Results

**11 - 19**of**19**### Galaxy Cutsets in Graphs

"... Abstract. Given a network G = (V, E), we say that a subset of vertices S ⊆ V has radius r if it is spanned by a tree of depth r. We are interested in determining whether G has a cutset that can be written as the union of k sets of radius r. This generalizes the notion of k-vertex connectivity, since ..."

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Abstract. Given a network G = (V, E), we say that a subset of vertices S ⊆ V has radius r if it is spanned by a tree of depth r. We are interested in determining whether G has a cutset that can be written as the union of k sets of radius r. This generalizes the notion of k-vertex connectivity, since in the special case r = 0, a set spanned by a tree of depth r is a single vertex. Our motivation for considering this problem is that it constitutes a simple model for virus-like malicious attacks on G: An attack occurs at a subset of k vertices and begins to spread through the network. Any vertex within distance r of one of the initially attacked vertices can be infected. Thus an attack corresponds to a subset of vertices that is spanned by k trees of depth at most r. The question we focus on is whether a given network has a cutset of this particular form. The main results of this paper are the following. If r = 1, an attack corresponds to a subset of vertices which is the union of at most k stars. We call such a set a galaxy of order k. We show that it is NP-hard to determine whether a given network contains a cutset which is a galaxy of order k, if k is part of the input. This is in stark contrast to the case r = 0, since testing whether a graph is k-vertex connected can be done in polynomial time, using standard maxflow-mincut type results. In contrast, testing whether a graph can be disconnected by a single attack (i.e. k = 1) can be done efficiently. Such an attack corresponds to a set of vertices spanned by a tree of depth r. We present an O(rnm) algorithm that determines if a given network contains such a set as a cutset.

### Cryptographic Accelerators on the UltraSPARC T2 with the Solaris Cryptographic Framework

"... As the both the requirement and demand for secure systems increases, so to will the ubiquitousness of cryptography. The most secure cryptographic schemes often involve complicated algorithms and are by no means cheap to implement on standard hardware, and it is this that has led to the development o ..."

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As the both the requirement and demand for secure systems increases, so to will the ubiquitousness of cryptography. The most secure cryptographic schemes often involve complicated algorithms and are by no means cheap to implement on standard hardware, and it is this that has led to the development of cryptographic hardware accelerators. Optimizing software to take advantage of these hardware devices is a problem akin to that of effective parallelization and this project aims to determine how these accelerators perform and under what conditions their use is cost-effective. Through the development of code designed to exercise the particular accelerators existing on the Solaris UltraSPARC T2 via the Solaris Cryptographic Framework (SCF) the system’s performance under a variety of different conditions was assessed. A suggestion for the possible design of a benchmark exclusively for hardware accelerated cryptography is also given. The results indicate that substantial performance gains can be had with

### BOOK REVIEW: INEVITABLE RANDOMNESS IN DISCRETE MATHEMATICS

"... The beauty and utility of randomness is more than matched by its mysteries. How can we tell if a putative source of randomness (such as the frequency of the emission of electrons from a decaying radioactive material) is truly random? Indeed, how does one define randomness? ..."

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The beauty and utility of randomness is more than matched by its mysteries. How can we tell if a putative source of randomness (such as the frequency of the emission of electrons from a decaying radioactive material) is truly random? Indeed, how does one define randomness?

### Thesis Supervisor Accepted by.......................................................................

, 2008

"... In this thesis, I investigate aspects of local Hamiltonians in quantum computing. First, I focus on the Adiabatic Quantum Computing model, based on evolution with a timedependent Hamiltonian. I show that to succeed using AQC, the Hamiltonian involved must have local structure, which leads to a resul ..."

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In this thesis, I investigate aspects of local Hamiltonians in quantum computing. First, I focus on the Adiabatic Quantum Computing model, based on evolution with a timedependent Hamiltonian. I show that to succeed using AQC, the Hamiltonian involved must have local structure, which leads to a result about eigenvalue gaps from information theory. I also improve results about simulating quantum circuits with AQC. Second, I look at classically simulating time evolution with local Hamiltonians and finding their ground state properties. I give a numerical method for finding the ground state of translationally invariant Hamiltonians on an infinite tree. This method is based on imaginary time evolution within the Matrix Product State ansatz, and uses a new method for bringing the state back to the ansatz after each imaginary time step. I then use it to investigate the phase transition in the transverse field Ising model on the Bethe lattice. Third, I focus on locally constrained quantum problems Local Hamiltonian and Quantum Satisfiability and prove several new results about their complexity. Finally, I define a Hamiltonian Quantum Cellular Automaton, a continuous-time model of computation which doesn’t require control

### Primeless Factoring-Based Cryptography –Solving the complexity bottlenecks of public-key encryption with ephemeral keys–

"... Abstract. Factoring-based public-key cryptosystems have an overall complexity which is dominated by the key-production algorithm, which requires the generation of prime numbers. This is most inconvenient in settings where the key-generation is not an one-off process, e.g., secure delegation of compu ..."

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Abstract. Factoring-based public-key cryptosystems have an overall complexity which is dominated by the key-production algorithm, which requires the generation of prime numbers. This is most inconvenient in settings where the key-generation is not an one-off process, e.g., secure delegation of computation or EKE password-based key exchange protocols. To this end, we extend the Goldwasser-Micali (GM) cryptosystem to a provably secure system, denoted SIS, where the generation of primes is bypassed. By developing on the correct choice of the parameters of SIS, we align SIS’s security guarantees (i.e., resistance to factoring of moduli, etc.) to those of other well-known factoring-based cryptosystems. Taking into consideration different possibilities to implement the fundamental operations, we explicitly compare and contrast the asymptotic complexity of well-known public-key cryptosystems (e.g., GM and/or RSA) with that of SIS’s. The latter shows that once we are ready to accept an increase in the size of the moduli, SIS offers a generally lower asymptotic complexity than, e.g., GM or even RSA (when scaling correctly the number of encrypted bits). This would yield most significant speed-ups to applications like the aforementioned secure delegation of computation or protocols where a fresh key needs to be generated with every new session, e.g., EKE password-based key exchange protocols. 1

### An Algorithm For Factoring Integers

"... seems new. Keywords ulus We propose an algorithm for factoring a composite number. The method 1. ..."

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seems new. Keywords ulus We propose an algorithm for factoring a composite number. The method 1.

### Construction of Extractors and other Pseudorandom objects

"... In this article, we study seedless extractors and other pseudorandom objects. We give constructions for seedless extractors with one bit output using expanders and error-correcting codes as black boxes. Further, we give an extractor based on higher order characters in Fp whose output length is optim ..."

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In this article, we study seedless extractors and other pseudorandom objects. We give constructions for seedless extractors with one bit output using expanders and error-correcting codes as black boxes. Further, we give an extractor based on higher order characters in Fp whose output length is optimal up to a factor. Subsequently, we also give definitions of colored expanders and show how they can be used for extractors with longer output length. In order to extend the relation between extractors with multiple sources and expanders, we use expander hypergraphs. While their definition in [FW95] was algebraic, we give corresponding combinatorial definitions of vertex expansion and random walks. Further, we show how algebraic definitions implies each of them. We also give construction of expander hypergraphs based on known expander graphs and show how well known examples of hypergraphs correspond to this black-box construction. We also define analogues of graph powering and tensor products while giving a possible candidate for zig-zag product. 1

### Proceedings of the Twenty-Third International Joint Conference on Artificial Intelligence Probabilistic Equivalence Verification Approach for Automatic Mathematical Solution Assessment

"... Automatic mathematical solution assessment checks the equivalence of mathematical expressions in the user answer and standard solution. It is a challenging problem as the semantics of mathematical expressions are highly symbolic and equivalent mathematical expressions can be expressed in different f ..."

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Automatic mathematical solution assessment checks the equivalence of mathematical expressions in the user answer and standard solution. It is a challenging problem as the semantics of mathematical expressions are highly symbolic and equivalent mathematical expressions can be expressed in different forms. In this paper, we propose an effective Probabilistic Equivalence Verification (PEV) approach for automatic mathematical solution assessment. The proposed PEV approach is a randomized method based on the probabilistic numerical equivalence testing of two mathematical expressions. It can avoid false negative errors completely while guaranteeing a small probability of false positive errors to occur. The performance results have shown that the proposed PEV approach has outperformed other popular techniques in Computer Algebra Systems such as Maple and Mathematica. 1