It is a longstanding open problem whether there is an algorithm to decide the Skolem Problem for linear recurrence sequences (LRS) over the integers, namely whether a given such sequence has a zero term (i.e., whether un = 0 for some n). A major breakthrough in the early 1980s established decidability for LRS of order 4 or less, i.e., for LRS in which every new term depends linearly on the previous four (or fewer) terms. The Skolem Problem for LRS of order 5 or more, in particular, remains a major open challenge to this day. Our main contributions in this paper are as follows: First, we show that the Skolem Problem is decidable for reversible LRS of order 7 or less. (An integer LRS is reversible if its unique extension to a bi-infinite LRS also takes exclusively integer values; a typical example is the classical Fibonacci sequence, whose bi-infinite extension is ⟨…, 5, −3, 2, −1, 1, 0, 1, 1, 2, 3, 5, …⟩.) Second, assuming the Skolem Conjecture (a central hypothesis in Diophantine analysis, also known as the Exponential Local-Global Principle), we show that the Skolem Problem for LRS of order 5 is decidable, and exhibit a concrete procedure for solving it.

The Skolem Problem asks, given a linear recurrence sequence (un), whether there exists n ∈ N such that un = 0. In this paper we consider the following specialisation of the problem: given in addition c ∈ N, determine whether there exists n ∈ N of the form n = lpk, with k, l ≤ c and p any prime number, such that un = 0.

Protecting software from malware injection is one of the biggest challenges of modern computer science. Despite intensive efforts by the scientific and engineering community, the number of successful attacks continues to increase. This work sets first footsteps towards a provably secure investigation of malware detection. We provide a formal model and cryptographic security definitions of attestation for systems with dynamic memory, and suggest novel provably secure attestation schemes. The key idea underlying our schemes is to use the very insertion of the malware itself to allow for the systems to detect it. This is, in our opinion, close in spirit to the quantum Observer Effect. The attackers, no matter how clever, no matter when they insert their malware, change the state of the system they are attacking. This fundamental idea can be a game changer. And our system does not rely on heuristics; instead, our scheme enjoys the unique property that it is proved secure in a formal and precise mathematical sense and with minimal and realistic CPU modification achieves strong provable security guarantees. We envision such systems with a formal mathematical security treatment as a venue for new directions in software protection.

Memory corruption attacks may lead to complete takeover of systems. There are numerous works offering protection mechanisms for this important problem. But the security guarantees that are offered by most works are only heuristic and, furthermore, most solutions are designed for protecting the local memory. In this paper we initiate the study of provably secure remote memory attestation; we concentrate on provably detecting heap-based overflow attacks and consider the setting where we aim to protect the memory in a remote system. We present two protocols offering various efficiency and security trade-offs (but all solutions are efficient enough for practical use as our implementation shows) that detect the presence of injected malicious code or data in remotely-stored heap memory. While our solutions offer protection only against a specific class of attacks, our novel formalization of threat models is general enough to cover a wide range of attacks and settings.

This introduction to quantum algorithms is concise but comprehensive, covering many key algorithms. It is mathematically rigorous but requires minimal background and assumes no knowledge of quantum theory or quantum mechanics. The book explains quantum computation in terms of elementary linear algebra; it assumes the reader will have some familiarity with vectors, matrices, and their basic properties, but offers a review of all the relevant material from linear algebra. By emphasizing computation and algorithms rather than physics, this primer makes quantum algorithms accessible to students and researchers in computer science without the complications of quantum mechanical notation, physical concepts, and philosophical issues. After explaining the development of quantum operations and computations based on linear algebra, the book presents the major quantum algorithms, from seminal algorithms by Deutsch, Jozsa, and Simon through Shor's and Grover's algorithms to recent quantum walks. It covers quantum gates, computational complexity, and some graph theory. Mathematical proofs are generally short and straightforward; quantum circuits and gates are used to illuminate linear algebra; and the discussion of complexity is anchored in computational problems rather than machine models. Quantum Algorithms via Linear Algebra is suitable for classroom use or as a reference for computer scientists and mathematicians.

We initiate the study of provably secure remote memory attestation. We present two protocols offering various efficiency and security trade-offs that detect the presence of injected malicious code in remotelystored heap memory. While our solutions offer protection only against a specific class of attacks, our novel formal security definitions are general enough to cover a wide range of attacks and settings, and should be useful for further research on the subject.

Protecting software from malware injection is the holy grail of modern computer security. Despite intensive efforts by the scientific and engineering community, the number of successful attacks continues to increase. We have a breakthrough novel approach to provably detect malware injection. The key idea is to use the very insertion of the malware itself to allow for the systems to detect it. This is, in our opinion, close in spirit to the famous Heisenberg Uncertainty Principle. The attackers, no matter how clever, no matter when or how they insert their malware, change the state of the system they are attacking. This fundamental idea is a game changer. And our system does not rely on heuristics; instead, our scheme enjoys the unique property that it is proved secure in a formal and precise mathematical sense and with minimal and realistic CPU modification achieves strong provable security guarantees. Thus, we anticipate our system and formal mathematical security treatment to open new directions in software protection.

This chapter contains sections titled: 11.1 Strategy, 11.2 Good Numbers, 11.3 Quantum Part of the Algorithm, 11.4 Analysis of the Quantum Part, 11.5 Probability of a Good Number, 11.6 Using a Good Number, 11.7 Continued Fractions, 11.8 Problems, 11.9 Summary and Notes

This chapter contains sections titled: 8.1 The Algorithm, 8.2 The Analysis, 8.3 Superdense Coding and Teleportation, 8.4 Problems, 8.5 Summary and Notes

It is shown that the Jacobian Conjecture (in all dimensions) is equivalent to the following statement: for almost all prime numbers p and each Keller map F∈Zp[X]n (i.e. det⁡JF=1), the induced map F¯:Fpn→Fpn is not the zero map.

This chapter contains sections titled: 16.1 The Class BQP, 16.2 Equations, Solutions, and Complexity, 16.3 A Circuit Labeling Algorithm, 16.4 Sum-Over-Paths and Polynomial Roots, 16.5 The Additive Polynomial Simulation, 16.6 Bounding BQP, 16.7 Problems, 16.8 Summary and Notes

This chapter contains sections titled: 13.1 Two Vectors, 13.2 The Algorithm, 13.3 The Analysis, 13.4 The General Case, with k Unknown, 13.5 Grover Approximate Counting, 13.6 Problems, 13.7 Summary and Notes

We present an electronics design for a tracking trigger for the CERN SLHC. The on detector part uses asynchronous logic so the only clock required is the LHC crossing clock. High Pt tracks are identified with a hierarchical method that finds track stubs using closely spaced pairs of detectors. Track segments (called tracklets) are then formed from pairs of stubs that are separated by 40 mm. This separation is close enough so that matching stubs is relatively easy but far enough apart so that the tracklets can be projected to another layer with mm accuracy. Matching segments in two or more layers then define a track.

Amplifying size–depth lower bounds. If CircEval has Boolean circuits of n k size and n 1−δ depth for some k and δ, then for every \({\epsilon > 0}\), there is a δ′ > 0 such that CircEval has circuits of \({n^{1 + \epsilon}}\) size and \({n^{1- \delta^{\prime}}}\) depth. Moreover, the resulting circuits require only \({\tilde{O}(n^{\epsilon})}\) bits of non-uniformity to construct. As a consequence, strong enough depth lower bounds for Circuit Evaluation imply a full separation of P and NC (even with a weak size lower bound).