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Automated theorem proving is a method to establish or disprove logical theorems. While these can be theorems in the classical mathematical sense, we are more concerned with logical encodings of properties of algorithms, hardware and software. Especially in the area of hardware verification, propositional logic is used widely in industry. Satisfiability Module Theories (SMT) is a set of logics which extend propositional logic with theories relevant for specific application domains. In particular, software verification has received much attention, and efficient algorithms have been devised for reasoning over arithmetic and data types. Built-in support for theories by decision procedures is often significantly more efficient than reductions to propositional logic (SAT). Most efficient SAT solvers are based on the DPLL architecture, which is also the basis for most efficient SMT solvers. The main shortcoming of both kinds of logics is the weak support for non-ground reasoning, which noticeably limits the applicability to real world systems. The Model Evolution Calculus (ME) was devised as a lifting of the DPLL architecture from the propositional setting to full first-order logic. In previous work, we created the solver Darwin as an implementation of ME, and showed how to adapt improvements from the DPLL setting. The first half of this thesis is concerned with ME and Darwin. First, we lift a further crucial ingredient of SAT and SMT solvers, lemma-learning, to Darwin and evaluate its benefits. Then, we show how to use Darwin for finite model finding, and how this application benefits from lemma-learning. In the second half of the thesis we present Model Evolution with Linear Integer Arithmetic (MELIA), a calculus combining function-free first-order logic with linear integer arithmetic (LIA). MELIA is based on ME and supports similar inference rules and redundancy criteria. We prove the correctness of the calculus, and show how to obtain complete proof procedures and decision procedures for some interesting classes of MELIA's logic. Finally, we explain in detail how MELIA can be implemented efficiently based on the techniques employed in SMT solvers and Darwin.

<p>The recent increase in performance of embedded processors has enabled the use of computationally heavy asymmetric cryptography in small and power efficient embedded systems. The goal of this thesis is to analyze whether it is possible to use this type of cryptography to enhance the security in access systems.</p><p>This report contains a literature study of the complications related to access systems and their functionality. Also a basic introduction to cryptography is included.</p><p>Several cryptographic algorithms were implemented using the public library LibTomCrypt and benchmarked on an ARM7-processor platform. The asymmetric coding schemes were ECC and RSA. The tested symmetric algorithms included AES, 3DES and Twofish among others. The benchmark considered both codesize and speed of the algorithms.</p><p>The two asymmetric algorithms, ECC and RSA, are possible to be used in an ARM7 based access system. Although, both technologies can be configured to finish the calculations within a reasonable time-frame of 10 Sec, ECC archives a higher security level for the same execution time. Therefore, an implementation of ECC would be preferable since it is faster and requires less resources. Some further suggestions of improvements to the implementation is discussed in the final chapters.</p>

The recent increase in performance of embedded processors has enabled the use of computationally heavy asymmetric cryptography in small and power efficient embedded systems. The goal of this thesis is to analyze whether it is possible to use this type of cryptography to enhance the security in access systems. This report contains a literature study of the complications related to access systems and their functionality. Also a basic introduction to cryptography is included. Several cryptographic algorithms were implemented using the public library LibTomCrypt and benchmarked on an ARM7-processor platform. The asymmetric coding schemes were ECC and RSA. The tested symmetric algorithms included AES, 3DES and Twofish among others. The benchmark considered both codesize and speed of the algorithms. The two asymmetric algorithms, ECC and RSA, are possible to be used in an ARM7 based access system. Although, both technologies can be configured to finish the calculations within a reasonable time-frame of 10 Sec, ECC archives a higher security level for the same execution time. Therefore, an implementation of ECC would be preferable since it is faster and requires less resources. Some further suggestions of improvements to the implementation is discussed in the final chapters.

Hypergoemetric Series are very important in mathematics and come up regularly when dealing with the precise definitions of constants such as <i>e</i>, &pi; and Apery's constant &sigmaf;(3). The evaluation of such series to high precision is traditionally done with multiple divisions, multiplications and factorials, which all takes a long time to compute, especially when the computation is done on a single machine. The interest lies in performing this computation in parallel and in a distributed fashion. In this thesis, we present a simple distributed toolkit for doing such computations by splitting the problem into smaller sub-problems, solving these sub-problems in parallel on distributed machines and then combining the result at the end. Our toolkit takes care of all the networking for the user; connectivity, dropped connections, management of the Clients and the Server. All the user has to provide is the definition of the problem; how to split the problem into sub-problems, how to solve the sub-problems and finally how to combine the sub-problems and produce a result. The toolkit records timings for computation as well as for communication. What is different about our application is that all the code is written in Java (which is completely machine independent) and all the Clients are Java Applets. This means that having a web browser in enough to take part in the computation when it is distributed over the internet. We are almost guaranteed that every computer on the internet has a web browser. The Java Plug-in (if unavailable) can easily be downloaded from Sun's web site. We present a comparison between Java's native BigInteger library and an FFT based Integer Library written by R. Howell of University of Kansas. This study is important since we are doing computations with very large integers. To test our system, we evaluate <i>e</i> to different number of digits of precision and show that our system truly works and is easy for anyone to use.

Hypergoemetric Series are very important in mathematics and come up regularly when dealing with the precise definitions of constants such as <i>e</i>, &pi; and Apery's constant &sigmaf;(3). The evaluation of such series to high precision is traditionally done with multiple divisions, multiplications and factorials, which all takes a long time to compute, especially when the computation is done on a single machine. The interest lies in performing this computation in parallel and in a distributed fashion. In this thesis, we present a simple distributed toolkit for doing such computations by splitting the problem into smaller sub-problems, solving these sub-problems in parallel on distributed machines and then combining the result at the end. Our toolkit takes care of all the networking for the user; connectivity, dropped connections, management of the Clients and the Server. All the user has to provide is the definition of the problem; how to split the problem into sub-problems, how to solve the sub-problems and finally how to combine the sub-problems and produce a result. The toolkit records timings for computation as well as for communication. What is different about our application is that all the code is written in Java (which is completely machine independent) and all the Clients are Java Applets. This means that having a web browser in enough to take part in the computation when it is distributed over the internet. We are almost guaranteed that every computer on the internet has a web browser. The Java Plug-in (if unavailable) can easily be downloaded from Sun's web site. We present a comparison between Java's native BigInteger library and an FFT based Integer Library written by R. Howell of University of Kansas. This study is important since we are doing computations with very large integers. To test our system, we evaluate <i>e</i> to different number of digits of precision and show that our system truly works and is easy for anyone to use.

Large integers are used in several key areas such as RSA (Rivest-Shamir-Adleman) public-key cryptographic system and elliptic curve public-key cryptographic system. To achieve higher levels of security requires larger key size and this becomes a limiting factor in prime finite field GF(p) arithmetic using large integers because operations on large integers suffer from the long carry propagation problem. Large integer representation has direct impact on the efficiency of the calculations and the hardware and software implementations. Attempts to use different representations such as residue number systems suffer from their own problems. In this dissertation, we propose a novel and efficient attribute-based large integer representation scheme capable of efficiently representing the large integers that are commonly used in cryptography such as the five NIST primes and the Pierpont primes used in supersingular isogeny Diffie-Hellman (SIDH) used in post-quantum cryptography. Moreover, we propose algorithms for this new representation to perform arithmetic operations such as conversions from and to binary representation, two’s complement, left-shift, numbers comparison, addition/subtraction, modular addition/subtraction, modular reduction, multiplication, and modular multiplication. Extensive numerical simulations and software implementations are done to verify the performance of the new number representation. Results show that the attribute-based large integer arithmetic operations are done faster in our proposed representation when compared with binary and residue number representations. This makes the proposed representation suitable for cryptographic applications on embedded systems and IoT devices with limited resources for better security level. / Graduate / 2020-07-04

This thesis studies the properties of finite automata recognizing sets of real vectors encoded in positional notation using an integer base. We consider both general infinite-word automata, and the restricted class of weak deterministic automata, used, in particular, as symbolic data structures for representing the sets of vectors definable in the first order additive theory of real and integer numbers. <br><br> In previous work, it has been established that all sets definable in the additive theory of reals and integers can be handled by weak deterministic automata regardless of the chosen numeration base. In this thesis, we address the reciprocal property, proving that the sets of vectors that are simultaneously recognizable in all bases, by either weak deterministic or Muller automata, are those definable in the additive theory of reals and integers. <br><br> Precisely, for weak deterministic automata, we establish that the sets of real vectors simultaneously recognizable in two multiplicatively independent bases are necessarily definable in the additive theory of reals and integers. For general automata, we show that the multiplicative independence is not sufficient, and we prove that, in this context, the sets of real vectors that are recognizable in two bases that do not share the same set of prime factors are exactly those definable in the additive theory of reals and integers. <br><br> Those results lead to a precise characterization of the sets of real vectors that are recognizable in multiple bases, and provide a theoretical justification to the use of weak automata as symbolic representations of sets. <br><br> As additional contribution, we also obtain valuable insight into the internal structure of automata recognizing sets of vectors definable in the additive theory of reals and integers.

This thesis tackles the problem of automatically proving the validity of mathematical formulas generated by program verification tools. In particular, it focuses on Satisfiability Modulo Theories (SMT): a young research topic that has seen great advances during the last decade. The solvers of this family have various applications in hardware design, program verification, model checking, etc.SMT solvers offer a good compromise between expressiveness and efficiency. They rely on a tight cooperation between a SAT solver and a combination of decision procedures for specific theories, such as the free theory of equality with uninterpreted symbols, linear arithmetic over integers and rationals, or the theory of arrays.This thesis aims at improving the efficiency and the expressiveness of the Alt-Ergo SMT solver. For that, we designed a new decision procedure for the theory of linear integer arithmetic. This procedure is inspired by Fourier-Motzkin's method, but it uses a rational simplex to perform computations in practice. We have also designed a new combination framework, capable of reasoning in the union of the free theory of equality, the AC theory of associative and commutativesymbols, and an arbitrary signature-disjoint Shostak theory. This framework is a modular and non-intrusive extension of the ground AC completion procedure with the given Shostak theory. In addition, we have extended Alt-Ergo with existing decision procedures to integrate additional interesting theories, such as the theory of enumerated data types and the theory of arrays. Finally, we have explored preprocessing techniques for formulas simplification as well as the enhancement of Alt-Ergo's SAT solver.

Cette thèse s'intéresse à la démonstration automatique de la validité de formules mathématiques issues de la preuve de programmes. Elle se focalise tout particulièrement sur la Satisfiabilité Modulo Théories (SMT): un jeune domaine de recherche qui a connu de grands progrès durant la dernière décennie. Les démonstrateurs de cette famille ont des applications diverses dans la conception de microprocesseurs, la preuve de programmes, le model-checking, etc.Les démonstrateurs SMT offrent un bon compromis entre l'expressivité et l'efficacité. Ils reposent sur une coopération étroite d'un solveur SAT avec une combinaison de procédures de décision pour des théories spécifiques comme la théorie de l'égalité libre avec des symboles non interprétés, l'arithmétique linéaire sur les entiers et les rationnels, et la théorie des tableaux.L'objectif de cette thèse est d'améliorer l'efficacité et l'expressivité du démonstrateur SMT Alt-Ergo. Pour cela, nous proposons une nouvelle procédure de décision pour la théorie de l'arithmétique linéaire sur les entiers. Cette procédure est inspirée par la méthode de Fourier-Motzkin, mais elle utilise un simplexe sur les rationnels pour effectuer les calculs en pratique. Nous proposons également un nouveau mécanisme de combinaison, capable de raisonner dans l'union de la théorie de l'égalité libre, la théorie AC des symboles associatifs et commutatifs et une théorie arbitraire deShostak. Ce mécanisme est une extension modulaire et non intrusive de la procédure de completion close modulo AC avec la théorie de Shostak. Aussi, nous avons étendu Alt-Ergo avec des procédures de décision existantes pour y intégrer d'autres théories intéressantes comme la théorie de types de données énumérés et la théorie des tableaux. Enfin, nous avons exploré des techniques de simplification de formules en amont et l'amélioration de son solveur SAT. / This thesis tackles the problem of automatically proving the validity of mathematical formulas generated by program verification tools. In particular, it focuses on Satisfiability Modulo Theories (SMT): a young research topic that has seen great advances during the last decade. The solvers of this family have various applications in hardware design, program verification, model checking, etc.SMT solvers offer a good compromise between expressiveness and efficiency. They rely on a tight cooperation between a SAT solver and a combination of decision procedures for specific theories, such as the free theory of equality with uninterpreted symbols, linear arithmetic over integers and rationals, or the theory of arrays.This thesis aims at improving the efficiency and the expressiveness of the Alt-Ergo SMT solver. For that, we designed a new decision procedure for the theory of linear integer arithmetic. This procedure is inspired by Fourier-Motzkin's method, but it uses a rational simplex to perform computations in practice. We have also designed a new combination framework, capable of reasoning in the union of the free theory of equality, the AC theory of associative and commutativesymbols, and an arbitrary signature-disjoint Shostak theory. This framework is a modular and non-intrusive extension of the ground AC completion procedure with the given Shostak theory. In addition, we have extended Alt-Ergo with existing decision procedures to integrate additional interesting theories, such as the theory of enumerated data types and the theory of arrays. Finally, we have explored preprocessing techniques for formulas simplification as well as the enhancement of Alt-Ergo's SAT solver.