The theme of the thesis is Physical Unclonable Functions (PUF). The following objectives were assigned: to provide a literature research concerning PUFs, to perform a property analysis to select appropriate type of PUF for implementation and to realize an authentication cryptosystem based on the chosen PUF. Based on the research, the cryptosystem was designed based on ring oscillator PUF. The proposed cryptosystem is tested in several scenarios with the maximal rate of successful authentication of 81%. The cryptosystem also allows to be used for device identification. Furthermore, the results were discussed and suitable improvements of design was proposed. Besides the cryptosystem itself, the thesis also introduced a unique comparison of existing types of PUFs.

Tout d’abord, mon sujet de recherche porte sur le cryptographie à clé publique, plus précisément la cryptographie basée sur la théorie des codes correcteurs d’erreurs. L’objectif principal de cette thèse est d’analyser la sécurité des systèmes de chiffrement. Pour cela j’étudie les propriétés structurelles des différentes familles de codes linéaires utilisées dans la pratique. Mon travail de recherche s’est orienté de maniéré naturelle, vers l’étude des deux dernières propositions de cryptosystèmes, plus exactement le schéma de McEliece à base des codes MDPC [MTSB13](moderate parity check codes) et des codes Polaires [SK14]. Dans le cas des codes MDPC on a mis en évidence une faiblesse importante au niveau des clés utilisées par les utilisateurs du système. En effet, on a proposé un algorithme très efficace qui permet de retrouver une clé privé à partir d’une clé publique. Ensuite on a compté le nombre des clés faibles et on a utilisé le problème d’équivalence de codes pour élargir le nombre de clés faibles. On a publié notre travail de recherche dans une conférence internationale en cryptographie [BDLO16]. Ensuite on a étudié les codes Polaires et leur application à la cryptographie à clé publique. Depuis leur découverte par E. Arikan [Arı09], les codes Polaires font partie des famille de codes les plus étudié du point de vue de le théorie de l’information. Ce sont des codes très efficaces en terme de performance car ils atteignent la capacité des canaux binaires symétriques et ils admettent des algorithmes d’encodage et décodage très rapides. Néanmoins, peu des choses sont connu sur leur propriétés structurelles. Dans ce cadre la, on a introduit un formalisme algébrique qui nous a permit de révéler unegrande partie de la structure de ces codes. En effet, on a réussi à répondre à des questions fondamentales concernant les codes Polaires comme : le dual ou la distance minimale d’un code Polaire, le groupe des permutations ou le nombre des mots de poids faible d’un code Polaire. On a publié nos résultats dans une conférence internationale en théorie de l’information [BDOT16]. Par la suite on a réussi à faire une cryptanalyse complète du schéma de McEliece à base des codes Polaires. Ce résultat a été une application directe des propriétés découvertes sur les codes Polaires et il a été publié dans une conférence internationale en cryptographie post-quantique [BCD+16]. / First of all, during my PhD I focused on the public key cryptography, more exactly on the code-based cryptography. The main motivation is to study the security of the latest encryption schemes. For that, I analyzed in detail the structural properties of the main code families. Thus, my research was naturally directed to the study of the McEliece based encryption schemes, among which the latest MDCP based variant [MTSB13] and Polar codes variant [SK14]. In the case of the MDPC based variant, we manage to reveal an important weakness regarding the key pairs that are used in the protocol. Indeed, we proposed an efficient algorithm that retrieves the private key given the public key of the scheme. Next we counted the proportion of weak keys and we used the code equivalence problem to extend the number of weak keys. We published our results in an international conference in cryptography [BDLO16]. Next we studied the Polar codes and their application to public key cryptography.Since they were discovered by Arikan [Arı09], Polar codes are part of the most studied from an information theory point of view, family of codes. In terms of performance they are really efficient since they are capacity achieving over the Binary Discrete Memoryless Channels and they allow extremely fast encoding and decoding algorithms. Nonetheless, few facts are known about their structure. In this context, we have introduced an algebraic formalism which allowed us to reveal a big part of the structure of Polar codes. Indeed, we have managed to answer fundamental questions regarding Polar codes such as the dual, the minimum distance, the permutation group and the number of minimum weight codewords of a Polar code. Our results were published in an international conference in information theory [BDOT16]. We also managed to completely cryptanalyze the McEliece variant using Polar codes. The attack was a direct application of the aforementioned results on the structural properties of Polar codes and it was published in an international conference in postquantum cryptography [BCD+16].

Em 2009, N. Heninger e H. Shacham apresentaram um algoritmo de reconstrução que permite recuperar a chave secreta sk do criptossistema RSA básico em tempo polinomial tendo em forma aleatória 27 % dos seus bits. Sabemos que podemos obter uma versão com erros (bits modicados) da chave secreta RSA graças aos ataques cold boot. O algoritmo apresentado por Heninger-Shacham corrige esses erros fazendo uso das relações matemáticas que existe entre as chaves pública e secreta do criptossistema RSA básico. O objetivo deste trabalho é estudar esse algoritmo para implementar e analisar seu análogo para o criptossistema RSA multi-primo. Os resultados obtidos mostram que para reconstruir a chave secreta sk do criptossistema RSA u-primos é preciso ter uma fração de bits corretos maior a 2 - 2^((u+2)/(2u+1)), mostrando assim que a segurança oferecida pelo criptossistema RSA multi-primo (u>/ 3) é maior com relação ao criptossistema RSA básico (u = 2). / In 2009, N. Heninger and H. Shacham presented an algoritm for reconstructing the secret key sk of the basic RSA cryptosystem in polynomial time With a fraction of random bits greater or equal to 0.27 of its bits. We know that secret key with errors sk can be obtained from DRAM using cold-boot attacks. The Heninger and Shacham\'s algorithm xes these errors using the redundancy of secret and public key of basic RSA cryptosystem. In this work, the topic is to study this algoritm to implement and analyze its analogous for the multi-prime RSA cryptosystem. Our obtained results show the secret key sk of multi-prime RSA cryptosystem can be Reconstructed having a fraction equal or greater than 2 - 2^((u+2)/(2u+1)) of random bits. therefore the security of multi-prime RSA cryptosystem (u >/ 3) is greater than basic RSA cryptosystem (u = 2).

Em 2009, N. Heninger e H. Shacham apresentaram um algoritmo de reconstrução que permite recuperar a chave secreta sk do criptossistema RSA básico em tempo polinomial tendo em forma aleatória 27 % dos seus bits. Sabemos que podemos obter uma versão com erros (bits modicados) da chave secreta RSA graças aos ataques cold boot. O algoritmo apresentado por Heninger-Shacham corrige esses erros fazendo uso das relações matemáticas que existe entre as chaves pública e secreta do criptossistema RSA básico. O objetivo deste trabalho é estudar esse algoritmo para implementar e analisar seu análogo para o criptossistema RSA multi-primo. Os resultados obtidos mostram que para reconstruir a chave secreta sk do criptossistema RSA u-primos é preciso ter uma fração de bits corretos maior a 2 - 2^((u+2)/(2u+1)), mostrando assim que a segurança oferecida pelo criptossistema RSA multi-primo (u>/ 3) é maior com relação ao criptossistema RSA básico (u = 2). / In 2009, N. Heninger and H. Shacham presented an algoritm for reconstructing the secret key sk of the basic RSA cryptosystem in polynomial time With a fraction of random bits greater or equal to 0.27 of its bits. We know that secret key with errors sk can be obtained from DRAM using cold-boot attacks. The Heninger and Shacham\'s algorithm xes these errors using the redundancy of secret and public key of basic RSA cryptosystem. In this work, the topic is to study this algoritm to implement and analyze its analogous for the multi-prime RSA cryptosystem. Our obtained results show the secret key sk of multi-prime RSA cryptosystem can be Reconstructed having a fraction equal or greater than 2 - 2^((u+2)/(2u+1)) of random bits. therefore the security of multi-prime RSA cryptosystem (u >/ 3) is greater than basic RSA cryptosystem (u = 2).

Title: Goppa codes and their applications Author: Bc. Jaroslav Kotil Department: Department of algebra Supervisor: prof. RNDr. Aleš Drápal, CSc., DSc. Abstract: In this diploma paper we introduce Goppa codes, describe their para- metres and inclusion in Alternant codes, which are residual Generalized Reed- Solomon codes, and Algebraic-geometry codes. Aftewards we demonstrate deco- ding of Goppa codes and introduce Wild Goppa codes. We also describe post- quantum cryptography member: McEliece cryptosystem for which no effective attacks with quantum computers are known. We outline a usage of this crypto- system with Goppa codes and describe the security of the cryptosystem together with possible attacks of which the most effective ones are based on information- set decoding. Keywords: Goppa codes, Generalized Reed-Solomon codes, Algebraic-geometry codes, Post-quantum cryptography, McEliece cryptosystem 1

This dissertation investigates the application of computational intelligence methods in the analysis of nonlinear chaotic systems in the framework of many known and newly designed complex systems. Parallel comparisons are made between these methods. This provides insight into the difficult challenges facing nonlinear systems characterization and aids in developing a generalized algorithm in computing algorithmic complexity measures, Lyapunov exponents, information dimension and topological entropy. These metrics are implemented to characterize the dynamic patterns of discrete and continuous systems. These metrics make it possible to distinguish order from disorder in these systems. Steps required for computing Lyapunov exponents with a reorthonormalization method and a group theory approach are formalized. Procedures for implementing computational algorithms are designed and numerical results for each system are presented. The advance-time sampling technique is designed to overcome the scarcity of phase space samples and the buffer overflow problem in algorithmic complexity measure estimation in slow dynamics feedback-controlled systems. It is proved analytically and tested numerically that for a quasiperiodic system like a Fibonacci map, complexity grows logarithmically with the evolutionary length of the data block. It is concluded that a normalized algorithmic complexity measure can be used as a system classifier. This quantity turns out to be one for random sequences and a non-zero value less than one for chaotic sequences. For periodic and quasi-periodic responses, as data strings grow their normalized complexity approaches zero, while a faster deceasing rate is observed for periodic responses. Algorithmic complexity analysis is performed on a class of certain rate convolutional encoders. The degree of diffusion in random-like patterns is measured. Simulation evidence indicates that algorithmic complexity associated with a particular class of 1/n-rate code increases with the increase of the encoder constraint length. This occurs in parallel with the increase of error correcting capacity of the decoder. Comparing groups of rate-1/n convolutional encoders, it is observed that as the encoder rate decreases from 1/2 to 1/7, the encoded data sequence manifests smaller algorithmic complexity with a larger free distance value.

Cryptography fascinates people of all generations and is increasingly presented as an example for the relevance and application of the mathematical sciences. Indeed, many principles of modern cryptography can be described at a secondary school level. In this context, the mathematical background is often only sparingly shown. In the worst case, giving mathematics this character of a tool reduces the application of mathematical insights to the message ”cryptography contains math”. This paper examines the question as to what else cryptography can offer to mathematics education. Using the RSA cryptosystem and related content, specific mathematical competencies are highlighted that complement standard teaching, can be taught with cryptography as an example, and extend and deepen key mathematical concepts.

Current needs of human communication came to status, when most of transferred messages are considered as private and transition over non-secured communication lines in open form is not possible. That originated a lot of different methods for securing of messages and transfers in ciphered form. Two mainstreams were established, symmetric cryptography and asymmetric cryptography. Second of mentioned groups is based on usage of two information – keys, when one of then is broadly known and is public and second, well protected and private. Using a public key it is possible to establish a cryptogram of message, but for deciphering it is necessary to know private key. Asymmetric methods are based on mathematical problems, for which there is not an effective computing algorithm. This thesis are focused for asymmetric cryptosystems based on discrete logarithm problem, where ciphering of message using public key is very easy and quick, but deciphering without knowledge of private key is extremely time consuming process. Work describes a mathematical base of discrete logarithm problem, its’ properties and methods developed for solving of this problem. Descriptions of particular cryptosystems are given, i.e. ElGamal cryptosystem, Diffie-Hellman protocol and DSA. Second part of thesis is focused for web application developed as study support of discrete logarithm problem and of cryptosystems using this problem. It describes functional and graphical interface, work with it and options given to user working with application. Mentions also lessons for user which should help with understanding of described problems and practicing.

Cette thèse aborde la conception et les contremesures permettant d'améliorer le calcul cryptographique matériel léger. Parce que la cryptographie (et la cryptanalyse) sont de nos jours de plus en plus omniprésentes dans notre vie quotidienne, il est crucial que les nouveaux systèmes développés soient suffisamment robustes pour faire face à la quantité croissante de données de traitement sans compromettre la sécurité globale. Ce travail aborde de nombreux sujets liés aux implémentations cryptographiques légères. Les principales contributions de cette thèse sont : - Un nouveau système d'accélération matérielle cryptographique appliqué aux codes BCH ; - Réduction de la consommation des systèmes embarqués et SoCs ; - Contre-mesures légères des attaques par canal auxiliaire applicables à l'algorithme de chiffrement reconfigurable AES ;- CSAC : Un pare-feu sécurisé sur la puce cryptographique ; - Attaques par analyse fréquentielle ; - Un nouveau protocole à divulgation nulle de connaissance appliquée aux réseaux de capteurs sans fil ; - OMD : Un nouveau schéma de chiffrement authentifié. / This thesis addresses lightweight hardware design and countermeasures to improve cryptographic computation. Because cryptography (and cryptanalysis) is nowadays becoming more and more ubiquitous in our daily lives, it is crucial that newly developed systems are robust enough to deal with the increasing amount of processing data without compromising the overall security. This work addresses many different topics related to lightweight cryptographic implementations. The main contributions of this thesis are: - A new cryptographic hardware acceleration scheme applied to BCH codes; - Hardware power minimization applied to SoCs and embedded devices; - Timing and DPA lightweight countermeasures applied to the reconfigurable AES block cipher; - CSAC: A cryptographically secure on-chip firewall; - Frequency analysis attack experiments; - A new zero-knowledge zero-knowledge protocol applied to wireless sensor networks; - OMD: A new authenticated encryption scheme.

Cryptography fascinates people of all generations and is increasingly presented as an example for the relevance and application of the mathematical sciences. Indeed, many principles of modern cryptography can be described at a secondary school level. In this context, the mathematical background is often only sparingly shown. In the worst case, giving mathematics this character of a tool reduces the application of mathematical insights to the message ”cryptography contains math”. This paper examines the question as to what else cryptography can offer to mathematics education. Using the RSA cryptosystem and related content, specific mathematical competencies are highlighted that complement standard teaching, can be taught with cryptography as an example, and extend and deepen key mathematical concepts.