods are validated by numerical case studies.Weltweit aber besonders in Europa steigt der Bedarf große Leistungen über weite Strecken zu transportieren. Dies ist hauptsächlich in der Energiewende und dem damit zusammenhängenden stark ansteigenden Anteil Erneuerbarer Energien und deren Erzeugungszentren begründet. Ein bedeutender Teil der Erneuerbaren Energien wird zukünftig weitab der Lastzentren produziert. Zur Lösung dieser daraus resultierenden neuen Transportaufgabe ist die Hochspannungsgleichstromübertragung (HGÜ) besonders geeignet. Eine redundante und damit auch wirtschaftliche Ausführung stellt das vermaschte HGÜ-Netz dar, das in der Energieversorgungsnetzhiearchie eine neue Netzebene dargestellt und somit als Overlay-HGÜ-Netz bezeichnet wird. Diese Arbeit widmet sich der Fragestellung der Betriebsführung eines Overlaynetzes. Dazu wird eine dreistufige Betriebsführung vorgeschlagen. In Anlehnung an die im europäischen AC-Verbundnetz bestehende Dreiteiligkeit wird eine Untergliederung in folgende Regelungsinstanzen vorgenommen: Tertiär-, Sekundär und Primärregelung. Die Tertiärregelung übernimmt die Koordinierungsaufgabe der Umrichter untereinander und mit dem unterlagerten AC-Netz im Rahmen einer Betriebsplanung. Es ist ein betriebstypisches Aktualisierungsintervall von 15 Minuten vorgesehen, indem die Umrichtersollwerte vorgegeben werden. Deren Bestimmung erfolgt durch ein auf dieses nichtlineare Problem zugeschnittenen AC/DC Optimal Power Flow. Dieses Verfahren fußt auf der Verfügbarkeit aller AC- und DC-Netzinformationen im Gebiet des Overlaynetzes. Im Falle einer föderalen Organisation eines HGÜ-Overlaynetzes in Europa müssen die Zielsetzungen mehrere Übertragungsnetzbetreiber (ÜNB) bei der Bestimmung eines Umrichtersollwertfahrplans berücksichtig werden. Für diesen Fall wird hier eine Methode vorgeschlagen, die mittels eines Aushandlungsprozesses die ÜNB spezifischen Kostenfunktionen für den Einsatz von HGÜ-Umrichtern in der entsprechenden Regelzone zu einer für das gesamte Overlaynetz gültigen Zielfunktion konsolidiert. Dabei werden Grenzwerte der einzelnen beteiligten ÜNB ebenso berücksichtigt wie lokale Zielfunktionen. Die Sekundärregelung passt die von der Tertiärregelung vorgegebenen Umrichtersollwerte innerhalb des 15-min-Betriebsintervalls vor allem im Fall von Störungen an. Dafür wird ein Verfahren vorgeschlagen, das sich der Informationen eines Weitbereichsüberwachungssystems bedient, um signifikante Abweichung der geplanten Leistungsflüsse zu erfassen. Die Umrichterwirkleistungssollwerte werden entsprechend angepasst. Eine Aufteilung von unplanmäßigen Leistungsflüssen zwischen AC und DC-Netz sorgt für eine Entlastung des AC-Netzes und beugt Betriebsmittelüberlastungen und dadurch verursachten Instabilitätsphänomenen vor. Die Primärregelung gewährleistet das Gleichgewicht zwischen ein- und ausgespeister Wirkleistung in das / aus dem HGÜ-Overlaynetz. Ist die diesbezügliche Leistungsbilanz ausgewogen, ist das Energiegleichgewicht, die sogenanntes Energiestabilität, gewahrt. Die DC-Zeitkonstanten sind klein. Nur eine dezentral (am Umrichterstandort) angeordnete Regelung kann zeitlich angemessen reagieren. Diese nutzt eine p-u-Regelcharakteristik, die die Umrichtersollleistung entsprechend der Abweichung von der DC-Sollspannung anpasst. Dafür werden eine kontinuierliche p-u-Charakteristik sowie Verfahren zu deren Parametrierung vorgeschlagen. Für die Bereitstellung von DC-Regelleistung besonders geeignete AC-Knoten können so angemessen für das HGÜ-Overlaynetz genutzt werden. Die Funktionalität des hier vorgeschlagenen dreiteiligen Bertriebsführungsverfahrens für vermaschte HGÜ-Netze wird anhand von numerischen Fallstudien auf Basis einer typischen Netztsituation in Zentraleuropa validiert.There is an increasing demand for long distance bulk power transmission worldwide and particularly in Europe. Energy turnaround from conventional to renewable energy generation is one of the main drivers. This implies that a significant percentage of electricity production is generated remotely from load centers, by huge wind farms, for example. This new transmission objective can be met with high voltage direct current (HVDC) transmission. An HVDC grid is favored for redundancy as well as economic reasons. As this HVDC grid will be a new network layer above the existing AC transmission layer, it is referred to as an “overlay” HVDC grid. This thesis proposes a three stage operation management strategy for future HVDC overlay grids. The architecture is comprised of tertiary, secondary and primary control instances which reflect the hierarchy of AC system operation. All control methods have been validated by numerical case studies on a reference grid which is a representative of a typical interconnected network situation in central Europe. The proposed tertiary control ensures coordination among all HVDC converters and with the underlaying AC system. It serves as an example of converter reference value determination in a 15 minutes time interval. Therefore a mixed AC/DC optimal power flow method is proposed which is capable of solving this nonlinear optimization problem based on a complete set of topological and other state information of the entire grid. In the event of having different transmission system operators (TSO) operating only a subset of converters of the HVDC overlay grid, the optimization problem becomes increasingly complex since each TSO might have its own optimization objectives. This problem is addressed by another multiple objective function approach. The proposed method superimposes particular cost functions of related TSO which yields system wide cost functions as a basis for AC/DC power flow optimization. The Secondary control instance adapts the tertiary control’s converter reference values within the 15 minute interval to the actual grid requirements, particularly in the event of grid disturbances. An algorithm is proposed that identifies significant deviations from the actual power flow schedule by a wide are monitoring system. Converter power references are adapted in order to optimally share the deviations between the AC system and the HVDC overlay grid. Since data availability is key for the robust operation of this method, backup mechanisms for data acquisition is also proposed. The Primary control ensures DC energy balance, which is referred to as the energy stability of HVDC grids. Converter reference values for active power need to be adjusted in the event of a mismatch between active power fed to and drawn from the HVDC grid. As the time constants within a DC grid are very small, this is a fast, local control based on p v characteristics; the converter’s power reference is adjusted in accordance with deviation of the DC node voltage from its reference. Furthermore, a continuous p v characteristic is proposed as well as two appropriate parameterization methods. One emulates already existing piecewise linear p v characteristics for DC node voltage control and the other performs an automatic parameterization according to available balancing power provision capabilities on related AC point of common couplings. The latter significantly reduces the additional loading of the AC transmission grid with DC balancing power flows as the AC nodes, which are the most technically feasible, are utilized to provide the most DC balancing power

Research and development program for non-linear structural modeling with advanced time-temperature dependent constitutive relationships

Results of a 20-month research and development program for nonlinear structural modeling with advanced time-temperature constitutive relationships are reported. The program included: (1) the evaluation of a number of viscoplastic constitutive models in the published literature; (2) incorporation of three of the most appropriate constitutive models into the MARC nonlinear finite element program; (3) calibration of the three constitutive models against experimental data using Hastelloy-X material; and (4) application of the most appropriate constitutive model to a three dimensional finite element analysis of a cylindrical combustor liner louver test specimen to establish the capability of the viscoplastic model to predict component structural response

Computing Igusa class polynomials

We bound the running time of an algorithm that computes the genus-two class polynomials of a primitive quartic CM-field K. This is in fact the first running time bound and even the first proof of correctness of any algorithm that computes these polynomials. Essential to bounding the running time is our bound on the height of the polynomials, which is a combination of denominator bounds of Goren and Lauter and our own absolute value bounds. The absolute value bounds are obtained by combining Dupont's estimates of theta constants with an analysis of the shape of CM period lattices. The algorithm is basically the complex analytic method of Spallek and van Wamelen, and we show that it finishes in time Otilde(Delta^(7/2)), where Delta is the discriminant of K. We give a complete running time analysis of all parts of the algorithm, and a proof of correctness including a rounding error analysis. We also provide various improvements along the way.Comment: 31 pages (Various improvements to the exposition suggested by the referee. For the most detailed exposition, see Chapter II of the author's thesis http://hdl.handle.net/1887/15572

Short addition sequences for theta functions

International audienceThe main step in numerical evaluation of classical Sl2 (Z) modular forms and elliptic functions is to compute the sum of the first N nonzero terms in the sparse q-series belonging to the Dedekind eta function or the Jacobi theta constants. We construct short addition sequences to perform this task using N + o(N) multiplications. Our constructions rely on the representability of specific quadratic progressions of integers as sums of smaller numbers of the same kind. For example, we show that every generalised pentagonal number c 5 can be written as c = 2a + b where a, b are smaller generalised pentagonal numbers. We also give a baby-step giant-step algorithm that uses O(N/ log r N) multiplications for any r > 0, beating the lower bound of N multiplications required when computing the terms explicitly. These results lead to speed-ups in practice

On the evaluation of some sparse polynomials

We give algorithms for the evaluation of sparse polynomials of the form P=p0 + p1 x + p2 x^4 + ... + p_{n-1} x^{(N-1)^2} for various choices of coefficients . First, we take p_i=p^i, for some fixed p; in this case, we address the question of fast evaluation at a given point in the base ring, and we obtain a cost quasi-linear in sqrt{N}. We present experimental results that show the good behavior of this algorithm in a floating-point context, for the computation of Jacobi theta functions. Next, we consider the case of arbitrary coefficients; for this problem, we study the question of multiple evaluation: we show that one can evaluate such a polynomial at N values in the base ring in subquadratic time