Homotopy-Theoretic Studies of Khovanov-Rozansky Homology

This dissertation applies homotopy-theoretic methods to the study of Khovanov-Rozansky homology, a generalization of Khovanov homology (which in turn categorifies the Jones polynomial) that is constructed using categories of matrix factorizations: these are variants of chain complexes for which we have δ2 ≠ 0 and for which therefore no homotopy theory in terms of ordinary derived categories is available. In this work I study alternative approaches to the construction of homotopy theories for matrix factorizations, and more generally categories of curved modules and singularity categories, and describe the use of these homotopy-theoretic considerations to the understanding of Khovanov-Rozansky homology. The dissertation consists of a knot-theoretic part focusing on concrete applications of homotopy-theoretic techniques to Khovanov-Rozansky homology, and a homotopy-theoretic part, in which these techniques are developed independently and with the aim of large generality in the context of the theory of abelian model structures. The central results of the knot-theoretic part are the following: Firstly, the development of a conceptual definition of stable Hochschild homology and the description of Khovanov-Rozansky homology as stable Hochschild homology of Rouquier complexes of Soergel bimodules. These are well-known from representation theory and also play an important role in the construction of other knot invariants. Further, the classical knowledge about the combinatorics of Rouquier complexes leads to a direct proof of the fact that Khovanov-Rozansky homology is indeed a knot invariant. Afterwards, the introduction of a combinatorial approximation to Khovanov-Rozansky homology through a diagrammatic calculus similar to ones that can be used for the definition of the Jones polynomial and Khovanov homology. The central results of the homotopy-theoretic part are the following: Firstly, the construction of abelian model structures for categories of curved modules and singularity categories, on the basis of general techniques for the localization and the proof of cofibrant generation of abelian model structures. Afterwards, the discussion of numerous examples and enrichments of classical equivalences and recollements between triangulated categories to the level of model categories. Finally, the construction of a realization functor from the derived category of an Grothendieck abelian category A to the homotopy category of any reasonable abelian model structure on A, by means of constructing a Quillen equivalent abelian model structure on the category of chain complexes over A.In dieser Dissertation widme ich mich der homotopietheoretischen Untersuchung der Khovanov-Rozansky Homologie, einer 2004 gefundenen Invariante von Knoten und Verschlingungen im Raum. Sie ist eine Verallgemeinerung der Khovanov-Homologie, welche wiederum das Jones-Polynom verfeinert, und wird mit Hilfe von Matrixfaktorisierungen konstruiert: das sind Varianten von Kettenkomplexen, in denen δ2 6 ≠ 0 gilt, und f ur die daher keine uber derivierte Kategorien erkl arte Homotopietheorie zur Verf ugung steht. Ich untersuche in dieser Arbeit alternative Ans atze zur Konstruktion von Homotopietheorien für Matrixfaktorisierungen, allgemeiner f ur Kategorien gekrümmter Moduln sowie Singularit atenkategorien, und beschreibe den Nutzen dieser homotopietheoretischen Betrachtungen für das Verständnis der Khovanov-Rozansky Homologie. Die Arbeit besteht daher aus einem knotentheoretischen Teil, in dem konkrete Anwendungen homotopietheoretischer Techniken auf die Khovanov-Rozansky Homologie diskutiert werden, sowie einem homotopietheoretischen Teil, in dem diese Techniken im Rahmen der Theorie abelscher Modellkategorien unabhängig und möglichst allgemein entwickelt werden. Die zentralen Ergebnisse des knotentheoretischen Teils sind die folgenden: Zunächst das Erarbeiten einer konzeptionellen Definition stabiler Hochschild Homologie sowie die Beschreibung von Khovanov-Rozansky Homologie als stabile Hochschild Homologie von Rouquier Komplexen von Soergel Bimoduln. Letztere sind aus der Darstellungstheorie wohlbekannt und spielen auch bei der Konstruktion anderer Knoteninvarianten eine zentrale Rolle. Ferner lässt sich das Wissen über die Kombinatorik der Rouquier Komplexe verwenden um einen direkten Nachweis darüber zu erbringen, dass Khovanov-Rozansky Homologie eine Knoteninvariante ist. Anschließend die Entwicklung einer kombinatorischen Approximation an die Khovanov-Rozansky Homologie durch einen diagrammatischen Kalkül ähnlich denjenigen, die zur Definition des Jones-Polynoms und der Khovanov Homologie herangezogen werden können. Die zentralen Ergebnisse des homotopietheoretischen Teils sind die folgenden: Zunächst die Konstruktion abelscher Modellstrukturen auf Kategorien gekrümmter Moduln sowie Singularit atenkategorien, auf Grundlage zuvor bereitgestellter allgemeiner Techniken zur Lokalisierung abelscher Modellstrukturen und zum Nachweis von deren kofasernder Erzeugtheit. Anschließend Diskussion zahlreicher Beispiele und Anreichern klassischer Äquivalenzen und Recollements zwischen triangulierten Kategorien zu Aussagen auf der Ebene der Modellkategorien. Zuletzt die Konstruktion eines Realisierungsfunktors D(A ) → Ho(M) zwischen der derivierten Kategorie einer Grothendieck-Kategorie A und der Homotopiekategorie einer "guten" abelschen Modellstruktur M auf A über die Konstruktion einer zu M äquivalenten Modellstruktur auf Ch(A )

Hybrid scalar/vector implementations of Keccak and SPHINCS+ on AArch64

This paper presents two new techniques for the fast implementation of the Keccak permutation on the A-profile of the Arm architecture: First, the elimination of explicit rotations in the Keccak permutation through Barrel shifting, applicable to scalar AArch64 implementations of Keccak-f1600. Second, the construction of hybrid implementations concurrently leveraging both the scalar and the Neon instruction sets of AArch64. The resulting performance improvements are demonstrated in the example of the hash-based signature scheme SPHINCS+, one of the recently announced winners of the NIST post-quantum cryptography project: We achieve up to 1.89× performance improvements compared to the state of the art. Our implementations target the Arm Cortex-{A55,A510,A78,A710,X1,X2} processors common in client devices such as mobile phones

Fast and Clean: Auditable high-performance assembly via constraint solving

Handwritten assembly is a widely used tool in the development of high-performance cryptography: By providing full control over instruction selection, instruction scheduling, and register allocation, highest performance can be unlocked. On the flip side, developing handwritten assembly is not only time-consuming, but the artifacts produced also tend to be difficult to review and maintain – threatening their suitability for use in practice. In this work, we present SLOTHY (Super (Lazy) Optimization of Tricky Handwritten assemblY), a framework for the automated superoptimization of assembly with respect to instruction scheduling, register allocation, and loop optimization (software pipelining): With SLOTHY, the developer controls and focuses on algorithm and instruction selection, providing a readable “base” implementation in assembly, while SLOTHY automatically finds optimal and traceable instruction scheduling and register allocation strategies with respect to a model of the target (micro)architecture. We demonstrate the flexibility of SLOTHY by instantiating it with models of the Cortex-M55, Cortex-M85, Cortex-A55 and Cortex-A72 microarchitectures, implementing the Armv8.1-M+Helium and AArch64+Neon architectures. We use the resulting tools to optimize three workloads: First, for Cortex-M55 and Cortex-M85, a radix-4 complex Fast Fourier Transform (FFT) in fixed-point and floating-point arithmetic, fundamental in Digital Signal Processing. Second, on Cortex-M55, Cortex-M85, Cortex-A55 and Cortex-A72, the instances of the Number Theoretic Transform (NTT) underlying CRYSTALS-Kyber and CRYSTALS-Dilithium, two recently announced winners of the NIST Post-Quantum Cryptography standardization project. Third, for Cortex-A55, the scalar multiplication for the elliptic curve key exchange X25519. The SLOTHY-optimized code matches or beats the performance of prior art in all cases, while maintaining compactness and readability

Efficient Multiplication of Somewhat Small Integers using Number-Theoretic Transforms

Conventional wisdom purports that FFT-based integer multiplication methods (such as the Schönhage-Strassen algorithm) begin to compete with Karatsuba and Toom-Cook only for integers of several tens of thousands of bits. In this work, we challenge this belief, leveraging recent advances in the implementation of number-theoretic transforms (NTT) stimulated by their use in post-quantum cryptography. We report on implementations of NTT-based integer arithmetic on two Arm Cortex-M CPUs on opposite ends of the performance spectrum: Cortex-M3 and Cortex-M55. Our results indicate that NTT-based multiplication is capable of outperforming the big-number arithmetic implementations of popular embedded cryptography libraries for integers as small as 2048 bits. To provide a realistic case study, we benchmark implementations of the RSA encryption and decryption operations. Our cycle counts on Cortex-M55 are about 10× lower than on Cortex-M3

Identification of novel mutations in X-linked retinitis pigmentosa families and implications for diagnostic testing

Contains fulltext : 69886.pdf (publisher's version ) (Open Access)PURPOSE: The goal of this study was to identify mutations in X-chromosomal genes associated with retinitis pigmentosa (RP) in patients from Germany, The Netherlands, Denmark, and Switzerland. METHODS: In addition to all coding exons of RP2, exons 1 through 15, 9a, ORF15, 15a and 15b of RPGR were screened for mutations. PCR products were amplified from genomic DNA extracted from blood samples and analyzed by direct sequencing. In one family with apparently dominant inheritance of RP, linkage analysis identified an interval on the X chromosome containing RPGR, and mutation screening revealed a pathogenic variant in this gene. Patients of this family were examined clinically and by X-inactivation studies. RESULTS: This study included 141 RP families with possible X-chromosomal inheritance. In total, we identified 46 families with pathogenic sequence alterations in RPGR and RP2, of which 17 mutations have not been described previously. Two of the novel mutations represent the most 3'-terminal pathogenic sequence variants in RPGR and RP2 reported to date. In exon ORF15 of RPGR, we found eight novel and 14 known mutations. All lead to a disruption of open reading frame. Of the families with suggested X-chromosomal inheritance, 35% showed mutations in ORF15. In addition, we found five novel mutations in other exons of RPGR and four in RP2. Deletions in ORF15 of RPGR were identified in three families in which female carriers showed variable manifestation of the phenotype. Furthermore, an ORF15 mutation was found in an RP patient who additionally carries a 6.4 kbp deletion downstream of the coding region of exon ORF15. We did not identify mutations in 39 sporadic male cases from Switzerland. CONCLUSIONS: RPGR mutations were confirmed to be the most frequent cause of RP in families with an X-chromosomal inheritance pattern. We propose a screening strategy to provide molecular diagnostics in these families