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37
Asymptotically optimal communication for torusbased cryptography
 In Advances in Cryptology (CRYPTO 2004), Springer LNCS 3152
, 2004
"... Abstract. We introduce a compact and efficient representation of elements of the algebraic torus. This allows us to design a new discretelog based publickey system achieving the optimal communication rate, partially answering the conjecture in [4]. For n the product of distinct primes, we construct ..."
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Abstract. We introduce a compact and efficient representation of elements of the algebraic torus. This allows us to design a new discretelog based publickey system achieving the optimal communication rate, partially answering the conjecture in [4]. For n the product of distinct primes, we construct efficient ElGamal signature and encryption schemes in a subgroup of F ∗ qn in which the number of bits exchanged is only a φ(n)/n fraction of that required in traditional schemes, while the security offered remains the same. We also present a DiffieHellman key exchange protocol averaging only φ(n) log2 q bits of communication per key. For the cryptographically important cases of n = 30 and n = 210, we transmit a 4/5 and a 24/35 fraction, respectively, of the number of bits required in XTR [14] and recent CEILIDH [24] cryptosystems. 1
Uncertainty in timefrequency representations on finite Abelian groups
, 2006
"... Classical and recent results on uncertainty principles for functions on finite Abelian groups relate the cardinality of the support of a function to the cardinality of the support of its Fourier transforms. We use these results and their proofs to obtain similar results relating the support sizes of ..."
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Cited by 9 (2 self)
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Classical and recent results on uncertainty principles for functions on finite Abelian groups relate the cardinality of the support of a function to the cardinality of the support of its Fourier transforms. We use these results and their proofs to obtain similar results relating the support sizes of functions and their short–time Fourier transforms. Further, we discuss applications of our results. For example, we use our results to construct a class of equal norm tight Gabor frames that are maximally robust to erasures and we discuss consequences of our findings to the theory of recovering and storing signals which have sparse time–frequency representations. 1.
Idempotents in complex group rings: theorems of Zalesskii and Bass revisited
 LIE THEORY 8 (1998), 219–228. ZBL 0911.16018 MR 1650345
, 1998
"... Let \Gamma be a group, and let C \Gamma be the group ring of \Gamma over C . We first give a simplified and selfcontained proof of Zalesskii's theorem [23] that the canonical trace on C \Gamma takes rational values on idempotents. Next, we contribute to the conjecture of idempotents by proving the ..."
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Cited by 6 (0 self)
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Let \Gamma be a group, and let C \Gamma be the group ring of \Gamma over C . We first give a simplified and selfcontained proof of Zalesskii's theorem [23] that the canonical trace on C \Gamma takes rational values on idempotents. Next, we contribute to the conjecture of idempotents by proving the following result: for a group \Gamma , denote by P \Gamma the set of primes p such that \Gamma embeds in a finite extension of a propgroup; if \Gamma is torsionfree and P \Gamma is infinite, then the only idempotents in C \Gamma are 0 and 1. This implies Bass' theorem [2] asserting that the conjecture of idempotents holds for torsionfree subgroups of GLn (C ) .
On communication over unknown sparse frequencyselective blockfading channels,” arXiv:1006.1548
, 2010
"... Abstract—This paper considers the problem of reliable communication over discretetime channels whose impulse responses have length and exactly nonzero coefficients, and whose support and coefficients remain fixed over blocks of channel uses but change independently from block to block. Here, it is ..."
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Cited by 4 (3 self)
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Abstract—This paper considers the problem of reliable communication over discretetime channels whose impulse responses have length and exactly nonzero coefficients, and whose support and coefficients remain fixed over blocks of channel uses but change independently from block to block. Here, it is assumed that the channel’s support and coefficient realizations are both unknown, although their statistics are known. Assuming Gaussian nonzerocoefficients and noise, and focusing on the highSNR regime, it is first shown that the ergodic noncoherent channel capacity has prelog factor 1 for any. It is then shown that, to communicate with arbitrarily small error probability at rates in accordance with the capacity prelog factor, it suffices to use pilotaided orthogonal frequencydivision multiplexing (OFDM) with pilots per fading block, in conjunction with an appropriate noncoherent decoder. Since the achievability result is proven using a noncoherent decoder whose complexity grows exponentially in the number of fading blocks, a simpler decoder, based on +1pilots, is also proposed. Itsachievable +1 rate is shown to have prelog factor equal to 1 with the previously considered channel, while its achievable rate is shown to +1 have prelog factor 1 when the support of the blockfading channel remains fixed over time. Index Terms—Bayes model averaging, compressed sensing, fading channels, noncoherent capacity, noncoherent communication, sparse channels. I.
Bounds for multiplicative cosets over fields of prime order
 Math. Comp
, 1997
"... Abstract. Let m be a positive integer and suppose that p is an odd prime with p ≡ 1modm. Suppose that a ∈ (Z/pZ) ∗ and consider the polynomial x m − a. If this polynomial has any roots in (Z/pZ) ∗ , where the coset representatives for Z/pZ are taken to be all integers u with u 
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Abstract. Let m be a positive integer and suppose that p is an odd prime with p ≡ 1modm. Suppose that a ∈ (Z/pZ) ∗ and consider the polynomial x m − a. If this polynomial has any roots in (Z/pZ) ∗ , where the coset representatives for Z/pZ are taken to be all integers u with u  <p/2, then these roots will form a coset of the multiplicative subgroup µm of (Z/pZ) ∗ consisting of the mth roots of unity mod p. Let C be a coset of µm in (Z/pZ) ∗, and define C  =maxu∈Cu. In the paper “Numbers Having m Small mth Roots mod p ” (Mathematics of Computation, Vol. 61, No. 203 (1993),pp. 393413), Robinson gives upper bounds for M1(m, p) = min C∈(Z/pZ) ∗ /µm C of the form M1(m, p) <Kmp 1−1/φ(m),whereφis the Euler phifunction. This paper gives lower bounds that are of the same form, and seeks to sharpen the constants in the upper bounds of Robinson. The upper bounds of Robinson areproventobeoptimalwhenmisapowerof2orwhenm=6.
The ppart of TateShafarevich groups of elliptic curves can be arbitrarily large
 J. Théor. Nomb. Bordeau
"... Abstract. In this paper it is shown that for every prime p> 5 the dimension of the ptorsion in the TateShafarevich group of E/K can be arbitrarily large, where E is an elliptic curve defined over a number field K, with [K: Q] bounded by a constant depending only on p. From this we deduce that the ..."
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Abstract. In this paper it is shown that for every prime p> 5 the dimension of the ptorsion in the TateShafarevich group of E/K can be arbitrarily large, where E is an elliptic curve defined over a number field K, with [K: Q] bounded by a constant depending only on p. From this we deduce that the dimension of the ptorsion in the TateShafarevich group of A/Q can be arbitrarily large, where A is an abelian variety, with dim A bounded by a constant depending only on p. 1.
On fixed points of permutations
 J. Algebraic Combin
"... Abstract. The number of fixed points of a random permutation of {1, 2,..., n} has a limiting Poisson distribution. We seek a generalization, looking at other actions of the symmetric group. Restricting attention to primitive actions, a complete classification of the limiting distributions is given. ..."
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Abstract. The number of fixed points of a random permutation of {1, 2,..., n} has a limiting Poisson distribution. We seek a generalization, looking at other actions of the symmetric group. Restricting attention to primitive actions, a complete classification of the limiting distributions is given. For most examples, they are trivial – almost every permutation has no fixed points. For the usual action of the symmetric group on ksets of {1, 2,..., n}, the limit is a polynomial in independent Poisson variables. This exhausts all cases. We obtain asymptotic estimates in some examples, and give a survey of related results. This paper is dedicated to the life and work of our colleague Manfred Schocker. 1.
History of Valuation Theory  Part I
"... The theory of valuations was started in 1912 by the Hungarian mathematician Josef Kursch'ak who formulated the valuation axioms as we are used today. The main motivation was to provide a solid foundation for the theory of padic fields as defined by Kurt Hensel. In the following decades we can o ..."
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The theory of valuations was started in 1912 by the Hungarian mathematician Josef Kursch'ak who formulated the valuation axioms as we are used today. The main motivation was to provide a solid foundation for the theory of padic fields as defined by Kurt Hensel. In the following decades we can observe a quick development of valuation theory, triggered mainly by the discovery that much of algebraic number theory could be better understood by using valuation theoretic notions and methods. An outstanding figure in this development was Helmut Hasse. Independent of the application to number theory, there were essential contributions to valuation theory given by Alexander Ostrowski, published 1934. About the same time Wolfgang Krull gave a more general, universal definition of valuation which turned out to be applicable also in many other mathematical disciplines such as algebraic geometry or functional analysis, thus opening a new era of valuation theory.
On a Theorem of Jordan
 BULLETIN (NEW SERIES) OF THE AMERICAN MATHEMATICAL SOCIETY
, 2003
"... The theorem of Jordan which I want to discuss here dates from 1872. It is an elementary result on finite groups of permutations. I shall first present its translations in Number Theory and Topology. ..."
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The theorem of Jordan which I want to discuss here dates from 1872. It is an elementary result on finite groups of permutations. I shall first present its translations in Number Theory and Topology.