Esta dissertação apresenta o desenvolvimento de um novo algoritmo de criptografia de chave pública. Este algoritmo apresenta duas características que o tornam único, e que foram tomadas como guia para a sua concepção. A primeira característica é que ele é semanticamente seguro. Isto significa que nenhum adversário limitado polinomialmente consegue obter qualquer informação parcial sobre o conteúdo que foi cifrado, nem mesmo decidir se duas cifrações distintas correspondem ou não a um mesmo conteúdo. A segunda característica é que ele depende, para qualquer tamanho de texto claro, de uma única premissa de segurança: que o logaritmo no grupo formado pelos pontos de uma curva elíptica de ordem prima seja computacionalmente intratável. Isto é obtido garantindo-se que todas as diferentes partes do algoritmo sejam redutíveis a este problema. É apresentada também uma forma simples de estendê-lo a fim de que ele apresente segurança contra atacantes ativos, em especial, contra ataques de texto cifrado adaptativos. Para tanto, e a fim de manter a premissa de que a segurança do algoritmo seja unicamente dependente do logaritmo elíptico, é apresentada uma nova função de resumo criptográfico (hash) cuja segurança é baseada no mesmo problema. / This dissertation presents the development of a new public key algorithm. This algorithm has two key features, which were taken to be a goal from the start. The first feature is that it is semantically secure. That means that no polynomially bounded adversary can extract any partial information about the plaintext from the ciphertext, not even decide if two different ciphertexts correspond to the same plaintext. The second feature of the algorithm is that it depends on only one security assumption: that it is computationally unfeasible to calculate the logarithm on the group formed by the points of a prime order elliptic curve. That is achieved by ensuring that all parts of the algorithm are reducible to that problem. Also, it is presented a way to extend the algorithm so that it the resists attacks of an active adversary, in special, against an adaptive chosen-ciphertext attack. In order to do that, and attain to the assumption that only the assumption of the logarithm is necessary, it is introduced a new hash function with strength based of the same problem.

Esta dissertação apresenta o desenvolvimento de um novo algoritmo de criptografia de chave pública. Este algoritmo apresenta duas características que o tornam único, e que foram tomadas como guia para a sua concepção. A primeira característica é que ele é semanticamente seguro. Isto significa que nenhum adversário limitado polinomialmente consegue obter qualquer informação parcial sobre o conteúdo que foi cifrado, nem mesmo decidir se duas cifrações distintas correspondem ou não a um mesmo conteúdo. A segunda característica é que ele depende, para qualquer tamanho de texto claro, de uma única premissa de segurança: que o logaritmo no grupo formado pelos pontos de uma curva elíptica de ordem prima seja computacionalmente intratável. Isto é obtido garantindo-se que todas as diferentes partes do algoritmo sejam redutíveis a este problema. É apresentada também uma forma simples de estendê-lo a fim de que ele apresente segurança contra atacantes ativos, em especial, contra ataques de texto cifrado adaptativos. Para tanto, e a fim de manter a premissa de que a segurança do algoritmo seja unicamente dependente do logaritmo elíptico, é apresentada uma nova função de resumo criptográfico (hash) cuja segurança é baseada no mesmo problema. / This dissertation presents the development of a new public key algorithm. This algorithm has two key features, which were taken to be a goal from the start. The first feature is that it is semantically secure. That means that no polynomially bounded adversary can extract any partial information about the plaintext from the ciphertext, not even decide if two different ciphertexts correspond to the same plaintext. The second feature of the algorithm is that it depends on only one security assumption: that it is computationally unfeasible to calculate the logarithm on the group formed by the points of a prime order elliptic curve. That is achieved by ensuring that all parts of the algorithm are reducible to that problem. Also, it is presented a way to extend the algorithm so that it the resists attacks of an active adversary, in special, against an adaptive chosen-ciphertext attack. In order to do that, and attain to the assumption that only the assumption of the logarithm is necessary, it is introduced a new hash function with strength based of the same problem.

Esta dissertação apresenta o desenvolvimento de um novo algoritmo de criptografia de chave pública. Este algoritmo apresenta duas características que o tornam único, e que foram tomadas como guia para a sua concepção. A primeira característica é que ele é semanticamente seguro. Isto significa que nenhum adversário limitado polinomialmente consegue obter qualquer informação parcial sobre o conteúdo que foi cifrado, nem mesmo decidir se duas cifrações distintas correspondem ou não a um mesmo conteúdo. A segunda característica é que ele depende, para qualquer tamanho de texto claro, de uma única premissa de segurança: que o logaritmo no grupo formado pelos pontos de uma curva elíptica de ordem prima seja computacionalmente intratável. Isto é obtido garantindo-se que todas as diferentes partes do algoritmo sejam redutíveis a este problema. É apresentada também uma forma simples de estendê-lo a fim de que ele apresente segurança contra atacantes ativos, em especial, contra ataques de texto cifrado adaptativos. Para tanto, e a fim de manter a premissa de que a segurança do algoritmo seja unicamente dependente do logaritmo elíptico, é apresentada uma nova função de resumo criptográfico (hash) cuja segurança é baseada no mesmo problema. / This dissertation presents the development of a new public key algorithm. This algorithm has two key features, which were taken to be a goal from the start. The first feature is that it is semantically secure. That means that no polynomially bounded adversary can extract any partial information about the plaintext from the ciphertext, not even decide if two different ciphertexts correspond to the same plaintext. The second feature of the algorithm is that it depends on only one security assumption: that it is computationally unfeasible to calculate the logarithm on the group formed by the points of a prime order elliptic curve. That is achieved by ensuring that all parts of the algorithm are reducible to that problem. Also, it is presented a way to extend the algorithm so that it the resists attacks of an active adversary, in special, against an adaptive chosen-ciphertext attack. In order to do that, and attain to the assumption that only the assumption of the logarithm is necessary, it is introduced a new hash function with strength based of the same problem.

Soit E une courbe elliptique sur C ayant multiplication complexe (CM) par l’ordre maximal OK d’un corps quadratique imaginaire K. Le premier théorème principal de la multiplication complexe affirme que le corps K(j(E)), obtenu en adjoignant à K le j-invariant de E, est égal au corps de classes de Hilbert de K, confer Cox [11, Theorem 11.1]. Notons que lorsque E est définie sur Q, le corps de classes de Hilbert K(j(E)) est égal à K et le groupe des classes ClK est trivial. Se pose alors le problème de déterminer les corps quadratiques totalement imaginaires K pour lesquels la courbe elliptique à multiplication complexe par OK correspondante est définie sur Q. De façon équivalente, il s’agit de trouver tous les corps quadratiques imaginaires dont le groupe des classes est trivial. Ce problème est connu sous le nom de problème du nombre de classes 1 de Gauss et a été résolu par Heegner en 1952 [16], Baker en 1967 [2] et Stark en 1967 [41]; les corps quadratiques imaginaires dont le groupe des classes est trivial sont les corps Q(racine carrée−d), où d e {3, 4, 7, 8, 11, 19, 43, 67, 163}. Dans les années ’50, Shimura et Taniyama [39] ont généralisé le premier théorème principal de la multiplication complexe aux variétés abéliennes. On dit qu’une variété abélienne A de dimension g a multiplication complexe si son anneau d’endomorphismes contient un ordre d’un corps CM de degré 2g. Soit K un corps CM de degré 2g et d’ordre maximal OK et soit un type CM de K. Soit A une variété abélienne complexe simplement polarisée de dimension g ayant multiplication complexe par OK. Le premier théorème principal de la multiplication complexe dans ce cadre affirme que le corps de classes H du corps du modules M de la variété abélienne simplement polarisée A est une extension non ramifiée du corps reflex Kr de K. De plus, le corps des classes H correspond au groupe d’idéaux I0(.r) (voir page 17) qui ne dépend que de (K,.), confer Théorème 1.5.6. Notons que le premier théorème de la multiplication complexe implique que si la variété abélienne polarisée A est définie sur Kr, le groupe des classes CM IKr/I0(.r) est trivial. Comme dans le cas des courbes elliptiques, on peut alors chercher à déterminer les couples CM (K,.) pour lesquels les variétés abéliennes correspondantes sont définies sur Kr. De fa¸con équivalente, il s’agit de déterminer les couples CM (K,.) dont le groupe des classes CM, IKr/I0(.r), est trivial. Dans cette thèse, on résout ce problème dans le cas des corps CM quartiques imaginaires (voir Chapitre 2) ainsi que dans celui des corps CM sextiques contenant un corps quadratique imaginaire (voir Chapitre 3). Enfin, on peut se demander quels sont les corps CM pour lesquels la variété abélienne simple à multiplication complexe admet Q comme corps de module. Murabayashi et Umegaki [31] ont déterminé les corps quartiques CM correspondant aux surfaces abéliennes simples à multiplication complexe de corps du module Q. Dans le chapitre 4, on détermine les corps CM sextiques correspondant aux variétés abéliennes simples à multiplication complexe de dimension 3 de corps du module Q. / Let E be an elliptic curve over C with complex multiplication (CM) by the maximal order OK of an imaginary quadratic field K. The first main theorem of complex multiplication for elliptic curves then states that the field extension K(j(E)), obtained by adjoining the j-invariant of E to K, is equal to the Hilbert class field of K, see Theorem 11.1 in Cox [11]. Note that if E is defined over Q, then the Hilbert class field K(j(E)) is equal to K, which implies that the class group ClK is trivial. We can ask for which imaginary quadratic fields K the corresponding elliptic curve with CM by OK is defined over Q. This is equivalent to asking to find all imaginary quadratic fields with trivial class group ClK. This problem is known as Gauss’ class number one problem, which was solved by Heegner in 1952 [16], Baker in 1967 [2], and Stark in 1967 [41]. The imaginary quadratic fields with trivial class group are the fields Q(V−d) with d E {3, 4, 7, 8, 11, 19, 43, 67, 163}. In the 1950’s, Shimura and Taniyama [39] generalized the first main theorem of CM for elliptic curves to abelian varieties. We say that an abelian variety A of dimension g has CM if the endomorphism ring of A contains an order of a CM field of degree 2g. Let K be a CM field of degree 2g with maximal order OK, and let K be a CM type of K. Let A be a polarized simple abelian variety over C of dimension g that has CM by OK. Then the first main theorem of CM says that the field of moduli M of the polarized simple abelian variety A gives an unramified class field H over the reflex field Kr of K. Moreover, the class field H corresponds to the ideal group I0(?r) (see page 17), which only depends on (K,?), see Theorem 1.5.6. Note that the first main theorem of CM implies that if the polarized abelian variety A is defined over Kr, then the CM class group IKr/I0(?r) is trivial. As in the elliptic curve case, we can ask for which CM pairs (K,?) the corresponding CM abelian varieties are defined over Kr. Equivalently, we can ask for which CM pairs (K,?) the CM class group IKr/I0(?r) is trivial. In this thesis we give an answer to this problem for quartic CM fields (see Chapter 2), and for sextic CM fields containing an imaginary quadratic field (see Chapter 3). Furthermore, we can ask for which CM fields the corresponding simple CM abelian varieties have field of moduli Q. Murabayashi and Umegak [31] determined the quartic CM fields that correspond to a simple CM abelian surface with field of moduli Q. In Chapter 4, we determine the sextic CM fields that correspond to a simple CM abelian threefold with field of moduli Q.

The conjecture of Birch and Swinnerton-Dyer is unquestionably one of the most important open problems in number theory today. Let $E$ be an elliptic curve defined over an imaginary quadratic field $K$ contained in $\mathbb{C}$, and suppose that $E$ has complex multiplication by the ring of integers of $K$. Let us assume the complex $L$-series $L(E/K,s)$ of $E$ over $K$ does not vanish at $s=1$. K. Rubin showed, using Iwasawa theory, that the $p$-part of Birch and Swinnerton-Dyer conjecture holds for $E$ for all prime numbers $p$ which do not divide the order of the group of roots of unity in $K$. In this thesis, we discuss extensions of this result. In Chapter $2$, we study infinite families of quadratic and cubic twists of the elliptic curve $A = X_0(27)$, so that they have complex multiplication by the ring of integers of $\mathbb{Q}(\sqrt{-3})$. For the family of quadratic twists, we establish a lower bound for the $2$-adic valuation of the algebraic part of the complex $L$-series at $s=1$, and, for the family of cubic twists, we establish a lower bound for the $3$-adic valuation of the algebraic part of the same $L$-value. We show that our lower bounds are precisely those predicted by Birch and Swinnerton-Dyer. In the remaining chapters, we let $K=\mathbb{Q}(\sqrt{-q})$, where $q$ is any prime number congruent to $7$ modulo $8$. Denote by $H$ the Hilbert class field of $K$. \mbox{B. Gross} proved the existence of an elliptic curve $A(q)$ defined over $H$ with complex multiplication by the ring of integers of $K$ and minimal discriminant $-q^3$. We consider twists $E$ of $A(q)$ by quadratic extensions of $K$. In the case $q=7$, we have $A(q)=X_0(49)$, and Gonzalez-Aviles and Rubin proved, again using Iwasawa theory, that if $L(E/\mathbb{Q},1)$ is nonzero then the full Birch--Swinnerton-Dyer conjecture holds for $E$. Suppose $p$ is a prime number which splits in $K$, say $p=\mathfrak{p}\mathfrak{p}^*$, and $E$ has good reduction at all primes of $H$ above $p$. Let $H_\infty=HK_\infty$, where $K_\infty$ is the unique $\mathbb{Z}_p$-extension of $K$ unramified outside $\mathfrak{p}$. We establish in this thesis the main conjecture for the extension $H_\infty/H$. Furthermore, we provide the necessary ingredients to state and prove the main conjecture for $E/H$ and $p$, and discuss its relation to the main conjecture for $H_\infty/H$ and the $p$-part of the Birch--Swinnerton-Dyer conjecture for $E/H$.

Cette thèse étudie le problème de minoration de la hauteur canonique sur les courbeselliptiques. Son résultat diophantien principal utilise des méthodes d’intersectionarithmétique pour retrouver un résultat de Laurent, qui démontrait la conjecturede Lehmer pour les courbes elliptiques à multiplications complexes à un exposant" près, tout en explicitant complètement sa dépendance en divers paramètres liésà la courbe elliptique ; une telle démarche peut être motivée par la conjecture deLang, qui présage une minoration possible de la hauteur canonique proportionnelle,essentiellement, à la hauteur de Faltings de la courbe.Notre dissertation commence toutefois par une partie dédiée à l’explicitation duthéorème de densité de Chebotarev, qui reprend les grandes lignes d’un travail deLagarias et Odlyzko, et s’avère être cruciale dans notre approche du problème deLehmer elliptique. On obtient également des majorations des zéros de Siegel et de lanorme du plus petit idéal premier entrant en jeu dans le théorème de Chebotarev. / In this thesis we consider the problem of lower bounds for the canonical height onelliptic curves, aiming for the conjecture of Lehmer. Our main diophantine result isan explicit version of a theorem of Laurent (who proved this conjecture for ellipticcurves with CM up to a " exponent) using arithmetic intersection, enlightening thedependence with parameters linked to the elliptic curve ; such a result can be motivatedby the conjecture of Lang, hoping for a lower bound proportional to, roughly,the Faltings height of the curve.Nevertheless, our dissertation begins with a part dedicated to a completely explicitversion of the density theorem of Chebotarev, along the lines of a previous workdue to Lagarias and Odlyzko, which will be crucial to investigate the elliptic Lehmerproblem. We also obtain upper bounds for Siegel zeros, and for the smallest primeideal whose Frobenius is in a fixed conjugacy class.