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272
SpaceTime Diversity Systems Based on Linear Constellation Precoding
 IEEE TRANS. WIRELESS COMMUN
, 2003
"... We present a unified approach to designing spacetime (ST) block codes using linear constellation precoding (LCP). Our designs are based either on parameterizations of unitary matrices, or on algebraic numbertheoretic constructions. With an arbitrary number of transmit and receiveantennas, STLCP ..."
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Cited by 128 (8 self)
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We present a unified approach to designing spacetime (ST) block codes using linear constellation precoding (LCP). Our designs are based either on parameterizations of unitary matrices, or on algebraic numbertheoretic constructions. With an arbitrary number of transmit and receiveantennas, STLCP achieves rate 1 symbol/s/Hz and enjoys diversity gain as high as over (possibly correlated) quasistatic and fast fading channels. As figures of merit, we use diversity and coding gains, as well as mutual information of the underlying multipleinputmultipleoutput system. We show that over quadratureamplitude modulation and pulseamplitude modulation, our LCP achieves the upper bound on the coding gain of all linear precoders for certain values of and comes close to this upper bound for other values of , in both correlated and independent fading channels. Compared with existing ST block codes adhering to an orthogonal design (STOD), STLCP offers not only better performance, but also higher mutual information for...
Quantum cryptanalysis of hidden linear functions
 in Proceedings of Crypto’95, Lecture Notes in Comput. Sci. 963
, 1995
"... Abstract. Recently there has been a great deal of interest in the power of \Quantum Computers " [4, 15, 18]. The driving force is the recent beautiful result of Shor that shows that discrete log and factoring are solvable in random quantum polynomial time [15]. We use a method similar to Shor&a ..."
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Cited by 75 (0 self)
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Abstract. Recently there has been a great deal of interest in the power of \Quantum Computers " [4, 15, 18]. The driving force is the recent beautiful result of Shor that shows that discrete log and factoring are solvable in random quantum polynomial time [15]. We use a method similar to Shor's to obtain a general theorem about quantum polynomial time. We show that any cryptosystem based on what we refer to as a `hidden linear form ' can be broken in quantum polynomial time. Our results imply that the discrete log problem is doable in quantum polynomial time over any group including Galois elds and elliptic curves. Finally, we introduce the notion of `junk bits ' which are helpful when performing classical computations that are not injective. 1
Congruences concerning Bernoulli numbers and Bernoulli polynomials
 Discrete Appl. Math
, 2000
"... Let {Bn(x)} denote Bernoulli polynomials. In this paper we generalize Kummer’s congruences by determining Bk(p−1)+b(x)=(k(p − 1) + b) (mod p n), where p is an odd prime, x is a pintegral rational number and p − 1 b. As applications we obtain explicit formulae for ∑p−1 x=1 (1=xk) (mod p 3); ∑ (p−1 ..."
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Cited by 53 (22 self)
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Let {Bn(x)} denote Bernoulli polynomials. In this paper we generalize Kummer’s congruences by determining Bk(p−1)+b(x)=(k(p − 1) + b) (mod p n), where p is an odd prime, x is a pintegral rational number and p − 1 b. As applications we obtain explicit formulae for ∑p−1 x=1 (1=xk) (mod p 3); ∑ (p−1)=2 (1=x
GENERAL CONGRUENCES FOR BERNOULLI POLYNOMIALS
 DISCRETE MATH. 262(2003), 253–276.
, 2003
"... In this paper we establish some explicit congruences for Bernoulli polynomials modulo a general positive integer. In particular Voronoi’s and Kummer’s congruences are vastly extended. ..."
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Cited by 38 (31 self)
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In this paper we establish some explicit congruences for Bernoulli polynomials modulo a general positive integer. In particular Voronoi’s and Kummer’s congruences are vastly extended.
Examples of genus two CM curves defined over the rationals
 Math. Comp
, 1999
"... Abstract. We present the results of a systematic numerical search for genus two curves defined over the rationals such that their Jacobians are simple and have endomorphism ring equal to the ring of integers of a quartic CM field. Including the wellknown example y 2 = x 5 − 1 we find 19 nonisomorp ..."
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Cited by 32 (1 self)
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Abstract. We present the results of a systematic numerical search for genus two curves defined over the rationals such that their Jacobians are simple and have endomorphism ring equal to the ring of integers of a quartic CM field. Including the wellknown example y 2 = x 5 − 1 we find 19 nonisomorphic such curves. We believe that these are the only such curves. 1.
BlochKato conjecture and Main Conjecture of Iwasawa theory for Dirichlet characters
, 2002
"... The Tamagawa number conjecture proposed by S. Bloch and K. Kato describes the “special values ” of Lfunctions in terms of cohomological data. The main conjecture of Iwasawa theory describes a padic Lfunction in terms of the structure of modules for the Iwasawa algebra. We give a complete proof of ..."
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Cited by 23 (3 self)
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The Tamagawa number conjecture proposed by S. Bloch and K. Kato describes the “special values ” of Lfunctions in terms of cohomological data. The main conjecture of Iwasawa theory describes a padic Lfunction in terms of the structure of modules for the Iwasawa algebra. We give a complete proof of both conjectures (up to the prime 2) for Lfunctions attached to Dirichlet characters. We use the insight of Kato and B. PerrinRiou that these two conjectures can be seen as incarnations of the same mathematical content. In particular, they imply each other. By a bootstrapping process using the theory of Euler systems and explicit reciprocity laws, both conjectures are reduced to the analytic class number formula. Technical problems with primes dividing the order of the character are avoided by using the correct cohomological formulation of the main conjecture.
Topological properties of Eschenburg spaces and 3Sasakian manifolds
 MATHEMATISCHE ANNALEN
, 2005
"... We examine topological properties of the sevendimensional positively curved Eschenburg biquotients and find many examples which are homeomorphic but not diffeomorphic. A special subfamily of these manifolds also carries a 3Sasakian metric. Among these we construct a pair of 3Sasakian spaces which ..."
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Cited by 22 (10 self)
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We examine topological properties of the sevendimensional positively curved Eschenburg biquotients and find many examples which are homeomorphic but not diffeomorphic. A special subfamily of these manifolds also carries a 3Sasakian metric. Among these we construct a pair of 3Sasakian spaces which are diffeomorphic to each other, thus giving rise to the first example of a manifold which carries two nonisometric 3Sasakian metrics.