In the paper we study sequences of random functions which are defined by some interpolation procedures for a given random function. We investigate the problem in what sense and under which conditions the sequences converge to the prescribed random function. Sufficient conditions for convergence of moment characteristics, of finite dimensional distributions and for weak convergence of distributions in spaces of continuous functions are given. The treatment of such questions is stimulated by an investigation of Monte Carlo simulation procedures for certain classes of random functions. In an appendix basic facts concerning weak convergence of probability measures in metric spaces are summarized.

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In the paper we study sequences of random functions which are defined by some interpolation procedures for a given random function. We investigate the problem in what sense and under which conditions the sequences converge to the prescribed random function. Sufficient conditions for convergence of moment characteristics, of finite dimensional distributions and for weak convergence of distributions in spaces of continuous functions are given. The treatment of such questions is stimulated by an investigation of Monte Carlo simulation procedures for certain classes of random functions. In an appendix basic facts concerning weak convergence of probability measures in metric spaces are summarized.

BFT consensus algorithms can ensure the consistency of distributed systems where nodes may behave arbitrarily due to faults or intentional malicious actions. However, most of the practical BFT consensus algorithms are leader-based. In an adversarial network, leader-based BFT consensus algorithms exhibit vulnerabilities and lack resilience. Byzantine leaders can pose a potential threat to the system; firstly, malicious leaders can actively downgrade the processing speed of handling proposals, thereby diminishing the system’s overall performance. Secondly, they can determine the submission order of received requests, which can be fatal in specific decentralized financial systems. Additionally, external attackers can compromise the system’s stability by conducting DDoS attacks on leader nodes, frequently triggering view changes and potentially causing the system to lose liveness altogether. We present BFT Baxos, a more robust and resilient BFT consensus protocol that equips a BFT random exponential backoff mechanism to ensure each node has the egalitarian right to propose. Employing random exponential backoff as a replacement for leader election eliminates the potential malicious actions of Byzantine leaders and prevents external attackers from conducting targeted DDoS attacks on the leader node within systems. We implemented and evaluated our BFT Baxos prototype. Our results indicate that BFT Baxos exhibits good performance and scalability in low-concurrency scenarios. Additionally, we illustrated the functioning of BFT Baxos even in extremely adverse network conditions by subjecting it to random DDoS attacks. / BFT-konsensusalgoritmer är utformade för att säkerställa konsistensen i distribuerade system där noder kan agera godtyckligt, antingen på grund av fel eller avsiktliga skadliga handlingar. Dock är de flesta praktiska BFT-konsensusalgoritmerna baserade på ledare. I en fientlig nätverksmiljö uppvisar ledar-baserade BFT-konsensusalgoritmer sårbarheter och brist på motståndskraft. Bysantinska ledare kan utgöra en potentiell hot mot systemet; för det första kan skadliga ledare aktivt sänka behandlingshastigheten för hantering av förslag och därigenom minska systemets totala prestanda. För det andra kan de bestämma ordningen för inskickning av mottagna begäranden, vilket kan vara ödesdigert i vissa decentraliserade finansiella system. Dessutom kan externa angripare kompromettera systemets stabilitet genom att genomföra DDoS-attacker mot ledarnoder, vilket ofta utlöser vynändringar och potentiellt orsakar att systemet förlorar livskraft helt och hållet. Vi presenterar BFT Baxos, en mer robust och motståndskraftig BFT-konsensusprotokoll som utrustar en BFT slumpmässig exponentiell backoff-mekanism för att säkerställa att varje nod har rätten att föreslå på ett egalitärt sätt. Genom att använda slumpmässig exponentiell backoff som ett alternativ till ledarval eliminerar det inte bara möjliga skadliga handlingar från bysantinska ledare utan förhindrar även externa angripare från att genomföra riktade DDoS-attacker mot ledarnoden inom system. Vi implementerade och utvärderade vår BFT Baxos-prototyp. Våra resultat visar att BFT Baxos uppvisar god prestanda och skalbarhet i scenarier med låg samtidighet. Dessutom illustrerade vi funktionen av BFT Baxos även under extremt ogynnsamma nätverksförhållanden genom att utsätta den för slumpmässiga DDoS-attacker.

L’aléatoire est un ingrédient clé en cryptographie. Par exemple, les nombres aléatoires sont utilisés pour générer des clés, pour le chiffrement et pour produire des nonces. Ces nombres sont générés par des générateurs pseudo-aléatoires et des fonctions pseudo-aléatoires dont les constructions sont basées sur des problèmes qui sont supposés difficiles. Dans cette thèse, nous étudions certaines mesures de complexité des fonctions pseudo-aléatoires de Naor-Reingold et Dodis-Yampolskiy et étudions la sécurité de certains générateurs pseudo-aléatoires (le générateur linéaire congruentiel et le générateur puissance basés sur les courbes elliptiques) et de certaines signatures à base de couplage basées sur le paradigme d’inversion. Nous montrons que la fonction pseudo-aléatoire de Dodis-Yampolskiy est uniformément distribué et qu’un polynôme multivarié de petit dégré ou de petit poids ne peut pas interpoler les fonctions pseudo-aléatoires de Naor-Reingold et de Dodis-Yampolskiy définies sur un corps fini ou une courbe elliptique. Le contraire serait désastreux car un tel polynôme casserait la sécurité de ces fonctions et des problèmes sur lesquels elles sont basées. Nous montrons aussi que le générateur linéaire congruentiel et le générateur puissance basés sur les courbes elliptiques sont prédictibles si trop de bits sont sortis à chaque itération. Les implémentations pratiques de cryptosystèmes souffrent souvent de fuites critiques d’informations à travers des attaques par canaux cachés. Ceci peut être le cas lors du calcul de l’exponentiation afin de calculer la sortie de la fonction pseudo-aléatoire de Dodis-Yampolskiy et plus généralement le calcul des signatures dans certains schémas de signatures bien connus à base de couplage (signatures de Sakai-Kasahara, Boneh-Boyen et Gentry) basées sur le paradigme d’inversion. Nous présentons des algorithmes (heuristiques) en temps polynomial à base des réseaux qui retrouvent le secret de celui qui signe le message dans ces trois schémas de signatures lorsque plusieurs messages sont signés sous l’hypothèse que des blocs consécutifs de bits des exposants sont connus de l’adversaire. / Randomness is a key ingredient in cryptography. For instance, random numbers are used to generate keys, for encryption and to produce nonces. They are generated by pseudo-random generators and pseudorandom functions whose constructions are based on problems which are assumed to be difficult. In this thesis, we study some complexity measures of the Naor-Reingold and Dodis-Yampolskiy pseudorandom functions and study the security of some pseudo-random generators (the linear congruential generator and the power generator on elliptic curves) and some pairing-based signatures based on exponentinversion framework. We show that the Dodis-Yampolskiy pseudo-random functions is uniformly distributed and that a lowdegree or low-weight multivariate polynomial cannot interpolate the Naor-Reingold and Dodis-Yampolskiy pseudo-random functions over finite fields and over elliptic curves. The contrary would be disastrous since it would break the security of these functions and of problems on which they are based. We also show that the linear congruential generator and the power generator on elliptic curves are insecure if too many bits are output at each iteration. Practical implementations of cryptosystems often suffer from critical information leakage through sidechannels. This can be the case when computing the exponentiation in order to compute the output of the Dodis-Yampolskiy pseudo-random function and more generally in well-known pairing-based signatures (Sakai-Kasahara signatures, Boneh-Boyen signatures and Gentry signatures) based on the exponent-inversion framework. We present lattice based polynomial-time (heuristic) algorithms that recover the signer’s secret in the pairing-based signatures when used to sign several messages under the assumption that blocks of consecutive bits of the exponents are known by the attacker.

Joseph Kampé de Fériet (1893–1982) est un mathématicien lillois, spécialiste international en mécanique des fluides et directeur de l'Institut de mécanique des fluides de Lille (IMFL) depuis sa création en 1929. En se familiarisant avec ce domaine et avec les questions expérimentales grâce à ses travaux de balistique pendant sa mobilisation scientifique à la Commission de Gâvre (1915-1919), ce savant a joué un triple rôle à l'institut. En tant que mathématicien, il a donné une contribution remarquable à la théorie statistique de la turbulence de Taylor-von Kármán à l'aide de la théorie des fonctions aléatoires de Kolmogorov, Khintchine, et Slutsky. En tant qu'expérimentateur, il a participé aux travaux expérimentaux de l'IMFL visant d'une part à étudier la turbulence atmosphérique et d’autre part à légitimer les idées de l'école de Philippe Wehrlé et Georges Dedebant, une école qui s'est constituée au sein de la Commission de la Turbulence Atmosphérique, créée par le ministère de l'Air en 1935. Enfin, en tant que directeur, il a valorisé les liens avec l'industrie et la société lilloise comme il a valorisé ses liens avec les officiers militaires pendant son expérience à Gâvre. Dans notre thèse, nous utiliserons le parcours scientifique et institutionnel de J. Kampé de Fériet - de sa mobilisation à Gâvre (1915) à l’année de sa démission de la direction de l’IMFL (1945) - en tant que prisme pour répondre à des questions plus générales concernant la mécanique des fluides en France pendant la première moitié du XXe siècle, dont certaines, mais pas toutes, apportent des éléments nouveaux qui sont communs à la balistique et aux autres domaines des mathématiques appliquées. / Joseph Kampé de Fériet (1893-1982), a French mathematician of Lille, was an international specialist in fluid mechanics and was director of the Institut de mécanique des fluides de Lille (IMFL) from its creation in 1929. By familiarizing himself with this field and by addressing questions of an experimental nature through his work on ballistics, during his scientific wartime service to the Gâvre Commission (1915-1919), this scientist played a triple role in the institute. As a mathematician, he made a remarkable contribution to Taylor-von Kármán's statistical theory of turbulence using the theory of random functions due to Kolmogorov, Khintchine, and Slutsky. As an experimental scientist, he took part in the experimental work of the IMFL aiming on one hand to study atmospheric turbulence and, on the other hand, to validate the ideas of the school of Philippe Wehrle and Georges Dedebant. This school was formed within the Atmospheric Turbulence Commission, created by the Minister of Air in 1935. Finally, as director of the institute, he strengthened links with industry and society in Lille, in the same way that he reinforced links with military officers during his work in Gâvre.In our thesis, we will use the scientific and institutional career path of J. Kampé de Fériet – from his service at Gâvre (1915) up until the year of his resignation as director of the IMFL (1945) - as a prism by which we will answer further questions of a more general nature regarding fluid mechanics in France during the first half of the twentieth century. Some but not all of these considerations bring to light new elements that are common to ballistics and to other areas of applied mathematics.

Thesis deals with the spectral analysis of 3D surface topography. The surface is described by a random function. Theoretical starting points necessary for right introduction and understanding of basic notions used within the framework of the surface topography spectral analysis are exactly formulated. They lie in the theory of random functions, the theory of the Fourier transform and the theory of signal processing. The notions mentioned are: the areal power spectral density (APSD) of a surface, the radial power spectral density (RPSD) of a surface and the angular power spectrum density (AnPSD) of a surface. These notions are introduced in their discrete form and generalized for the two-dimensional case. The thorough discussion of possible mistakes and inaccuracies which can be done during the application of spectral analysis techniques in a surface topography evaluation is performed. The procedure of APSD estimation by means of the periodogram method combined with the Welch´s method is discussed. The principle and capabilities of the optical profilometer MicroProf?FRT used for the surface topography measurement are described. Our original computer program computing APSD, RPSD and AnPSD is described too. The 3D spectral analyses is applied to surfaces generated by AWJ cutting, plane grinding and casting. We have focused our attention to AWJ cut surfaces, 3D spectral analyses of which brings new still unpublished opportunities of the surfaces topography evaluation. The influence of technological parameters on these AWJ cut surfaces topography is studied. The conclusion of the study is that results of the spectral analyses of these surfaces topography strongly depend on the technological conditions of the surfaces generation. An original procedure of the ASPD shape evaluation within individual regions of its frequency domain is mentioned. This procedure brings new substantial knowledge concerning the topography of surfaces. Results obtained from surfaces generated by plane grinding and casting are presented like examples of results from non-isotropic and isotropic surfaces.