Zeta and L-functions of elliptic curves

In questa tesi si studiano alcune proprietà fondamentali delle funzioni Zeta e L associate ad una curva ellittica. In particolare, si dimostra la razionalità della funzione Zeta e l'ipotesi di Riemann per due famiglie specifiche di curve ellittiche. Si studia poi il problema dell'esistenza di un prolungamento analitico al piano complesso della funzione L di una curva ellittica con moltiplicazione complessa, attraverso l'analisi diretta di due casi particolari.

Crittografia basata sull'identità e curve ellittiche

Questa tesi ha lo scopo di fornire una panoramica generale sulle curve ellittiche e il loro utilizzo nella crittografia moderna. L'ultima parte è invece focalizzata a descrivere uno specifico sistema per lo scambio sicuro di messaggi: la crittografia basata sull'identità. Quest'ultima utilizza uno strumento molto interessante, il pairing di Weil, che sarà introdotto nel contesto della teoria dei divisori di funzioni razionali sulle curve ellittiche.

Base pairing and miscoding properties of 1,N(6)-ethenoadenine- and 3,N(4)-ethenocytosine-containing RNA oligonucleotides

Two RNA phosphoramidites containing the bases 1,N(6)-ethenoadenine (εA) and 3,N(4)-ethenocytosine (εC) were synthesized. These building blocks were incorporated into two 12-mer oligoribonucleotides for evaluation of the base pairing properties of these base lesions by UV melting curve (Tm) and circular dichroism measurements. The Tm data of the resulting duplexes with the etheno modifications opposing all natural bases showed a substantial destabilization compared to the corresponding natural duplexes, confirming their inability to form base pairs. The coding properties of these lesions were further investigated by introducing them into 31-mer oligonucleotides and assessing their ability to serve as templates in primer extension reactions with HIV, AMV, and MMLV reverse transcriptases (RT). Primer extension reactions showed complete arrest of the incorporation process using MMLV RT and AMV RT, while HIV RT preferentially incorporates dAMP opposite εA and dAMP as well as dTMP opposite εC. The properties of these RNA lesions are discussed in the context of its putative biological role.

Watson-Crick base-pairing properties of tricyclo-DNA

Tricyclo-DNA belongs to the family of conformationally restricted oligodeoxynucleotide analogues. It differs structurally from DNA by an additional ethylene bridge between the centers C(3') and C(5') of the nucleosides, to which a cyclopropane unit is fused for further enhancement of structural rigidity. The synthesis of the hitherto unknown tricyclodeoxynucleosides containing the bases cytosine and guanine and of the corresponding phosphoramidite building blocks is described, as well as a structural description of a representative of an alpha- and a beta-tricyclodeoxynucleoside by X-ray analysis. Tricyclodeoxynucleoside building blocks of all four bases were used for the synthesis of fully modified mixed-base oligonucleotides. Their Watson-Crick pairing properties with complementary DNA, RNA, and with itself were investigated by UV melting curves, CD spectroscopy, and molecular modeling. Tricyclo-DNA was found to be a very stable Watson-Crick base-pairing system. A UV melting curve analysis of the decamers tcd(pcgtgacagtt) and tcd(paactgtcacg) showed increased thermal stabilities of up to DeltaT(m)/mod. = +1.2 degrees C with complementary DNA and +2.4 degrees C with complementary RNA. With itself, tricyclo-DNA showed an increase in stability of +3.1 degrees C/base pair relative to DNA. Investigations into the thermodynamic properties of these decamers revealed an entropic stabilization and an enthalpic destabilization for the tricyclo-DNA/DNA duplexes. CD spectroscopic structural investigations indicated that tricyclo-DNA containing duplexes preferrably exist in an A-conformation, a fact which is in agreement with results from molecular modeling

hp-DGFEM for second-order mixed elliptic problems polyhedra

We prove exponential rates of convergence of hp-version discontinuous Galerkin (dG) interior penalty finite element methods for second-order elliptic problems with mixed Dirichlet-Neumann boundary conditions in axiparallel polyhedra. The dG discretizations are based on axiparallel, σ-geometric anisotropic meshes of mapped hexahedra and anisotropic polynomial degree distributions of μ-bounded variation. We consider piecewise analytic solutions which belong to a larger analytic class than those for the pure Dirichlet problem considered in [11, 12]. For such solutions, we establish the exponential convergence of a nonconforming dG interpolant given by local L 2 -projections on elements away from corners and edges, and by suitable local low-order quasi-interpolants on elements at corners and edges. Due to the appearance of non-homogeneous, weighted norms in the analytic regularity class, new arguments are introduced to bound the dG consistency errors in elements abutting on Neumann edges. The non-homogeneous norms also entail some crucial modifications of the stability and quasi-optimality proofs, as well as of the analysis for the anisotropic interpolation operators. The exponential convergence bounds for the dG interpolant constructed in this paper generalize the results of [11, 12] for the pure Dirichlet case.

Optical digital chaos cryptography

In this work we present a new way to mask the data in a one-user communication system when direct sequence - code division multiple access (DS-CDMA) techniques are used. The code is generated by a digital chaotic generator, originally proposed by us and previously reported for a chaos cryptographic system. It is demonstrated that if the user's data signal is encoded with a bipolar phase-shift keying (BPSK) technique, usual in DS-CDMA, it can be easily recovered from a time-frequency domain representation. To avoid this situation, a new system is presented in which a previous dispersive stage is applied to the data signal. A time-frequency domain analysis is performed, and the devices required at the transmitter and receiver end, both user-independent, are presented for the optical domain.

Approximate Relational Reasoning for Probabilistic Programs

La seguridad verificada es una metodología para demostrar propiedades de seguridad de los sistemas informáticos que se destaca por las altas garantías de corrección que provee. Los sistemas informáticos se modelan como programas probabilísticos y para probar que verifican una determinada propiedad de seguridad se utilizan técnicas rigurosas basadas en modelos matemáticos de los programas. En particular, la seguridad verificada promueve el uso de demostradores de teoremas interactivos o automáticos para construir demostraciones completamente formales cuya corrección es certificada mecánicamente (por ordenador). La seguridad verificada demostró ser una técnica muy efectiva para razonar sobre diversas nociones de seguridad en el área de criptografía. Sin embargo, no ha podido cubrir un importante conjunto de nociones de seguridad “aproximada”. La característica distintiva de estas nociones de seguridad es que se expresan como una condición de “similitud” entre las distribuciones de salida de dos programas probabilísticos y esta similitud se cuantifica usando alguna noción de distancia entre distribuciones de probabilidad. Este conjunto incluye destacadas nociones de seguridad de diversas áreas como la minería de datos privados, el análisis de flujo de información y la criptografía. Ejemplos representativos de estas nociones de seguridad son la indiferenciabilidad, que permite reemplazar un componente idealizado de un sistema por una implementación concreta (sin alterar significativamente sus propiedades de seguridad), o la privacidad diferencial, una noción de privacidad que ha recibido mucha atención en los últimos años y tiene como objetivo evitar la publicación datos confidenciales en la minería de datos. La falta de técnicas rigurosas que permitan verificar formalmente este tipo de propiedades constituye un notable problema abierto que tiene que ser abordado. En esta tesis introducimos varias lógicas de programa quantitativas para razonar sobre esta clase de propiedades de seguridad. Nuestra principal contribución teórica es una versión quantitativa de una lógica de Hoare relacional para programas probabilísticos. Las pruebas de correción de estas lógicas son completamente formalizadas en el asistente de pruebas Coq. Desarrollamos, además, una herramienta para razonar sobre propiedades de programas a través de estas lógicas extendiendo CertiCrypt, un framework para verificar pruebas de criptografía en Coq. Confirmamos la efectividad y aplicabilidad de nuestra metodología construyendo pruebas certificadas por ordendor de varios sistemas cuyo análisis estaba fuera del alcance de la seguridad verificada. Esto incluye, entre otros, una meta-construcción para diseñar funciones de hash “seguras” sobre curvas elípticas y algoritmos diferencialmente privados para varios problemas de optimización combinatoria de la literatura reciente. ABSTRACT The verified security methodology is an emerging approach to build high assurance proofs about security properties of computer systems. Computer systems are modeled as probabilistic programs and one relies on rigorous program semantics techniques to prove that they comply with a given security goal. In particular, it advocates the use of interactive theorem provers or automated provers to build fully formal machine-checked versions of these security proofs. The verified security methodology has proved successful in modeling and reasoning about several standard security notions in the area of cryptography. However, it has fallen short of covering an important class of approximate, quantitative security notions. The distinguishing characteristic of this class of security notions is that they are stated as a “similarity” condition between the output distributions of two probabilistic programs, and this similarity is quantified using some notion of distance between probability distributions. This class comprises prominent security notions from multiple areas such as private data analysis, information flow analysis and cryptography. These include, for instance, indifferentiability, which enables securely replacing an idealized component of system with a concrete implementation, and differential privacy, a notion of privacy-preserving data mining that has received a great deal of attention in the last few years. The lack of rigorous techniques for verifying these properties is thus an important problem that needs to be addressed. In this dissertation we introduce several quantitative program logics to reason about this class of security notions. Our main theoretical contribution is, in particular, a quantitative variant of a full-fledged relational Hoare logic for probabilistic programs. The soundness of these logics is fully formalized in the Coq proof-assistant and tool support is also available through an extension of CertiCrypt, a framework to verify cryptographic proofs in Coq. We validate the applicability of our approach by building fully machine-checked proofs for several systems that were out of the reach of the verified security methodology. These comprise, among others, a construction to build “safe” hash functions into elliptic curves and differentially private algorithms for several combinatorial optimization problems from the recent literature.

Accurate Parabolic Navier-Stokes solutions of the supersonic flow around and elliptic cone

Flows of relevance to new generation aerospace vehicles exist, which are weakly dependent on the streamwise direction and strongly dependent on the other two spatial directions, such as the flow around the (flattened) nose of the vehicle and the associated elliptic cone model. Exploiting these characteristics, a parabolic integration of the Navier-Stokes equations is more appropriate than solution of the full equations, resulting in the so-called Parabolic Navier-Stokes (PNS). This approach not only is the best candidate, in terms of computational efficiency and accuracy, for the computation of steady base flows with the appointed properties, but also permits performing instability analysis and laminar-turbulent transition studies a-posteriori to the base flow computation. This is to be contrasted with the alternative approach of using order-of-magnitude more expensive spatial Direct Numerical Simulations (DNS) for the description of the transition process. The PNS equations used here have been formulated for an arbitrary coordinate transformation and the spatial discretization is performed using a novel stable high-order finite-difference-based numerical scheme, ensuring the recovery of highly accurate solutions using modest computing resources. For verification purposes, the boundary layer solution around a circular cone at zero angle of attack is compared in the incompressible limit with theoretical profiles. Also, the recovered shock wave angle at supersonic conditions is compared with theoretical predictions in the same circular-base cone geometry. Finally, the entire flow field, including shock position and compressible boundary layer around a 2:1 elliptic cone is recovered at Mach numbers 3 and 4

Vector Boolean Functions: Applications in Symmetric Cryptography

Esta tesis establece los fundamentos teóricos y diseña una colección abierta de clases C++ denominada VBF (Vector Boolean Functions) para analizar funciones booleanas vectoriales (funciones que asocian un vector booleano a otro vector booleano) desde una perspectiva criptográfica. Esta nueva implementación emplea la librería NTL de Victor Shoup, incorporando nuevos módulos que complementan a las funciones de NTL, adecuándolas para el análisis criptográfico. La clase fundamental que representa una función booleana vectorial se puede inicializar de manera muy flexible mediante diferentes estructuras de datas tales como la Tabla de verdad, la Representación de traza y la Forma algebraica normal entre otras. De esta manera VBF permite evaluar los criterios criptográficos más relevantes de los algoritmos de cifra en bloque y de stream, así como funciones hash: por ejemplo, proporciona la no-linealidad, la distancia lineal, el grado algebraico, las estructuras lineales, la distribución de frecuencias de los valores absolutos del espectro Walsh o del espectro de autocorrelación, entre otros criterios. Adicionalmente, VBF puede llevar a cabo operaciones entre funciones booleanas vectoriales tales como la comprobación de igualdad, la composición, la inversión, la suma, la suma directa, el bricklayering (aplicación paralela de funciones booleanas vectoriales como la empleada en el algoritmo de cifra Rijndael), y la adición de funciones coordenada. La tesis también muestra el empleo de la librería VBF en dos aplicaciones prácticas. Por un lado, se han analizado las características más relevantes de los sistemas de cifra en bloque. Por otro lado, combinando VBF con algoritmos de optimización, se han diseñado funciones booleanas cuyas propiedades criptográficas son las mejores conocidas hasta la fecha. ABSTRACT This thesis develops the theoretical foundations and designs an open collection of C++ classes, called VBF, designed for analyzing vector Boolean functions (functions that map a Boolean vector to another Boolean vector) from a cryptographic perspective. This new implementation uses the NTL library from Victor Shoup, adding new modules which complement the existing ones making VBF better suited for cryptography. The fundamental class representing a vector Boolean function can be initialized in a flexible way via several alternative types of data structures such as Truth Table, Trace Representation, Algebraic Normal Form (ANF) among others. This way, VBF allows the evaluation of the most relevant cryptographic criteria for block and stream ciphers as well as for hash functions: for instance, it provides the nonlinearity, the linearity distance, the algebraic degree, the linear structures, the frequency distribution of the absolute values of the Walsh Spectrum or the Autocorrelation Spectrum, among others. In addition, VBF can perform operations such as equality testing, composition, inversion, sum, direct sum, bricklayering (parallel application of vector Boolean functions as employed in Rijndael cipher), and adding coordinate functions of two vector Boolean functions. This thesis also illustrates the use of VBF in two practical applications. On the one hand, the most relevant properties of the existing block ciphers have been analysed. On the other hand, by combining VBF with optimization algorithms, new Boolean functions have been designed which have the best known cryptographic properties up-to-date.

Cryptographic techniques for personal communication systems security

The advent of personal communication systems within the last decade has depended upon the utilization of advanced digital schemes for source and channel coding and for modulation. The inherent digital nature of the communications processing has allowed the convenient incorporation of cryptographic techniques to implement security in these communications systems. There are various security requirements, of both the service provider and the mobile subscriber, which may be provided for in a personal communications system. Such security provisions include the privacy of user data, the authentication of communicating parties, the provision for data integrity, and the provision for both location confidentiality and party anonymity. This thesis is concerned with an investigation of the private-key and public-key cryptographic techniques pertinent to the security requirements of personal communication systems and an analysis of the security provisions of Second-Generation personal communication systems is presented. Particular attention has been paid to the properties of the cryptographic protocols which have been employed in current Second-Generation systems. It has been found that certain security-related protocols implemented in the Second-Generation systems have specific weaknesses. A theoretical evaluation of these protocols has been performed using formal analysis techniques and certain assumptions made during the development of the systems are shown to contribute to the security weaknesses. Various attack scenarios which exploit these protocol weaknesses are presented. The Fiat-Sharmir zero-knowledge cryptosystem is presented as an example of how asymmetric algorithm cryptography may be employed as part of an improved security solution. Various modifications to this cryptosystem have been evaluated and their critical parameters are shown to be capable of being optimized to suit a particular applications. The implementation of such a system using current smart card technology has been evaluated.

Developments towards practical free-space quantum cryptography

We describe a free space quantum cryptography system which is designed to allow continuous unattended key exchanges for periods of several days, and over ranges of a few kilometres. The system uses a four-laser faint-pulse transmission system running at a pulse rate of 10MHz to generate the required four alternative polarization states. The receiver module similarly automatically selects a measurement basis and performs polarization measurements with four avalanche photodiodes. The controlling software can implement the full key exchange including sifting, error correction, and privacy amplification required to generate a secure key.

Improved Cryptoanalysis of the Self-shrinking P-adic Cryptographic Generator

The Self-shrinking p-adic cryptographic generator (SSPCG) is a fast software stream cipher. Improved cryptoanalysis of the SSPCG is introduced. This cryptoanalysis makes more precise the length of the period of the generator. The linear complexity and the cryptography resistance against most recently used attacks are invesigated. Then we discuss how such attacks can be avoided. The results show that the sequence generated by a SSPCG has a large period, large linear complexity and is stable against the cryptographic attacks. This gives the reason to consider the SSPSG as suitable for critical cryptographic applications in stream cipher encryption algorithms.

Cryptographic Primitives with Quasigroup Transformations

AMS Subj. Classification: Primary 20N05, Secondary 94A60

Ribosomal frameshifting and transcriptional slippage: from genetic steganography and cryptography to adventitious use

Genetic decoding is not ‘frozen’ as was earlier thought, but dynamic. One facet of this is frameshifting that often results in synthesis of a C-terminal region encoded by a new frame. Ribosomal frameshifting is utilized for the synthesis of additional products, for regulatory purposes and for translational ‘correction’ of problem or ‘savior’ indels. Utilization for synthesis of additional products occurs prominently in the decoding of mobile chromosomal element and viral genomes. One class of regulatory frameshifting of stable chromosomal genes governs cellular polyamine levels from yeasts to humans. In many cases of productively utilized frameshifting, the proportion of ribosomes that frameshift at a shift-prone site is enhanced by specific nascent peptide or mRNA context features. Such mRNA signals, which can be 5′ or 3′ of the shift site or both, can act by pairing with ribosomal RNA or as stem loops or pseudoknots even with one component being 4 kb 3′ from the shift site. Transcriptional realignment at slippage-prone sequences also generates productively utilized products encoded trans-frame with respect to the genomic sequence. This too can be enhanced by nucleic acid structure. Together with dynamic codon redefinition, frameshifting is one of the forms of recoding that enriches gene expression.