The modularity theorem implies that for every elliptic curve E /Q there exist rational maps from the modular curve X_0(N) to E, where N is the conductor of E. These maps may be expressed in terms of pairs of modular functions X(z) and Y(z) that satisfy the Weierstrass equation for E as well as a certain differential equation. Using these two relations, a recursive algorithm can be constructed to calculate the q - expansions of these parameterizations at any cusp. These functions are algebraic over Q(j(z)) and satisfy modular polynomials where each of the coefficient functions are rational functions in j(z). Using these functions, we determine the divisor of the parameterization and the preimage of rational points on E. We give a sufficient condition for when these preimages correspond to CM points on X_0(N). We also examine a connection between the algebras generated by these functions for related elliptic curves, and describe sufficient conditions to determine congruences in the q-expansions of these objects.

<pre>This dissertation concerns two questions involving the injectivity of specialization homomorphisms for elliptic surfaces. We primarily focus on elliptic surfaces over the projective line defined over the rational numbers. The specialization theorem of Silverman proven in 1983 says that, for a fixed surface, all but finitely many specialization homomorphisms are injective. Given a subgroup of the group of rational sections with explicit generators, we thus ask the following.</pre><pre>Given some rational number, how can we effectively determine whether or not the associated specialization map is injective?</pre><pre>What is the set of rational numbers such that the corresponding specialization maps are injective?</pre><pre>The classical specialization theorem of Neron proves that there is a set S which differs from a Hilbert subset of the rational numbers by finitely many elements such that for each number in S the associated specialization map is injective. We expand this into an effective procedure that determines if some rational number is in S, yielding a partial answer to question 1. Computing the Hilbert set provides a partial answer to question 2, and we carry this out for some examples. We additionally expand an effective criterion of Gusic and Tadic to include elliptic surfaces with a rational 2-torsion curve.<br></pre>

Let E/ℚ be an elliptic curve. Silverman and Stange define primes p and q to be an elliptic, amicable pair if #E(Fp) = q and #E(Fq) = p. More generally, they define the notion of aliquot cycles for elliptic curves. Here, we study the same notion in the case that the elliptic curve is defined over a number field K. We focus on proving the existence of an elliptic curve E/K with aliquot cycle (p1,⋯, pn) where the pi are primes of K satisfying mild conditions.

The thesis deals with arithmetics of elliptic curves over finite fields and methods to improve those calculations. In the first part, algebraic geometry helps to define elliptic curves and derive their basic properties including the group law. The second chapter seeks ways to speed up these calculations by means of time-memory tradeoff, i.e. adding redundancy. At last, the third part introduces a wholly new curve form, which is particularly effective for such purposes.

The thesis deals with arithmetics of elliptic curves over finite fields and methods to improve those calculations. In the first part, algebraic geometry helps to define elliptic curves and derive their basic properties including the group law. The second chapter seeks ways to speed up these calculations by means of time-memory tradeoff, i.e. adding redundancy. At last, the third part introduces a wholly new curve form, which is particularly effective for such purposes.

This thesis deals with creation of elliptic curves generator. MIRACL library and C++ language are used. One of important issues is to determine the order of the elliptic curve group. SEA algorithm (Schoof–Elkies–Atkin) is used for point counting on the elliptic curve. Method with this algorithm is called as counting points method, SEA method etc. Next method is CM method. Both methods are available in the generator. The measurements of dependency of basic operations speed on the group size and parameters were done. ECIES hybrid scheme was implemented. It is practical verification of proper functionality of the generator. Another benchmarks measured dependency of ECIES encryption and decryption on various parameters, e.g. size of the curve, generating method, message size etc.

The aim of the thesis is to desribe and implement the elliptic curve factorization method using curves in Edwards form. The thesis can be notionally divided into two parts. The first part deals with the theory of Edwards curves especially with properties of elliptic function fields. The second part deals with the factorization algorithm using Edwards form both formally and practically in the way the algorithm is really implemented. The contribution of this thesis is the enclosed implementation of the elliptic curve factorisation algorithm which can be run on a graphic card and which is faster than the state-of-the-art implementation GMP-ECM. Powered by TCPDF (www.tcpdf.org)

This thesis describes the theory behind elliptic-curve Diffie-Hellman key exchanges. All the way from the definition of a group until how the operator over an elliptic curve forms an abelian group. This is illustrated with clear examples. After that a smaller study is made to determine if there is a connection betweenthe size of the underlying field, the amount of points on the curve and the order of the points to determine how hard it is to find out the secret key in elliptic-curve Diffie-Hellman key exchanges. No clear connection is found. Since elliptic curves over extension fields have more computational heavy operations, it is concluded that these curves serve no practical use in elliptic-curve Diffie-Hellman key exchange. / Denna rapport går igenom teorin bakom Diffie-Hellmans nyckelutbyte över elliptiska kurvor. Från definitionen av en grupp hela vägen till hur operatorn över en elliptisk kurva utgör en abelsk grupp gås igenom och görs tydligt med konstruktiva exempel. Sedan görs en mindre undersökning av sambandet mellan storleken av den underliggande kroppen, antal punkter på kurvan och ordning av punkterna på kurvan, det vill säga svårigheten att hitta den hemliga nyckeln framtagen med Diffie-Hellmans nyckelutbyte för elliptiska kurvor. Ingen tydlig koppling hittas. Då elliptiska kurvor över utvidgade kroppar har mer beräkningstunga operationer dras slutsatsen att dessa kurvor inte är praktiska inom Diffie-Hellman nyckelutbyte över elliptiska kurvor.

Sistemas criptográficos como o RSA e o Diffie-Hellman sobre Curvas Elípticas (DHCE) têm fundamento em problemas computacionais considerados difíceis, por exemplo, o problema do logaritmo (PLD) e o problema da fatoração de inteiros (PFI). Diversos trabalhos têm relacionado a segurança desses sistemas com os problemas subjacentes. Também é investigada a segurança do LSB (bit menos significativo) da chave secreta no DHCE (no RSA é o LSB da mensagem) com relação à segurança de toda a chave. Nesses trabalhos são apresentados algoritmos que conseguem inverter os sistemas criptográficos citados fazendo uso de oráculos que predizem o LSB. Nesta dissertação, fazemos a implementação de dois desses algoritmos. Identificamos parâmetros críticos e mudamos a amostragem do formato original. Com essa mudança na amostragem conseguimos uma melhora significativa nos tempos de execução. Um dos algoritmos (ACGS), para valores práticos do RSA, era mais lento que a solução para o PFI, com nosso resultado passou a ser mais veloz. Ainda, mostramos como provas teóricas podem não definir de maneira precisa o tempo de execução de um algoritmo. / Cryptographic systems like RSA and Elliptic Curve Diffie-Hellman (DHCE) is based on computational problems that are considered hard, e.g. the discrete logarithm (PLD) and integer factorization (PFI) problems. Many papers investigated the relationship between the security of these systems to the computational difficulty of the underlying problems. Moreover, they relate the bit security, actually the LSB (Least Significant Bit), of the secret key in the DHCE and the LSB of the message in the RSA, to the security of the whole key. In these papers, algorithms are presented to invert these cryptographic systems making use of oracles that predict the LSB. In this dissertation we implement two of them. Critical parameters are identified and the original sampling is changed. With the modified sampling we achieve an improvement in the execution times. For practical values of the RSA, the algorithm ACGS becomes faster than the PFI. Moreover, we show how theoretical proofs may lead to inaccurate timing estimates.

Actuellement, la technologie RFID (Radio Frequency Identification) est utilisée dans plusieurs domaines d'applications allant de l'identification dans les chaines d'approvisionnement à l'authentification dans les applications les plus sensibles telles que: les titres de transport, la médicine, les systèmes de surveillance, les cartes de crédit ou encore le passeport biométrique. Cependant, la nature sans-fil des données échangées rend cette technologie vulnérable à un certain nombre d'attaques et de nouvelles menaces. Ceci, engendre deux principaux problèmes à savoir; celui lié à la sécurité des informations échangées et celui lié à l'atteinte à la vie privée du propriétaire de l'étiquette. Par conséquent, cette technologie nécessite l'emploi de mécanismes de sécurité pour lutter contre tout type d'attaque et menace, ce qui peut se donner par le service authentification. Néanmoins, il se trouve que les étiquettes RFID imposent de fortes contraintes en termes de ressources matérielles telles que le temps de calcul, l'espace de stockage et l'énergie consommée et communication, ainsi, les primitives cryptographiques classiques telles que l'AES (Advanced Encryption Standard), le RSA (Rivest, Shamir and Adleman), etc. ne peuvent plus être employées. C'est pourquoi, dans la plupart des applications RFID, que nous retrouvons sur le marché, adoptent la cryptographie légère ou ultralégère (à faible coût) qui s'avère la principale solution pour résoudre le problème de capacité limitée, mais son limitation réside dans son niveau de sécurité. Ainsi, avec les moyens de calcul que nous disposons aujourd'hui, ces systèmes deviennent de plus en plus vulnérables à un nombre important d'attaques, d'où la nécessité de chercher des primitives cryptographiques qui soient à la fois robustes et sûres envers tout type d'attaques et en conformités avec contraintes imposées par les étiquettes RFID. Par conséquent, l'étude et l'exploitation des applications RFID est d'intérêt primordial afin de mieux comprendre et maitriser les menaces et les risques de cette technologie. A travers les travaux de recherche présentés dans cette thèse, nous nous sommes inscrits dans cette compétition qui consiste à chercher des solutions permettant de résoudre les problèmes liés à la sécurité de systèmes RFID, allant de l'authentification ultralégère à celle adoptant les courbes elliptiques (ECC) pour les étiquettes actives ou encore à la génération de clés de chiffrement pour les crypto-systèmes ECC. Parmi les tâches développées au cours de ces travaux de thèse, nous avons proposé de nouveaux protocoles d'authentification RFID en utilisant les concepts des courbes elliptiques qui présentent plus d'efficacité, de sécurité et de robustesse. Dans part, nous avons cryptanalyé, développé et proposé des protocoles d'authentification légers et ultralégers convenables aux étiquettes à faible coût. Plus loin, une autre contribution importante qui rentre dans le cadre de la génération aléatoire des clés de chiffrement, nous avons proposé un nouveau générateur pseudo aléatoire (PRNG) construit à base de plusieurs courbes elliptiques auto-sélectionnées convenable aux applications de type sécurisation des systèmes embarqués, la sécurité et simulations informatiques ou plus encore pour les systèmes miniaturisés à base des ECC tels que les cartes à puces et les étiquettes RFID. / The deployment and use of radio-frequency identification (RFID) technology is growing rapidly in different aspects of our daily life. This technology is used not only in traditional applications such as access control and container identification but also in security services such as in biometric passports, medicine, RFID-embedded cards. However, the main drawback of exchanging data wirelessly is the security issue. These systems are especially vulnerable to different attacks such as, eavesdropping attack, tracking attack, active attacks. For these reasons, the security and privacy of the RFID systems are to be addressed seriously and considered as a crucial matter before deploying this technology. These security mechanisms may be given by the authentication service. However, it turns out that RFID tags impose challenging constraints in terms of storage requirements, computing power, bandwidth and computational cost, thus, it is hard for them to implement or to adapt the existing custom cryptographic primitives and protocols or modern ciphers, such as AES (Advanced Encryption standard), RSA (Rivest, Shamir and Adleman), etc., which require a huge computational workload and storage space. Hence only lightweight cryptographic primitives can be implemented. Therefore, with the development of the calculation means, these systems are becoming increasingly vulnerable to a significant number of attacks. Consequently, the need for strong and secure cryptographic primitives compliant with the tag's challenging constraints must be addressed seriously. In addition, the study and the exploitation of the RFID applications is paramount interest in order to understand and master the threats and risks of this technology. Through the research presented in this thesis, we entered in this competition which consists to find solutions and solving problems related to the RFID systems security, ranging from the use of the lightweight authentication to those adopting elliptic curves cryptography. Among the tasks developed in the thesis works, we have proposed new RFID authentication protocols using the elliptic curves concepts that present more efficiency, security and robustness. In the other hand, we have cryptanalyzed, developed and proposed efficient lightweight and ultra-lightweight authentication protocols suitable for low cost RFID tags. Further, another important contribution which comes within the framework of the random generation of encryption keys, we have proposed a new pseudo-random generator (PRNG) constructed by randomly selecting points from elliptic curves, suitable for applications such as security systems, computer physic simulations, cryptographic applications and control coding.