Computing normalizers of permutation groups efficiently using isomorphisms of association schemes

ABSTRACT This note presents an algorithm to speed up the computation of normalizers of permutation groups. It is an application of computation of isomorphisms of association schemes

Computing normalisers of highly intransitive groups

We investigate the normaliser problem, that is, given , ≤ ₙ, compute [sub](). The fastest known theoretical algorithm for this problem is simply exponential, but more efficient algorithms are known for some restriction of classes for and . In this thesis, we will focus on highly intransitive groups, which are groups with many orbits. We give new algorithms to compute [sub](ₙ)() for highly intransitive groups ≤ ₙ and for some subclasses that perform substantially faster than previous implementations in the computer algebra system GAP."This work was supported by the University of St Andrews (School of Computer Science and St Leonard’s College Scholarship)." -- Fundin

Q(sqrt(-3))-Integral Points on a Mordell Curve

We use an extension of quadratic Chabauty to number fields,recently developed by the author with Balakrishnan, Besser and M ̈uller,combined with a sieving technique, to determine the integral points overQ(√−3) on the Mordell curve y2 = x3 − 4

P-Schemes and Deterministic Polynomial Factoring Over Finite Fields

We introduce a family of mathematical objects called P-schemes, where P is a poset of subgroups of a finite group G. A P-scheme is a collection of partitions of the right coset spaces H\G, indexed by H∈P, that satisfies a list of axioms. These objects generalize the classical notion of association schemes [BI84] as well as the notion of m-schemes [IKS09]. Based on P-schemes, we develop a unifying framework for the problem of deterministic factoring of univariate polynomials over finite field under the generalized Riemann hypothesis (GRH). More specifically, our results include the following: We show an equivalence between m-scheme as introduced in [IKS09] and P-schemes in the special setting that G is an multiply transitive permutation group and P is a poset of pointwise stabilizers, and therefore realize the theory of m-schemes as part of the richer theory of P-schemes. We give a generic deterministic algorithm that computes the factorization of the input polynomial ƒ(X) ∈ Fq[X] given a "lifted polynomial" ƒ~(X) of ƒ(X) and a collection F of "effectively constructible" subfields of the splitting field of ƒ~(X) over a certain base field. It is routine to compute ƒ~(X) from ƒ(X) by lifting the coefficients of ƒ(X) to a number ring. The algorithm then successfully factorizes ƒ(X) under GRH in time polynomial in the size of ƒ~(X) and F, provided that a certain condition concerning P-schemes is satisfied, for P being the poset of subgroups of the Galois group G of ƒ~(X) defined by F via the Galois correspondence. By considering various choices of G, P and verifying the condition, we are able to derive the main results of known (GRH-based) deterministic factoring algorithms [Hua91a; Hua91b; Ron88; Ron92; Evd92; Evd94; IKS09] from our generic algorithm in a uniform way. We investigate the schemes conjecture in [IKS09] and formulate analogous conjectures associated with various families of permutation groups, each of which has applications on deterministic polynomial factoring. Using a technique called induction of P-schemes, we establish reductions among these conjectures and show that they form a hierarchy of relaxations of the original schemes conjecture. We connect the complexity of deterministic polynomial factoring with the complexity of the Galois group G of ƒ~(X). Specifically, using techniques from permutation group theory, we obtain a (GRH-based) deterministic factoring algorithm whose running time is bounded in terms of the noncyclic composition factors of G. In particular, this algorithm runs in polynomial time if G is in Γk for some k=2O(√(log n), where Γk denotes the family of finite groups whose noncyclic composition factors are all isomorphic of subgroups of the symmetric group of degree k. Previously, polynomial-time algorithms for Γk were known only for bounded k. We discuss various aspects of the theory of P-schemes, including techniques of constructing new P-schemes from old ones, P-schemes for symmetric groups and linear groups, orbit P-schemes, etc. For the closely related theory of m-schemes, we provide explicit constructions of strongly antisymmetric homogeneous m-schemes for m≤3. We also show that all antisymmetric homogeneous orbit 3-schemes have a matching for m≥3, improving a result in [IKS09] that confirms the same statement for m≥4. In summary, our framework reduces the algorithmic problem of deterministic polynomial factoring over finite fields to a combinatorial problem concerning P-schemes, allowing us to not only recover most of the known results but also discover new ones. We believe progress in understanding P-schemes associated with various families of permutation groups will shed some light on the ultimate goal of solving deterministic polynomial factoring over finite fields in polynomial time.</p

Normalizer Circuits and Quantum Computation

(Abridged abstract.) In this thesis we introduce new models of quantum computation to study the emergence of quantum speed-up in quantum computer algorithms. Our first contribution is a formalism of restricted quantum operations, named normalizer circuit formalism, based on algebraic extensions of the qubit Clifford gates (CNOT, Hadamard and $\pi/4$-phase gates): a normalizer circuit consists of quantum Fourier transforms (QFTs), automorphism gates and quadratic phase gates associated to a set $G$, which is either an abelian group or abelian hypergroup. Though Clifford circuits are efficiently classically simulable, we show that normalizer circuit models encompass Shor's celebrated factoring algorithm and the quantum algorithms for abelian Hidden Subgroup Problems. We develop classical-simulation techniques to characterize under which scenarios normalizer circuits provide quantum speed-ups. Finally, we devise new quantum algorithms for finding hidden hyperstructures. The results offer new insights into the source of quantum speed-ups for several algebraic problems. Our second contribution is an algebraic (group- and hypergroup-theoretic) framework for describing quantum many-body states and classically simulating quantum circuits. Our framework extends Gottesman's Pauli Stabilizer Formalism (PSF), wherein quantum states are written as joint eigenspaces of stabilizer groups of commuting Pauli operators: while the PSF is valid for qubit/qudit systems, our formalism can be applied to discrete- and continuous-variable systems, hybrid settings, and anyonic systems. These results enlarge the known families of quantum processes that can be efficiently classically simulated. This thesis also establishes a precise connection between Shor's quantum algorithm and the stabilizer formalism, revealing a common mathematical structure in several quantum speed-ups and error-correcting codes.Comment: PhD thesis, Technical University of Munich (2016). Please cite original papers if possible. Appendix E contains unpublished work on Gaussian unitaries. If you spot typos/omissions please email me at JLastNames at posteo dot net. Source: http://bit.ly/2gMdHn3. Related video talk: https://www.perimeterinstitute.ca/videos/toy-theory-quantum-speed-ups-based-stabilizer-formalism Posted on my birthda

Computing Galois cohomology and forms of linear algebraic groups

Om op een e±ciÄente manier te rekenen met groepen is een geschikte voorstelling nodig van de groepselementen. Een groep heeft vaak een intrinsieke de¯ni- tie, dat wil zeggen dat zij impliciet gede¯nieerd wordt door een beschrijving van de eigenschappen van de elementen (bijv.: de vaste punt ondergroep van een groep). Een dergelijke de¯nitie is voor berekeningen met groepselementen niet erg handig aangezien het, afgezien van de identiteit, geen construeerbare groepselementen geeft. In dergelijke gevallen dient men te beschikken over een extrinsieke de¯nitie van de groep, zoals een voorstelling. Wij ontwerpen en implementeren algoritmen voor berekeningen aan gedraaide groepen van Lie-type, waaronder begrepen zijn de groepen die niet quasi-gesple- ten zijn. Algoritmen voor het rekenen met elementen in de Steinberg voorstelling voor ongedraaide groepen van Lie-type en algoritmen voor de overgang tussen deze voorstelling en de lineaire representatie worden gegeven in [12] (gebaseerd op werk van [15] en [26]). Dit werk wordt in diverse richtingen uitgebreid. De gedraaide groepen van Lie-type zijn groepen van rationale punten van gedraaide vormen van reductieve lineaire algebraijsche groepen. De gedraaide vormen zijn geclassi¯ceerd door Galoiscohomologie. Ten einde de Galoisco- homologie te berekenen ontwerpen we een methode voor het berekenen van de cohomologie van een eindig voortgebrachte groep ¡ op een groep A. Deze meth- ode is op zichzelf van belang. De methode wordt toegepast op de berekening van de Galoiscohomologie van een reductieve lineaire algebraijsche groep. Laat G een reductieve lineaire algebraijsche groep gede¯nieerd over een lichaam k zijn. Een gedraaide groep van Lie-type G®(k) wordt uniek bepaald door de co- cykel ® van de Galois groep van K op AutK(G), en de groep van K-algebraijsche automor¯smen waar K de eindige Galoisuitbreiding over k is. Algoritmen voor de berekening van het relatieve wortelsysteem op G®(k), voor de wortelonder- groepen en de wortelelementen worden gegeven. Daarnaast worden ook algo- ritmen voor de berekening van onderlinge relaties, zoals de commutatorrelaties en producten gegeven. Dit maakt het mogelijk om te rekenen binnen de nor- male ondergroep G®(k)y van G®(k) voortgebracht door de wortelelementen. We passen het algoritme toe op diverse voorbeelden, waaronder 2E6;1(k) en 3;6D4;1(k). Een toepassing is een algoritme, ontworpen voor de berekening van alle gedraai- de maximale tori van een eindige groep van Lie-type. De orde van zo'n torus wordt berekend als een polynoom in q, de orde van het lichaam k. Daarnaast berekenen we de ordes van de faktoren in de decompositie van de torus als een direkt product van cyklische ondergroepen. Voor een gegeven lichaam k, worden de maximale tori van G¯(k) berekend als ondergroepen van G¯(K) over een uitbreidingslichaam K en daarna wordt de e®ectieve versie van Lang's Theorem [11] gebruikt om de torus te conjugeren tot een k-torus, wat een ondergroep van G¯(k) is. Gebruikmakend van deze informatie over maximale tori, geven we een algo- ritme voor de berekening van alle Sylowondergroepen van de groep van Lie-type. Als p niet de karakteristiek van het lichaam is, wordt de Sylowondergroep berek- end als een ondergroep van de normalisator van de k-torus. Alle hier besproken algoritmen zijn geijmplementeerd in Magma [5]

LIPIcs, Volume 251, ITCS 2023, Complete Volume

LIPIcs, Volume 251, ITCS 2023, Complete Volum

LIPIcs, Volume 244, ESA 2022, Complete Volume

LIPIcs, Volume 244, ESA 2022, Complete Volum

Quaternion Algebras

This open access textbook presents a comprehensive treatment of the arithmetic theory of quaternion algebras and orders, a subject with applications in diverse areas of mathematics. Written to be accessible and approachable to the graduate student reader, this text collects and synthesizes results from across the literature. Numerous pathways offer explorations in many different directions, while the unified treatment makes this book an essential reference for students and researchers alike. Divided into five parts, the book begins with a basic introduction to the noncommutative algebra underlying the theory of quaternion algebras over fields, including the relationship to quadratic forms. An in-depth exploration of the arithmetic of quaternion algebras and orders follows. The third part considers analytic aspects, starting with zeta functions and then passing to an idelic approach, offering a pathway from local to global that includes strong approximation. Applications of unit groups of quaternion orders to hyperbolic geometry and low-dimensional topology follow, relating geometric and topological properties to arithmetic invariants. Arithmetic geometry completes the volume, including quaternionic aspects of modular forms, supersingular elliptic curves, and the moduli of QM abelian surfaces. Quaternion Algebras encompasses a vast wealth of knowledge at the intersection of many fields. Graduate students interested in algebra, geometry, and number theory will appreciate the many avenues and connections to be explored. Instructors will find numerous options for constructing introductory and advanced courses, while researchers will value the all-embracing treatment. Readers are assumed to have some familiarity with algebraic number theory and commutative algebra, as well as the fundamentals of linear algebra, topology, and complex analysis. More advanced topics call upon additional background, as noted, though essential concepts and motivation are recapped throughout