6,814 research outputs found
Perfect Space–Time Block Codes
In this paper, we introduce the notion of perfect space–time block codes (STBCs). These codes have full-rate, full-diversity, nonvanishing constant minimum determinant for increasing spectral efficiency, uniform average transmitted energy per antenna and good shaping. We present algebraic constructions of perfect STBCs for 2, 3, 4, and 6 antennas
Hard isogeny problems over RSA moduli and groups with infeasible inversion
We initiate the study of computational problems on elliptic curve isogeny
graphs defined over RSA moduli. We conjecture that several variants of the
neighbor-search problem over these graphs are hard, and provide a comprehensive
list of cryptanalytic attempts on these problems. Moreover, based on the
hardness of these problems, we provide a construction of groups with infeasible
inversion, where the underlying groups are the ideal class groups of imaginary
quadratic orders.
Recall that in a group with infeasible inversion, computing the inverse of a
group element is required to be hard, while performing the group operation is
easy. Motivated by the potential cryptographic application of building a
directed transitive signature scheme, the search for a group with infeasible
inversion was initiated in the theses of Hohenberger and Molnar (2003). Later
it was also shown to provide a broadcast encryption scheme by Irrer et al.
(2004). However, to date the only case of a group with infeasible inversion is
implied by the much stronger primitive of self-bilinear map constructed by
Yamakawa et al. (2014) based on the hardness of factoring and
indistinguishability obfuscation (iO). Our construction gives a candidate
without using iO.Comment: Significant revision of the article previously titled "A Candidate
Group with Infeasible Inversion" (arXiv:1810.00022v1). Cleared up the
constructions by giving toy examples, added "The Parallelogram Attack" (Sec
5.3.2). 54 pages, 8 figure
Special points on products of modular curves
We prove the Andre-Oort conjecture on special points of Shimura varieties for
arbitrary products of modular curves, assuming the Generalized Riemann
Hypothesis. More explicitly, this means the following. Let n be a positive
integer, and let S be a subset of C^n (with C the complex numbers) consisting
of points all of whose coordinates are j-invariants of elliptic curves with
complex multiplications. Then we prove (under GRH) that the irreducible
components of the Zariski closure of S are ``special subvarieties'', i.e.,
determined by isogeny conditions on coordinates and pairs of coordinates. A
weaker variant is proved unconditionally.Comment: 21 pages, referee's remarks have been taken into account, some
references updated, to appear in Duke Mathematical Journa
Computing the torsion of the -ramified module
We fix a prime number and \K a number field, we denote by the
maximal abelian -extension of \Ko unramified outside . The aim of this
paper is to study the -module \gal(M/\Ko) and to give a method to
effectively compute its structure as a -module. Then we give numerical
results, for real quadratic fields, together with interpretations via
Cohen-Lenstra's heuristics
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