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Cube Testers and Key Recovery Attacks On Reducedround MD6 and Trivium
"... CRYPTO 2008 saw the introduction of the hash function MD6 and of cube attacks, a type of algebraic attack applicable to cryptographic functions having a lowdegree algebraic normal form over GF(2). This paper applies cube attacks to reduced round MD6, finding the full 128bit key of a 14round MD6 w ..."
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CRYPTO 2008 saw the introduction of the hash function MD6 and of cube attacks, a type of algebraic attack applicable to cryptographic functions having a lowdegree algebraic normal form over GF(2). This paper applies cube attacks to reduced round MD6, finding the full 128bit key of a 14round MD6 with complexity 2 22 (which takes less than a minute on a single PC). This is the best key recovery attack announced so far for MD6. We then introduce a new class of attacks called cube testers, based on efficient propertytesting algorithms, and apply them to MD6 and to the stream cipher Trivium. Unlike the standard cube attacks, cube testers detect nonrandom behavior rather than performing key extraction, but they can also attack cryptographic schemes described by nonrandom polynomials of relatively high degree. Applied to MD6, cube testers detect nonrandomness over 18 rounds in 2 17 complexity; applied to a slightly modified version of the MD6 compression function, they can distinguish 66 rounds from random in 2 24 complexity. Cube testers give distinguishers on Trivium reduced to 790 rounds from random with 2 30 complexity and detect nonrandomness over 885 rounds in 2 27, improving on the original 767round cube attack.
What is good mathematics
, 2007
"... Abstract. Some personal thoughts and opinions on what “good quality mathematics” is, and whether one should try to define this term rigorously. As a case study, the story of Szemerédi’s theorem is presented. 1. The many aspects of mathematical quality We all agree that mathematicians should strive t ..."
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Abstract. Some personal thoughts and opinions on what “good quality mathematics” is, and whether one should try to define this term rigorously. As a case study, the story of Szemerédi’s theorem is presented. 1. The many aspects of mathematical quality We all agree that mathematicians should strive to produce good mathematics. But how does one define “good mathematics”, and should one even dare to try at all? Let us first consider the former question. Almost immediately one realises that there are many different types of mathematics which could be designated “good”. For instance, “good mathematics ” could refer (in no particular order) to (i) Good mathematical problemsolving (e.g. a major breakthrough on an important mathematical problem); (ii) Good mathematical technique (e.g. a masterful use of existing methods, or the development of new tools); (iii) Good mathematical theory (e.g. a conceptual framework or choice of notation which systematically unifies and generalises an existing body of results);
ON NOWHERE DENSE GRAPHS
"... A set A of vertices of a graph G is called dscattered in G if no two dneighborhoods of (distinct) vertices of A intersect. In other words, A is dscattered if no two distinct vertices of A have distance at most 2d. This notion was isolated in the context of finite model theory by Gurevich and rec ..."
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A set A of vertices of a graph G is called dscattered in G if no two dneighborhoods of (distinct) vertices of A intersect. In other words, A is dscattered if no two distinct vertices of A have distance at most 2d. This notion was isolated in the context of finite model theory by Gurevich and recently it played a prominent role in the study of homomorphism preservation theorems for special classes of structures (such as minor closed families). This in turn led to the notions of wide, semiwide and quasiwide classes of graphs. It has been proved previously that minor closed classes and classes of graphs with locally forbidden minors are examples of such classes and thus (relativised) homomorphism preservation theorem holds for them. In this paper we show that (more general) classes with bounded expansion and (newly defined) classes with bounded local expansion and even (very general) classes of nowhere dense graphs are quasi wide. This not only strictly generalizes the previous results and solves several open problems but it also provides new proofs. It appears that bounded expansion and nowhere dense classes are perhaps a proper setting for investigation of widetype classes as in several instances we obtain a structural characterization. This also puts classes of bounded expansion in the new context and we are able to prove a trichotomy result which separates classes of graphs which are dense (somewhere dense), nowhere dense and finite. Our motivation stems from finite dualities. As a corollary we obtain that any homomorphism closed first order definable property restricted to a bounded expansion class is a duality.
Asymptotic Abelianness, weak mixing, and property T
"... Abstract. LetG be a second countable locally compact group andH a closed subgroup. We characterize the lack of Kazhdan’s property T for the pair (G,H) by the genericity of Gactions on the hyperfinite II1 factor with a certain asymptotic Abelianness property relative toH, as well as by the genericit ..."
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Abstract. LetG be a second countable locally compact group andH a closed subgroup. We characterize the lack of Kazhdan’s property T for the pair (G,H) by the genericity of Gactions on the hyperfinite II1 factor with a certain asymptotic Abelianness property relative toH, as well as by the genericity of measurepreserving Gactions on a nonatomic standard probability space that are weakly mixing for H. The latter furnishes a definitive generalization of a classical theorem of Halmos for single automorphisms and strengthens a recent result of Glasner, Thouvenot, and Weiss on generic ergodicity. We also establish a weak mixing version of Glasner and Weiss’s characterization of property T for discrete G in terms of the invariant state space of a Bernoulli shift and show that on the CAR algebra a type of norm asymptotic Abelianness is generic for Gactions when G is discrete and admits a nontorsion Abelian quotient. 1.
A simple regularization of hypergraphs
"... Abstract. We give a simple and natural construction of hypergraph regularization. It yields a short proof of a hypergraph regularity lemma. Consequently, as an example of its applications, we have a short selfcontained proof of Szemerédi’s classic theorem on arithmetic progressions (1975) as well a ..."
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Abstract. We give a simple and natural construction of hypergraph regularization. It yields a short proof of a hypergraph regularity lemma. Consequently, as an example of its applications, we have a short selfcontained proof of Szemerédi’s classic theorem on arithmetic progressions (1975) as well as its multidimensional extension by FurstenbergKatznelson (1978). 1.
Removal lemma for infinitelymany forbidden hypergraphs and property testing
, 2008
"... We prove a removal lemma for infinitelymany forbidden hypergraphs. It affirmatively settles a question on property testing raised by Alon and Shapira (2005) [2, 3]. All monotone hypergraph properties and all hereditary partite hypergraph properties are testable. Our proof constructs a constanttim ..."
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Cited by 6 (5 self)
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We prove a removal lemma for infinitelymany forbidden hypergraphs. It affirmatively settles a question on property testing raised by Alon and Shapira (2005) [2, 3]. All monotone hypergraph properties and all hereditary partite hypergraph properties are testable. Our proof constructs a constanttime probabilistic algorithm to edit a small number of edges. It also gives a quantitative bound in terms of a coloring number of the property. It is based on a new hypergraph regularity lemma [14].
COMBINATORIAL INDEPENDENCE IN MEASURABLE DYNAMICS
"... Abstract. We develop a finescale local analysis of measure entropy and measure sequence entropy based on combinatorial independence. The concepts of measure IEtuples and measure INtuples are introduced and studied in analogy with their counterparts in topological dynamics. Local characterizations ..."
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Abstract. We develop a finescale local analysis of measure entropy and measure sequence entropy based on combinatorial independence. The concepts of measure IEtuples and measure INtuples are introduced and studied in analogy with their counterparts in topological dynamics. Local characterizations of the Pinsker von Neumann algebra and its sequence entropy analogue are given in terms of combinatorial independence, ℓ1 geometry, and Voiculescu’s completely positive approximation entropy. Among the novel features of our local study is the treatment of general discrete acting groups, with the structural assumption of amenability in the case of entropy. 1.