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.

Les problématiques de respect de la vie privée sont aujourd'hui indissociables des technologies modernes. Dans ce contexte, cette thèse s'intéresse plus particulièrement aux outils cryptographiques et à la façon de les utiliser pour répondre à ces nouvelles questions.Dans ce mémoire, je m'intéresserai tout d'abord aux preuves de connaissance sans divulgation qui permettent notamment d'obtenir la propriété d'anonymat pour les usagers de services de télécommunications. Je proposerai ainsi une nouvelle solution de preuve de connaissance d'un secret appartenant à un intervalle, ainsi que la première étude comparative des preuves existantes sur ce sujet. Je décrirai ensuite une nouvelle méthode permettant de vérifier efficacement un ensemble de preuves de type "Groth-Sahaï'', accélérant ainsi considérablement le travail du vérifieur pour de telles preuves. Dans un second temps, je m'intéresserai aux signatures caméléons. Celles-ci permettent de modifier, sous certaines conditions, un message signé. Ainsi, pour ces schémas, il est possible d'exhiber, à l'aide d'une trappe, une signature valide du signataire initial sur le message modifié. Je proposerai d'abord un nouveau schéma qui est à ce jour le plus efficace dans le modèle simple. Je m'intéresserai ensuite à certaines extensions de ce modèle qui ont pour vocation de donner au signataire les moyens de garder un certain contrôle sur les modifications faites a posteriori sur le message initial. Je décrirai ainsi à la fois le nouveau modèle de sécurité et les schémas associés prenant en compte ces nouvelles extensions. Enfin, je présenterai un ensemble d'applications se basant sur les briques cryptographiques introduites ci-dessus et qui permettent d'améliorer la protection de la vie privée des utilisateurs. J'aborderai tout particulièrement les problématiques d'abonnement, d'utilisation ou de facturation de services, ainsi que la gestion de contenus protégés dans un groupe hiérarchisé. / Privacy is, nowadays, inseparable from modern technology. This is the context in which the present thesis proposes new cryptographic tools to meet current challenges.Firstly, I will consider zero-knowledge proofs of knowledge, which allow in particular to reach the anonymity property. More precisely, I will propose a new range proof system and next give the first comparison between all existing solutions to this problem. Then, I will describe a new method to verify a set of ``Groth-Sahaï'' proofs, which significantly decreases the verification time for such proofs.In a second part, I will consider sanitizable signatures which allow, under some conditions, to manipulate (we say ``sanitize'') a signed message while keeping a valid signature of the initial signer. I will first propose a new scheme in the classical case. Next, I will introduce several extensions which enable the signer to obtain better control of the modifications done by the ``sanitizer''. In particular, I will propose a new security model taking into account these extensions and give different schemes achieving those new properties.Finally, I will present different applications of the above cryptographic tools that enhance customer privacy. In particular, I will consider the questions of subscription, use and billing of services and also address the issue of managing protected content in a hierarchical group.

Cryptography is science of secure message transmission. Cryptanalysis is involved with breaking these encrypted messages. Both cryptography and cryptanalysis constitute together to form cryptology. Anonymity means namelessness i.e., the quality or state of being unknown while authenticity translates to the quality or condition of being authentic or genuine. Anonymity and authenticity are two different embodiments of personal secrecy. Modern power has increased in its capacity to designate individuals, due to which they find it inconvenient to continue communicating, remaining anonymous. In this thesis we are going to describe an anonymous system which consists of a number of entities which are anonymous and are communicating with each other without revealing their identity and at the same time maintaining their authenticity such that an anonymous entity(sayE1)will be able to verify that, the message it received from another anonymous entity(sayE2)subsequent to an initial message from E2, are in fact from E2 itself. Later when E2 tries to recommend a similar communication to E1 with another anonymous entity E3 in the system, E1 must be able to verify that recommendation, without E2 losing its authenticity of its communication with E1 to E3. This thesis is divided into four chapters. The first chapter is an introduction to cryptography, symmetric key cryptography and public key cryptography. It also summarizes the contribution of this thesis. The second chapter gives various protocol for the above problem ’Anonymity with Authenticity’ along with its extension. Totally six protocols are proposed for the above problem. In third chapter all these six protocols are realized using four different schemes, where each scheme has its own pros and cons. The fourth and final chapter concludes with a note on what possible factors these four different realization schemes need to be chosen and other possible realization schemes.

Dans cette thèse, j'étudie deux aspects distincts de la cryptographie basée sur les courbes elliptiques et hyperelliptiques. Dans une première partie, je confronte deux méthodes de calcul de couplages, originales car ne reposant pas sur le traditionnel algorithme de Miller. Ainsi, dans [42], K. Stange calcula le couplage de Tate sur une courbe elliptique à partir d'un nouvel outil, les elliptic nets. Y. Uchida et S. Uchiyama généralisèrent ces objets au cas hyperelliptique ([47]), mais ne donnèrent un algorithme pour le calcul de couplages que dans le cas des courbes de genre 2. Mon premier travail dans cette thèse fut de donner cet algorithme pour le cas général. De leur côté, D. Lubicz et D. Robert donnèrent dans [28] une autre méthode de calcul de couplage, basée sur les fonctions thêta. Le second résultat de ma thèse est de réunifier ces deux méthodes : je montre que la formule de récurrence à la base des nets est une conséquence des formules d'addition des fonctions thêta utilisées dans l'algorithme de Lubicz et Robert. Dans la seconde partie de ma thèse, je me suis intéressé à l'algorithme de calcul d'index attaquant le problème du logarithme discret sur les courbes elliptiques et hyperelliptiques. Dans le cas elliptique, une des étapes principales de cette attaque repose sur les polynômes de Semaev. Je donne une nouvelle construction ces polynômes en utilisant la fonction sigma de Weierstrass, pour pouvoir ensuite les généraliser pour la première fois au cas hyperelliptique. / In this thesis, I study two different aspects of elliptic and hyperelliptic curves based cryptography.In the first part, I confront two methods of pairings computation, whose original feature is that they are not based the traditional Miller algorithm. Therefore, in [42], K. Stange computed Tate pairings on elliptic curves using a new tool, the elliptic nets. Y. Uchida and S. Uchiyama generalized these objects to hyperelliptic case ([47]), but they gave an algorithm for pairing computation only for the genus 2 case. My first work in this thesis was to give this algorithm for the general case. Meanwhile, D. Lubicz and D. Robert gave in [28] an other pairing computation method, based on theta functions. The second result of my thesis is the reunification of these two methods : I show that the recurrence equation which is the basis of nets theory is a consequence of the addition law of theta functions used in the Lubicz and Robert’s algorithm. In the second part, I study the index calculus algorithm attacking the elliptic and hyperelliptic curve discrete logarithm problem. In the elliptic case, one of the main steps of this attack requires the Semaev polynomials. I reconstruct these polynomials using Weierstrass sigma function, with the purpose of giving their first hyperelliptic generalization.

Cryptography based on semirings can be one of the possible approaches for the post-quantum cryptography in the public-key schemes. In our work, we are interested in only one concrete semiring - tropical algebra. We are examining one concrete scheme for the key-agreement protocol - tropical Stickel's protocol. Although there was introduced an attack on it, we have implemented this attack and more importantly, stated its complexity. Further, we propose other variants of Stickel's protocol and we are investigating their potential for practical usage. During the process, we came across the theory of tropical matrix powers, thus we want to make an overview of it due to the use in cryptography based on matrices over the tropical algebra semiring. 1

Background. People use the Internet for communication, work, online banking and more. Public-key cryptography enables this use to be secure by providing confidentiality and trust online. Though these algorithms may be secure from attacks from classical computers, future quantum computers may break them using Shor’s algorithm. Post-quantum algorithms are therefore being developed to mitigate this issue. The National Institute of Standards and Technology (NIST) has started a standardization process for these algorithms. Objectives. In this work, we analyze what specialized features applicable for post-quantum algorithms are available in the mainframe architecture IBM Z. Furthermore, we study the performance of these algorithms on various hardware in order to understand what techniques may increase their performance. Methods. We apply a literature study to identify the performance characteristics of post-quantum algorithms as well as what features of IBM Z may accommodate and accelerate these. We further apply an experimental study to analyze the practical performance of the two prominent finalists NTRU and Classic McEliece on consumer, cloud and mainframe hardware. Results. IBM Z was found to be able to accelerate several key symmetric primitives such as SHA-3 and AES via the Central Processor Assist for Cryptographic Functions (CPACF). Though the available Hardware Security Modules (HSMs) did not support any of the studied algorithms, they were found to be able to accelerate them via a Field-Programmable Gate Array (FPGA). Based on our experimental study, we found that computers with support for the Advanced Vector Extensions (AVX) were able to significantly accelerate the execution of post-quantum algorithms. Lastly, we identified that vector extensions, Application-Specific Integrated Circuits (ASICs) and FPGAs are key techniques for accelerating these algorithms. Conclusions. When considering the readiness of hardware for the transition to post-quantum algorithms, we find that the proposed algorithms do not perform nearly as well as classical algorithms. Though the algorithms are likely to improve until the post-quantum transition occurs, improved hardware support via faster vector instructions, increased cache sizes and the addition of polynomial instructions may significantly help reduce the impact of the transition. / Bakgrund. Människor använder internet för bland annat kommunikation, arbete och bankärenden. Asymmetrisk kryptering möjliggör att detta sker säkert genom att erbjuda sekretess och tillit online. Även om dessa algoritmer förväntas vara säkra från attacker med klassiska datorer, riskerar framtida kvantdatorer att knäcka dem med Shors algoritm. Därför utvecklas kvantsäkra krypton för att mitigera detta problem. National Institute of Standards and Technology (NIST) har påbörjat en standardiseringsprocess för dessa algoritmer. Syfte. I detta arbete analyserar vi vilka specialiserade funktioner för kvantsäkra algoritmer som finns i stordator-arkitekturen IBM Z. Vidare studerar vi prestandan av dessa algoritmer på olika hårdvara för att förstå vilka tekniker som kan öka deras prestanda. Metod. Vi utför en litteraturstudie för att identifiera vad som är karaktäristiskt för kvantsäkra algoritmers prestanda samt vilka funktioner i IBM Z som kan möta och accelerera dessa. Vidare applicerar vi en experimentell studie för att analysera den praktiska prestandan av de två framträdande finalisterna NTRU och Classic McEliece på konsument-, moln- och stordatormiljöer. Resultat. Vi fann att IBM Z kunde accelerera flera centrala symmetriska primitiver så som SHA-3 och AES via en hjälpprocessor för kryptografiska funktioner (CPACF). Även om befintliga hårdvarusäkerhetsmoduler inte stödde några av de undersökta algoritmerna, fann vi att de kan accelerera dem via en på-plats-programmerbar grind-matris (FPGA). Baserat på vår experimentella studie, fann vi att datorer med stöd för avancerade vektorfunktioner (AVX) möjlggjorde en signifikant acceleration av kvantsäkra algoritmer. Slutligen identifierade vi att vektorfunktioner, applikationsspecifika integrerade kretsar (ASICs) och FPGAs är centrala tekniker som kan nyttjas för att accelerera dessa algortmer. Slutsatser. Gällande beredskapen hos hårdvara för en övergång till kvantsäkra krypton, finner vi att de föreslagna algoritmerna inte presterar närmelsevis lika bra som klassiska algoritmer. Trots att det är sannolikt att de kvantsäkra kryptona fortsatt förbättras innan övergången sker, kan förbättrat hårdvarustöd för snabbare vektorfunktioner, ökade cachestorlekar och tillägget av polynomoperationer signifikant bidra till att minska påverkan av övergången till kvantsäkra krypton.

In the presented work we focus on applications of decision problems from combinatorial group theory. Namely we analyse the Shpilrain-Zapata pro- tocol. We give formal proof that small cancellation groups are good platform for the protocol because the word problem is solvable in linear time and they are generic. We also analyse the complexity of the brute force attack on the protocol and show that in a theoretical way the protocol is immune to attack by adversary with arbitrary computing power.

As quantum computers continue to develop, they pose a threat to cryptography since many popular cryptosystems will be rendered vulnerable. This is because the security of most currently used asymmetric systems requires the computational hardness of the integer factorization problem, the discrete logarithm or the elliptic curve discrete logarithm problem. However, there are still some cryptosystems that resist quantum computing. We will look at code-based cryptography in general and the McEliece cryptosystem specifically. Our goal is to understand the structure behind the McEliece scheme, including the encryption and decryption processes, and what some advantages and disadvantages are that the system has to offer. In addition, using the results from Courtois, Finiasz, and Sendrier's paper in 2001, we will discuss a digital signature scheme based on the McEliece cryptosystem. We analyze one classical algebraic attack against the security analysis of the system based on the distinguishing problem whether the public key of the McEliece scheme is generated from a generating matrix of a binary Goppa code or a random binary matrix. The idea of the attack involves solving an algebraic system of equations and we examine the dimension of the solution space of the linearized system of equations. With the assistance from a paper in 2010 by Faugere, Gauthier-Umana, Otmani, Perret, Tillich, we will see the parameters needed for the intractability of the distinguishing problem. / Includes bibliography. / Thesis (M.S.)--Florida Atlantic University, 2015. / FAU Electronic Theses and Dissertations Collection

Secure communication is critical to many applications. To this end, various security goals can be achieved using elliptic/hyperelliptic curve and pairing based cryptography. Polynomial multiplication is used in the underlying operations of these protocols. Therefore, as part of this thesis different recursive algorithms are studied; these algorithms include Karatsuba, Toom, and Bernstein. In this thesis, we investigate algorithms and implementation techniques to improve the performance of the cryptographic protocols. Common factors present in explicit formulae in elliptic curves operations are utilized such that two multiplications are replaced by a single multiplication in a higher field. Moreover, we utilize the idea based on common factor used in elliptic curves and generate new explicit formulae for hyperelliptic curves and pairing. In the case of hyperelliptic curves, the common factor method is applied to the fastest known even characteristic hyperelliptic curve operations, i.e. divisor addition and divisor doubling. Similarly, in pairing we observe the presence of common factors inside the Miller loop of Eta pairing and the theoretical results show significant improvement when applying the idea based on common factor method. This has a great advantage for applications that require higher speed.