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18
A probabilistic proof of the Rogers–Ramanujan identities
 Bull. London Math. Soc
"... The asymptotic probability theory of conjugacy classes of the finite general groups leads to a probability measure on the set of all partitions of natural numbers. A simple method of understanding these measures in terms of Markov chains is given, leading to an elementary probabilistic proof of the ..."
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Cited by 9 (5 self)
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The asymptotic probability theory of conjugacy classes of the finite general groups leads to a probability measure on the set of all partitions of natural numbers. A simple method of understanding these measures in terms of Markov chains is given, leading to an elementary probabilistic proof of the RogersRamanujan identities. This is compared with work on the uniform measure. The main case of Bailey’s lemma is interpreted as finding eigenvectors of the transition matrix of a Markov chain. It is shown that the viewpoint of Markov chains extends to quivers.
Conjugacy class properties of the extension of gl(n; Fq
 Journal of Algebra
, 2004
"... Abstract. Letting τ denote the inverse transpose automorphism of GL(n, q), a formula is obtained for the number of g in GL(n, q) so that gg τ is equal to a given element h. This generalizes a result of Gow and Macdonald for the special case that h is the identity. We conclude that for g random, gg τ ..."
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Cited by 7 (1 self)
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Abstract. Letting τ denote the inverse transpose automorphism of GL(n, q), a formula is obtained for the number of g in GL(n, q) so that gg τ is equal to a given element h. This generalizes a result of Gow and Macdonald for the special case that h is the identity. We conclude that for g random, gg τ behaves like a hybrid of symplectic and orthogonal groups. It is shown that our formula works well with both cycle index generating functions and asymptotics, and is related to the theory of random partitions. The derivation makes use of models of representation theory of GL(n, q) and of symmetric function theory, including a new identity for HallLittlewood polynomials. We obtain information about random elements of finite symplectic groups in even characteristic, and explicit bounds for the number of conjugacy classes and centralizer sizes in the extension of GL(n, q) generated by the inverse transpose automorphism. We give a second approach to these results using the theory of bilinear forms over a field. The results in this paper are key tools in forthcoming work of the authors on derangements in actions of almost simple groups, and we give a few examples in this direction. 1.
Bounds on the number and sizes of conjugacy classes in finite Chevalley groups
"... Abstract. We present explicit upper bounds for the number and size of conjugacy classes in finite Chevalley groups and their variations. These results have been used by many authors to study zeta functions associated to representations of finite simple groups, random walks on Chevalley groups, the f ..."
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Cited by 3 (0 self)
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Abstract. We present explicit upper bounds for the number and size of conjugacy classes in finite Chevalley groups and their variations. These results have been used by many authors to study zeta functions associated to representations of finite simple groups, random walks on Chevalley groups, the final solution to the Ore conjecture about commutators in finite simple groups and other similar problems. In this paper, we solve a strong version of the BostonShalev conjecture on derangements in simple groups for most of the families of primitive permutation group representations of finite simple groups (the remaining cases are settled in two other papers of the authors and applications are given in a third). 1.
Practical Cryptanalysis of the Identification Scheme Based on the Isomorphism of Polynomial with One Secret Problem
"... Abstract. This paper presents a practical cryptanalysis of the Identification Scheme proposed by Patarin at Crypto 1996. This scheme relies on the hardness of the Isomorphism of Polynomial with One Secret (IP1S), and enjoys shorter key than many other schemes based on the hardness of a combinatorial ..."
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Cited by 3 (2 self)
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Abstract. This paper presents a practical cryptanalysis of the Identification Scheme proposed by Patarin at Crypto 1996. This scheme relies on the hardness of the Isomorphism of Polynomial with One Secret (IP1S), and enjoys shorter key than many other schemes based on the hardness of a combinatorial problem (as opposed to numbertheoretic problems). Patarin proposed concrete parameters that have not been broken faster than exhaustive search so far. On the theoretical side, IP1S has been shown to be harder than Graph Isomorphism, which makes it an interesting target. We present two new deterministic algorithms to attack the IP1S problem, and we rigorously analyze their complexity and success probability. We show that they can solve a (big) constant fraction of all the instances of degree two in polynomial time. We verified that our algorithms are very efficient in practice. All the parameters with degree two proposed by Patarin are now broken in a few seconds. The parameters with degree three can be broken in less than a CPUmonth. The identification scheme is thus quite badly broken. 1
Integer Sequences and Matrices Over Finite
, 2006
"... In this expository article we collect the integer sequences that count several different types of matrices over finite fields and provide references to the Online Encyclopedia of Integer Sequences (OEIS). Section 1 contains the sequences, their generating functions, and examples. Section 2 contains ..."
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Cited by 1 (0 self)
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In this expository article we collect the integer sequences that count several different types of matrices over finite fields and provide references to the Online Encyclopedia of Integer Sequences (OEIS). Section 1 contains the sequences, their generating functions, and examples. Section 2 contains the proofs of the formulas for the coefficients and the generating functions of those sequences if the proofs are not easily available in the literature. The cycle index for matrices is an essential ingredient in most of the derivations.
On the Capacity of a Binary MIMO Channel with Random Interference
"... We study the capacity of a binary multiple input multiple output (MIMO) channel with interference. Interference is modelled as a nonlinear relation between the input and the output words of the system. Namely, we assume that the fading coefficients in the channel matrix are binary and drawn with i. ..."
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We study the capacity of a binary multiple input multiple output (MIMO) channel with interference. Interference is modelled as a nonlinear relation between the input and the output words of the system. Namely, we assume that the fading coefficients in the channel matrix are binary and drawn with i.i.d. Bernoulli probabilities, and any element of the output word is given by the minimum value (the AND function) of the input elements connected to that output. We find the expected number of output codewords for a given dimension of the MIMO system and interference level. Using this result and Jensen’s inequality, we derive an upper bound on the capacity of this system. We show by numerical simulations that the capacity upper bound is tight for most of the interference levels.
Isomorphism of Polynomials: New Results
"... Abstract. In this paper, we investigate the difficulty of the Isomorphism of Polynomials (IP) Problem as well as one of its variant IP1S. The Isomorphism of Polynomials is a wellknown problem studied in multivariate cryptography. It is related to the hardness of the key recovery of some cryptosyste ..."
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Abstract. In this paper, we investigate the difficulty of the Isomorphism of Polynomials (IP) Problem as well as one of its variant IP1S. The Isomorphism of Polynomials is a wellknown problem studied in multivariate cryptography. It is related to the hardness of the key recovery of some cryptosystems. The problem is the following: given two families of multivariate polynomials a and b, find two invertible linear (or affine) mappings S and T such that b = T ◦a◦S. For IP1S, we suppose that T is the identity. It is known that the difficulty of such problems depends on the structure of the polynomials (i.e., homogeneous, or not) and the nature of the transformations (affine, or linear). Here, we analyze the different cases and propose improved algorithms. We precisely describe the situation in term of complexity and sufficient conditions so that the algorithms work. The algorithms presented here combine linear algebra techniques, including the use of differentials, together with Gröbner bases. We show that random instances of IP1S with quadratic polynomials can be broken in time O ` n 6 ´ , where n is the number of variables, independently of the number of polynomials. For IP1S with cubic polynomials, as well as for IP, we propose new algorithms of complexity O ` n 6 ´ if the polynomials of a are inhomogeneous and S, T linear. In all the other cases, we propose an algorithm that requires O ` n 6 q n ´ computation. Finally, if a and b have a small number of nontrivial zeros, the complexity solving the IP instance is reduced to O ` n 6 + q n ´. This allows to break a publickey authentication scheme based on IP1S, and to break all the IP challenges proposed by Patarin in 1996 in practical time: the more secure parameters require less than 6 months of computations on 10 inexpensive GPUs. A consequence of our results is that HFE can be broken in polynomial time if the secret transforms S and T are linear and if the internal polynomial is made public and contains linear and constant terms. 1
The Expected Order of a Random Unitary Matrix (Preliminary Version)
, 2008
"... Let U(n, q) be the group consisting of those invertible matrices A = (ai,j) 1≤i,j≤n whose inverse is the conjugate transpose with respect to the involution c ↦ → c q of the finite field Fq2. ∑ In other words, the i, j’th en. Let µn = Order(A) be the average of try of A −1 is a q j,i 1 U(n,q) A∈U( ..."
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Let U(n, q) be the group consisting of those invertible matrices A = (ai,j) 1≤i,j≤n whose inverse is the conjugate transpose with respect to the involution c ↦ → c q of the finite field Fq2. ∑ In other words, the i, j’th en. Let µn = Order(A) be the average of try of A −1 is a q j,i 1 U(n,q) A∈U(n,q) the orders of the elements in this finite group. We prove the following conjecture of Fulman: for any fixed q, as n → ∞, log µn = n log(q) − log n + oq(log n).
IMS Collections
, 2008
"... Brownian motion on disconnected sets, basic hypergeometric functions, and some continued fractions of Ramanujan ..."
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Brownian motion on disconnected sets, basic hypergeometric functions, and some continued fractions of Ramanujan