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77
Every monotone graph property has a sharp threshold
 Proc. Amer. Math. Soc
, 1996
"... Abstract. In their seminal work which initiated random graph theory Erdös and Rényi discovered that many graph properties have sharp thresholds as the number of vertices tends to infinity. We prove a conjecture of Linial that every monotone graph property has a sharp threshold. This follows from the ..."
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Cited by 131 (14 self)
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Abstract. In their seminal work which initiated random graph theory Erdös and Rényi discovered that many graph properties have sharp thresholds as the number of vertices tends to infinity. We prove a conjecture of Linial that every monotone graph property has a sharp threshold. This follows from the following theorem. Let Vn(p) ={0,1} n denote the Hamming space endowed with the probability measure µp defined by µp(ɛ1,ɛ2,...,ɛn) = pk ·(1 − p) n−k,where k = ɛ1+ ɛ2+ ···+ ɛn. Let A be a monotone subset of Vn. We say that A is symmetric if there is a transitive permutation group Γ on {1, 2,...,n} such that A is invariant under Γ. Theorem. For every symmetric monotone A,ifµp(A)>ɛthen µq(A)> 1−ɛ for q = p + c1 log(1/2ɛ) / log n. (c1isan absolute constant.) 1. Graph properties A graph property is a property of graphs which depends only on their isomorphism class. Let P be a monotone graph property; that is, if a graph G satisfies P
On the Construction of PseudoRandom Permutations: LubyRackoff Revisited
 JOURNAL OF CRYPTOLOGY
, 1997
"... Luby and Rackoff [27] showed a method for constructing a pseudorandom permutation from a pseudorandom function. The method is based on composing four (or three for weakened security) so called Feistel permutations, each of which requires the evaluation of a pseudorandom function. We reduce somewh ..."
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Cited by 93 (8 self)
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Luby and Rackoff [27] showed a method for constructing a pseudorandom permutation from a pseudorandom function. The method is based on composing four (or three for weakened security) so called Feistel permutations, each of which requires the evaluation of a pseudorandom function. We reduce somewhat the complexity of the construction and simplify its proof of security by showing that two Feistel permutations are sufficient together with initial and final pairwise independent permutations. The revised construction and proof provide a framework in which similar constructions may be brought up and their security can be easily proved. We demonstrate this by presenting some additional adjustments of the construction that achieve the following:  Reduce the success probability of the adversary.  Provide a construction of pseudorandom permutations with large input size using pseudorandom functions with small input size.
Algorithms in algebraic number theory
 Bull. Amer. Math. Soc
, 1992
"... Abstract. In this paper we discuss the basic problems of algorithmic algebraic number theory. The emphasis is on aspects that are of interest from a purely mathematical point of view, and practical issues are largely disregarded. We describe what has been done and, more importantly, what remains to ..."
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Cited by 40 (3 self)
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Abstract. In this paper we discuss the basic problems of algorithmic algebraic number theory. The emphasis is on aspects that are of interest from a purely mathematical point of view, and practical issues are largely disregarded. We describe what has been done and, more importantly, what remains to be done in the area. We hope to show that the study of algorithms not only increases our understanding of algebraic number fields but also stimulates our curiosity about them. The discussion is concentrated of three topics: the determination of Galois groups, the determination of the ring of integers of an algebraic number field, and the computation of the group of units and the class group of that ring of integers. 1.
Influences Of Variables And Threshold Intervals Under Group Symmetries
 Funct. Anal
"... Introduction. A subset A of f0; 1g n is called monotone provided if x 2 A; x 0 2 f0; 1g n ; x i x 0 i for i = 1; : : : ; n then x 0 2 A. For 0 p 1, define p the product measure on f0; 1g n with weights 1 \Gamma p at 0 and p at 1. Thus p (fxg) = (1 \Gamma p) n\Gammaj p j where ..."
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Cited by 27 (8 self)
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Introduction. A subset A of f0; 1g n is called monotone provided if x 2 A; x 0 2 f0; 1g n ; x i x 0 i for i = 1; : : : ; n then x 0 2 A. For 0 p 1, define p the product measure on f0; 1g n with weights 1 \Gamma p at 0 and p at 1. Thus p (fxg) = (1 \Gamma p) n\Gammaj p j where j = #fi = 1; : : : ; njx i = 1g: (0.1) If
The affine permutation groups of rank three
 Proc. London Math. Soc
, 1987
"... Introduction and statement of results Finite primitive permutation groups of rank 3 have been the subject of much study in the past twenty years, leading on the one hand to interesting new groups (for instance, some sporadic groups) and on the other to new techniques in the theory of permutation gro ..."
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Cited by 24 (0 self)
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Introduction and statement of results Finite primitive permutation groups of rank 3 have been the subject of much study in the past twenty years, leading on the one hand to interesting new groups (for instance, some sporadic groups) and on the other to new techniques in the theory of permutation groups. It is readily seen that if G is a primitive rank 3
Fast management of permutation groups I
, 1997
"... We present new algorithms for permutation group manipulation. Our methods result in an improvement of nearly an order of magnitude in the worstcase analysis for the fundamental problems of finding strong generating sets and testing membership. The normal structure of the group is brought into play ..."
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Cited by 21 (3 self)
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We present new algorithms for permutation group manipulation. Our methods result in an improvement of nearly an order of magnitude in the worstcase analysis for the fundamental problems of finding strong generating sets and testing membership. The normal structure of the group is brought into play even for such elementary issues. An essential element is the recognition of large alternating composition factors of the given group and subsequent extension of the permutation domain to display the natural action of these alternating groups. Further new features include a novel fast handling of alternating groups and the sifting of defining relations in order to link these and other analyzed factors with the rest of the group. The analysis of the algorithm depends on the classification of finite simple groups. In a sequel to this paper, using an enhancement of the present method, we shall achieve a further order of magnitude improvement.
Computing the Composition Factors of a Permutation Group in Polynomial Time
 Combinatorica
, 1987
"... Givengenerators for a group of permutations, it is shown that generators for the subgroups in a composition series can be found in polynomial time. The procedure also yields permutation representations of the composition factors. ..."
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Cited by 21 (2 self)
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Givengenerators for a group of permutations, it is shown that generators for the subgroups in a composition series can be found in polynomial time. The procedure also yields permutation representations of the composition factors.
Derandomized constructions of kwise (almost) independent permutations
 In Proceedings of the 9th Workshop on Randomization and Computation (RANDOM
, 2005
"... Abstract Constructions of kwise almost independent permutations have been receiving a growingamount of attention in recent years. However, unlike the case of kwise independent functions,the size of previously constructed families of such permutations is far from optimal. This paper gives a new met ..."
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Cited by 18 (3 self)
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Abstract Constructions of kwise almost independent permutations have been receiving a growingamount of attention in recent years. However, unlike the case of kwise independent functions,the size of previously constructed families of such permutations is far from optimal. This paper gives a new method for reducing the size of families given by previous constructions. Ourmethod relies on pseudorandom generators for spacebounded computations. In fact, all we need is a generator, that produces "pseudorandom walks " on undirected graphs with a consistent labelling. One such generator is implied by Reingold's logspace algorithm for undirected connectivity [35, 36]. We obtain families of kwise almost independent permutations, with anoptimal description length, up to a constant factor. More precisely, if the distance from uniform for any k tuple should be at most ffi, then the size of the description of a permutation inthe family is O(kn + log 1ffi). 1 Introduction In explicit constructions of pseudorandom objects, we are interested in simulating a large randomobject using a succinct one and would like to capture some essential properties of the former. A natural way to phrase such a requirement is via limited access. Suppose the object that we areinterested in simulating is a random function f: {0, 1}n 7! {0, 1}n and we want to come up witha small family of functions G that simulates it. The kwise independence requirement in this caseis that a function g chosen at random from G be completely indistinguishable from a function fchosen at random from the set of all functions, for any process that receives the value of either
An O’NanScott Theorem for finite quasiprimitive permutation groups and an application to 2arc transitive graphs
 J. London Math. Soc
, 1993
"... A permutation group is said to be quasiprimitive if each of its nontrivial normal subgroups is transitive. A structure theorem for finite quasiprimitive permutation groups is proved, along the lines of the O'NanScott Theorem for finite primitive permutation groups. It is shown that every finite, no ..."
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Cited by 18 (7 self)
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A permutation group is said to be quasiprimitive if each of its nontrivial normal subgroups is transitive. A structure theorem for finite quasiprimitive permutation groups is proved, along the lines of the O'NanScott Theorem for finite primitive permutation groups. It is shown that every finite, nonbipartite, 2arc transitive graph is a cover of a quasiprimitive 2arc transitive graph. The structure theorem for quasiprimitive groups is used to investigate the structure of quasiprimitive 2arc transitive graphs, and a new construction is given for a family of such graphs. 1.
On the Probability That a Group Element is PSingular
 JOURNAL OF ALGEBRA
, 1995
"... Suppose that G is a permutation group of degree n and let p be a prime divisor of jGj. In computational group theory it is a natural and important problem to nd an element of G of order p. A polynomialtime (but impractical) algorithm for this is given in Ka]. In practice, an element of the desired ..."
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Cited by 16 (5 self)
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Suppose that G is a permutation group of degree n and let p be a prime divisor of jGj. In computational group theory it is a natural and important problem to nd an element of G of order p. A polynomialtime (but impractical) algorithm for this is given in Ka]. In practice, an element of the desired type is obtained by \randomly" choosing elements of G and computing their orders. After a few tries, and with some luck, a psingular element (i.e., one of order divisible by p) frequently turns up. The purpose of this note is to make it clear just how well this procedure can be expected to work.