TOPICS IN COMPUTATIONAL NUMBER THEORY AND CRYPTANALYSIS - On Simultaneous Chinese Remaindering, Primes, the MiNTRU Assumption, and Functional Encryption

This thesis reports on four independent projects that lie in the intersection of mathematics, computer science, and cryptology: Simultaneous Chinese Remaindering: The classical Chinese Remainder Problem asks to find all integer solutions to a given system of congruences where each congruence is defined by one modulus and one remainder. The Simultaneous Chinese Remainder Problem is a direct generalization of its classical counterpart where for each modulus the single remainder is replaced by a non-empty set of remainders. The solutions of a Simultaneous Chinese Remainder Problem instance are completely defined by a set of minimal positive solutions, called primitive solutions, which are upper bounded by the lowest common multiple of the considered moduli. However, contrary to its classical counterpart, which has at most one primitive solution, the Simultaneous Chinese Remainder Problem may have an exponential number of primitive solutions, so that any general-purpose solving algorithm requires exponential time. Furthermore, through a direct reduction from the 3-SAT problem, we prove first that deciding whether a solution exists is NP-complete, and second that if the existence of solutions is guaranteed, then deciding whether a solution of a particular size exists is also NP-complete. Despite these discouraging results, we studied methods to find the minimal solution to Simultaneous Chinese Remainder Problem instances and we discovered some interesting statistical properties. A Conjecture On Primes In Arithmetic Progressions And Geometric Intervals: Dirichlet’s theorem on primes in arithmetic progressions states that for any positive integer q and any coprime integer a, there are infinitely many primes in the arithmetic progression a + nq (n ∈ N), however, it does not indicate where those primes can be found. Linnik’s theorem predicts that the first such prime p0 can be found in the interval [0;q^L] where L denotes an absolute and explicitly computable constant. Albeit only L = 5 has been proven, it is widely believed that L ≤ 2. We generalize Linnik’s theorem by conjecturing that for any integers q ≥ 2, 1 ≤ a ≤ q − 1 with gcd(q, a) = 1, and t ≥ 1, there exists a prime p such that p ∈ [q^t;q^(t+1)] and p ≡ a mod q. Subsequently, we prove the conjecture for all sufficiently large exponent t, we computationally verify it for all sufficiently small modulus q, and we investigate its relation to other mathematical results such as Carmichael’s totient function conjecture. On The (M)iNTRU Assumption Over Finite Rings: The inhomogeneous NTRU (iNTRU) assumption is a recent computational hardness assumption, which claims that first adding a random low norm error vector to a known gadget vector and then multiplying the result with a secret vector is sufficient to obfuscate the considered secret vector. The matrix inhomogeneous NTRU (MiNTRU) assumption essentially replaces vectors with matrices. Albeit those assumptions strongly remind the well-known learning-with-errors (LWE) assumption, their hardness has not been studied in full detail yet. We provide an elementary analysis of the corresponding decision assumptions and break them in their basis case using an elementary q-ary lattice reduction attack. Concretely, we restrict our study to vectors over finite integer rings, which leads to a problem that we call (M)iNTRU. Starting from a challenge vector, we construct a particular q-ary lattice that contains an unusually short vector whenever the challenge vector follows the (M)iNTRU distribution. Thereby, elementary lattice reduction allows us to distinguish a random challenge vector from a synthetically constructed one. A Conditional Attack Against Functional Encryption Schemes: Functional encryption emerged as an ambitious cryptographic paradigm supporting function evaluations over encrypted data revealing the result in plain. Therein, the result consists either in a valid output or a special error symbol. We develop a conditional selective chosen-plaintext attack against the indistinguishability security notion of functional encryption. Intuitively, indistinguishability in the public-key setting is based on the premise that no adversary can distinguish between the encryptions of two known plaintext messages. As functional encryption allows us to evaluate functions over encrypted messages, the adversary is restricted to evaluations resulting in the same output only. To ensure consistency with other primitives, the decryption procedure of a functional encryption scheme is allowed to fail and output an error. We observe that an adversary may exploit the special role of these errors to craft challenge messages that can be used to win the indistinguishability game. Indeed, the adversary can choose the messages such that their functional evaluation leads to the common error symbol, but their intermediate computation values differ. A formal decomposition of the underlying functionality into a mathematical function and an error trigger reveals this dichotomy. Finally, we outline the impact of this observation on multiple DDH-based inner-product functional encryption schemes when we restrict them to bounded-norm evaluations only

Modern Cryptography Volume 2

This open access book covers the most cutting-edge and hot research topics and fields of post-quantum cryptography. The main purpose of this book is to focus on the computational complexity theory of lattice ciphers, especially the reduction principle of Ajtai, in order to fill the gap that post-quantum ciphers focus on the implementation of encryption and decryption algorithms, but the theoretical proof is insufficient. In Chapter 3, Chapter 4 and Chapter 6, author introduces the theory and technology of LWE distribution, LWE cipher and homomorphic encryption in detail. When using random analysis tools, there is a problem of "ambiguity" in both definition and algorithm. The greatest feature of this book is to use probability distribution to carry out rigorous mathematical definition and mathematical demonstration for various unclear or imprecise expressions, so as to make it a rigorous theoretical system for classroom teaching and dissemination. Chapters 5 and 7 further expand and improve the theory of cyclic lattice, ideal lattice and generalized NTRU cryptography. This book is used as a professional book for graduate students majoring in mathematics and cryptography, as well as a reference book for scientific and technological personnel engaged in cryptography research

Modern Cryptography Volume 2

This open access book covers the most cutting-edge and hot research topics and fields of post-quantum cryptography. The main purpose of this book is to focus on the computational complexity theory of lattice ciphers, especially the reduction principle of Ajtai, in order to fill the gap that post-quantum ciphers focus on the implementation of encryption and decryption algorithms, but the theoretical proof is insufficient. In Chapter 3, Chapter 4 and Chapter 6, author introduces the theory and technology of LWE distribution, LWE cipher and homomorphic encryption in detail. When using random analysis tools, there is a problem of "ambiguity" in both definition and algorithm. The greatest feature of this book is to use probability distribution to carry out rigorous mathematical definition and mathematical demonstration for various unclear or imprecise expressions, so as to make it a rigorous theoretical system for classroom teaching and dissemination. Chapters 5 and 7 further expand and improve the theory of cyclic lattice, ideal lattice and generalized NTRU cryptography. This book is used as a professional book for graduate students majoring in mathematics and cryptography, as well as a reference book for scientific and technological personnel engaged in cryptography research

Evaluation Methodologies in Software Protection Research

Man-at-the-end (MATE) attackers have full control over the system on which the attacked software runs, and try to break the confidentiality or integrity of assets embedded in the software. Both companies and malware authors want to prevent such attacks. This has driven an arms race between attackers and defenders, resulting in a plethora of different protection and analysis methods. However, it remains difficult to measure the strength of protections because MATE attackers can reach their goals in many different ways and a universally accepted evaluation methodology does not exist. This survey systematically reviews the evaluation methodologies of papers on obfuscation, a major class of protections against MATE attacks. For 572 papers, we collected 113 aspects of their evaluation methodologies, ranging from sample set types and sizes, over sample treatment, to performed measurements. We provide detailed insights into how the academic state of the art evaluates both the protections and analyses thereon. In summary, there is a clear need for better evaluation methodologies. We identify nine challenges for software protection evaluations, which represent threats to the validity, reproducibility, and interpretation of research results in the context of MATE attacks

Air Force Institute of Technology Research Report 2012

This report summarizes the research activities of the Air Force Institute of Technology’s Graduate School of Engineering and Management. It describes research interests and faculty expertise; lists student theses/dissertations; identifies research sponsors and contributions; and outlines the procedures for contacting the school. Included in the report are: faculty publications, conference presentations, consultations, and funded research projects. Research was conducted in the areas of Aeronautical and Astronautical Engineering, Electrical Engineering and Electro-Optics, Computer Engineering and Computer Science, Systems and Engineering Management, Operational Sciences, Mathematics, Statistics and Engineering Physics

The major and the minor on political aesthetics in the control society

This thesis examines the crucial diagnostic and productive roles that the concepts of minor and major practice, two interrelated modes of cultural production set out by Gilles Deleuze and Félix Guattari in Kafka: toward a Minor Literature (1975), have to play in the present era of ubiquitous digital technology and informatics that Deleuze himself has influentially described as the control society. In first establishing the conditions of majority and majority, Deleuze and Guattari’s historical focus in Kafka is the early twentieth century period of Franz Kafka’s writing, a period which, for Deleuze, marks the start of a transition between two types of society – the disciplinary society described by Michel Foucault in Discipline and Punish and the control society that is set apart by its distribution, indifferent technical processes and the replacement of the individual with the dividual in social and political thought. Because of their unique conceptual location, at the transition between societies, the concepts of majority and minority present an essential framework for understanding the impact of ubiquitous digital technology and informatics on cultural production in the twentieth century and beyond. In order to determine the conditions of contemporary major and minor practice across the transition from disciplinary to control societies, the thesis is comprised of two interconnecting threads corresponding to majority and minority respectively. Drawing on the theoretical work of Deleuze and Guattari, Friedrich Kittler and Fredric Jameson alongside pioneering figures in the historical development of computation and informatics (Alan Turing, Claude Shannon and others), material observation on the technical function of digital machines, and the close examination of emblematic cultural forms, I determine the specific conditions of majority that emerge through the development of the contemporary control era. Alongside this delineation of the conditions of majority I examine the prospective tactics, corresponding to the characteristics of minority set out by Deleuze and Guattari in Kafka, which emerge as a contemporary counter-practice within the control-era. This is carried out through the close observation of key examples of cultural production in the fields of literature, film, video, television and the videogame that manifest prospective tactics for a control-era minor practice within the overarching technical characteristics of the control-era major. Through an examination of these interrelated threads the thesis presents a framework for both addressing the significant political and cultural changes that ubiquitous computation effects in constituting the contemporary control society and determining the ways in which these changes can be addressed and countered through cultural production

Modified Mclaren-marsaglia Pseudo-random Number Generator and Stochastic Key Agreement

A discussion of problems in cryptographic applications, with a brief survey of pseudo-random number generators (PRNG) used as synchronous stream ciphers, leads to a discussion of the McClaren-Marsaglia shuffling PRNG, and some means of altering its structure to both provide a more secure PRNG and to provide effective means by which to inject aperiodicity into a modified form of McClaren-Marsaglia. A discussion of two closely related protocols using this modified form of McClaren-Marsaglia as means by which correspondents may agree upon a set of random bits in a manner suitable for use in cryptographic applications is then presented, with implementation in the C programming language of the second protocol. Analysis of the protocols concludes that a reasonable expectation of confidentiality and cryptographic strength in the agreed bit-sequence is obtained.Computer Science Departmen