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Publication# Round Compression for Parallel Matching Algorithms

Résumé

For over a decade now we have been witnessing the success of massive parallel computation (MPC) frameworks, such as MapReduce, Hadoop, Dryad, or Spark. One of the reasons for their success is the fact that these frameworks are able to accurately capture the nature of large-scale computation. In particular, compared to the classic distributed algorithms or PRAM models, these frameworks allow for much more local computation. The fundamental question that arises in this context is though: can we leverage this additional power to obtain even faster parallel algorithms? A prominent example here is the maximum matching problem-one of the most classic graph problems. It is well known that in the PRAM model one can compute a 2-approximate maximum matching in O(logn) rounds. However, the exact complexity of this problem in the MPC framework is still far from understood. Lattanzi et al. (SPAA 2011) showed that if each machine has n(1+Omega(1)) memory, this problem can also be solved 2-approximately in a constant number of rounds. These techniques, as well as the approaches developed in the follow up work, seem though to get stuck in a fundamental way at roughly O(logn) rounds once we enter the (at most) near-linear memory regime. It is thus entirely possible that in this regime, which captures in particular the case of sparse graph computations, the best MPC round complexity matches what one can already get in the PRAM model, without the need to take advantage of the extra local computation power. In this paper, we finally refute that possibility. That is, we break the above O(logn) round complexity bound even in the case of slightly sublinear memory per machine. In fact, our improvement here is almost exponential: we are able to deliver a (2+epsilon)-approximate maximum matching, for any fixed constant epsilon > 0, in O (log logn)(2)) rounds. To establish our result we need to deviate from the previous work in two important ways that are crucial for exploiting the power of the MPC model, as compared to the PRAM model. Firstly, we use vertex-based graph partitioning, instead of the edge-based approaches that were utilized so far. Secondly, we develop a technique of round compression. This technique enables one to take a (distributed) algorithm that computes an O(1)-approximation of maximum matching in O(logn) independent PRAM phases and implement a super-constant number of these phases in only a constant number of MPC rounds.

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The security of public-key cryptography relies on well-studied hard problems, problems for which we do not have efficient algorithms. Factorization and discrete logarithm are the two most known and used hard problems. Unfortunately, they can be easily solved on a quantum computer by Shor's algorithm. Also, the research area of cryptography demands for crypto-diversity which says that we should offer a range of hard problems for public-key cryptography. If one hard problem proves to be easy, we should be able to provide alternative solutions. Some of the candidates for post-quantum hard problems, i.e. problems which are believed to be hard even on a quantum computer, are the Learning Parity with Noise (LPN), the Learning with Errors (LWE) and the Shortest Vector Problem (SVP). A thorough study of these problems is needed in order to assess their hardness. In this thesis we focus on the algorithmic study of LPN. LPN is a hard problem that is attractive, as it is believed to be post-quantum resistant and suitable for lightweight devices. In practice, it has been employed in several encryption schemes and authentication protocols. At the beginning of this thesis, we take a look at the existing LPN solving algorithms. We provide the theoretical analysis that assesses their complexity. We compare the theoretical results with practice by implementing these algorithms. We study the efficiency of all LPN solving algorithms which allow us to provide secure parameters that can be used in practice. We push further the state of the art by improving the existing algorithms with the help of two new frameworks. In the first framework, we split an LPN solving algorithm into atomic steps. We study their complexity, how they impact the other steps and we construct an algorithm that optimises their use. Given an LPN instance that is characterized by the noise level and the secret size, our algorithm provides the steps to follow in order to solve the instance with optimal complexity. In this way, we can assess if an LPN instance provides the security we require and we show what are the secure instances for the applications that rely on LPN. The second framework handles problems that can be decomposed into steps of equal complexity. Here, we assume that we have an adversary that has access to a finite or infinite number of instances of the same problem. The goal of the adversary is to succeed in just one instance as soon as possible. Our framework provides the strategy that achieves this. We characterize an LPN solving algorithm in this framework and show that we can improve its complexity in the scenario where the adversary is restricted. We show that other problems, like password guessing, can be modeled in the same framework.

The graph coloring problem is one of the most famous problems in graph theory and has a large range of applications. It consists in coloring the vertices of an undirected graph with a given number of colors such that two adjacent vertices get different colors. This thesis deals with some variations of this basic coloring problem which are related to scheduling and discrete tomography. These problems may also be considered as partitioning problems. In Chapter 1 basic definitions of computational complexity and graph theory are presented. An introduction to graph coloring and discrete tomography is given. In the next chapter we discuss two coloring problems in mixed graphs (i.e., graphs having edges and arcs) arising from scheduling. In the first one (strong mixed graph coloring problem) we have to cope with disjunctive constraints (some pairs of jobs cannot be processed simultaneously) as well as with precedence constraints (some pairs of jobs must be executed in a given order). It is known that this problem is NP-complete in mixed bipartite graphs. In this thesis we strengthen this result by proving that for k = 3 colors the strong mixed graph coloring problem is NP-complete even if the mixed graph is planar bipartite with maximum degree 4 and each vertex incident to at least one arc has maximum degree 2 or if the mixed graph is bipartite and has maximum degree 3. Furthermore we show that the problem is polynomially solvable in partial p-trees, for fixed p, as well as in general graphs with k = 2 colors. We also give upper bounds on the strong mixed chromatic number or even its exact value for some classes of graphs. In the second problem (weak mixed graph coloring problem), we allow jobs linked by precedence constraints to be executed at the same time. We show that for k = 3 colors this problem is NP-complete in mixed planar bipartite graphs of maximum degree 4 as well as in mixed bipartite graphs of maximum degree 3. Again, for partial p-trees, p fixed, and for general graphs with k = 2 colors, we prove that the weak mixed graph coloring problem is polynomially solvable. We consider in Chapter 3 the problem of characterizing in an undirected graph G = (V, E) a minimum set R of edges for which maximum matchings M can be found with specific values of p = |M ∩ R|. We obtain partial results for some classes of graphs and show in particular that for odd cacti with triangles only and for forests one can determine in polynomial time whether there exists a minimum set R for which there are maximum matchings M such that p= |R ∩ M|, for p= 0,1, ..., ν(G). The remaining chapters deal with some coloring (or partitioning) problems related to the basic image reconstruction problem in discrete tomography. In Chapter 4 we consider a generalization of the vertex coloring problem associated with the basic image reconstruction problem. We are given an undirected graph and a family of chains covering its vertices. For each chain the number of occurrences of each color is given. We then want to find a coloring respecting these occurrences. We are interested in both, arbitrary and proper colorings and give complexity results. In particular we show that for arbitrary colorings the problem is NP-complete with two colors even if the graph is a tree of maximum degree 3. We also consider the edge coloring version of both problems. Again we present some complexity results. We consider in Chapter 5 some generalized neighborhoods instead of chains. For each vertex x we are given the number of occurrences of each color in its open neighborhood Nd(x) (resp. closed neighborhood Nd+(x)), representing the set of vertices which are at distance d from x (resp. at distance at most d from x). We are interested in arbitrary colorings as well as proper colorings. We present some complexity results and we show in particular that for d = 1 the problems are polynomially solvable in trees using a dynamic programming approach. For the open neighborhood and d = 2 we obtain a polynomial time algorithm for quatrees (i.e. trees where all internal vertices have degree at least 4). We also examine the bounded version of these problems, i.e., instead of the exact number of occurrences of each color we are given upper bounds on these occurrences. In particular we show that the problem for proper colorings is NP-complete in bipartite graphs of maximum degree 3 with four colors and each color appearing at most once in the neighborhood N(x) of each vertex x. This result implies that the L(1,1)-labelling problem is NP-complete in this class of graphs for four colors. Finally in Chapter 6 we consider the edge partitioning version of the basic image reconstruction problem, i.e., we have to partition the edge set of a complete bipartite graph into k subsets such that for each vertex there must be a given number of edges of each set of the partition incident to this vertex. For k = 3 the complexity status is still open. Here we present a new solvable case for k = 3. Then we examine some variations where the union of two subsets E1, E2 has to satisfy some additional constraints as for example it must form a tree or a collection of disjoint chains. In both cases we give necessary and sufficient conditions for a solution to exist. We also consider the case where we have a complete graph instead of a complete bipartite graph. We show that the edge partitioning problem in a complete graph is at least as difficult as in a complete bipartite graph. We also give necessary and sufficient conditions for a solution to exist if E1 ∪ E2 form a tree or if they form a Hamiltonian cycle in the case of a complete graph. Finally we examine for both, complete and complete bipartite graphs, the case where each one of the sets E1 and E2 is structured (two disjoint Hamiltonian chains, two edge disjoint cycles) and present necessary and sufficient conditions.

Modern cryptography pushed forward the need of having provable security. Whereas ancient cryptography was only relying on heuristic assumptions and the secrecy of the designs, nowadays researchers try to make the security of schemes to rely on mathematical problems which are believed hard to solve. When doing these proofs, the capabilities of potential adversaries are modeled formally. For instance, the black-box model assumes that an adversary does not learn anything from the inner-state of a construction. While this assumption makes sense in some practical scenarios, it was shown that one can sometimes learn some information by other means, e.g., by timing how long the computation take. In this thesis, we focus on two different areas of cryptography. In both parts, we take first a theoretical point of view to obtain a result. We try then to adapt our results so that they are easily usable for implementers and for researchers working in practical cryptography. In the first part of this thesis, we take a look at post-quantum cryptography, i.e., at cryptographic primitives that are believed secure even in the case (reasonably big) quantum computers are built. We introduce HELEN, a new public-key cryptosystem based on the hardness of the learning from parity with noise problem (LPN). To make our results more concrete, we suggest some practical instances which make the system easily implementable. As stated above, the design of cryptographic primitives usually relies on some well-studied hard problems. However, to suggest concrete parameters for these primitives, one needs to know the precise complexity of algorithms solving the underlying hard problem. In this thesis, we focus on two recent hard-problems that became very popular in post-quantum cryptography: the learning with error (LWE) and the learning with rounding problem (LWR). We introduce a new algorithm that solves both problems and provide a careful complexity analysis so that these problems can be used to construct practical cryptographic primitives. In the second part, we look at leakage-resilient cryptography which studies adversaries able to get some side-channel information from a cryptographic primitive. In the past, two main disjoint models were considered. The first one, the threshold probing model, assumes that the adversary can put a limited number of probes in a circuit. He then learns all the values going through these probes. This model was used mostly by theoreticians as it allows very elegant and convenient proofs. The second model, the noisy-leakage model, assumes that every component of the circuit leaks but that the observed signal is noisy. Typically, some Gaussian noise is added to it. According to experiments, this model depicts closely the real behaviour of circuits. Hence, this model is cherished by the practical cryptographic community. In this thesis, we show that making a proof in the first model implies a proof in the second model which unifies the two models and reconciles both communities. We then look at this result with a more practical point-of-view. We show how it can help in the process of evaluating the security of a chip based solely on the more standard mutual information metric.