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be a maximal order in a definite quaternion algebra over
of prime discriminant
a small prime. We describe a probabilistic algorithm which, for a given left
-ideal, computes a representative in its left ideal class of
-power norm. In practice the algorithm is efficient and, subject to heuristics on expected distributions of primes, runs in expected polynomial time. This solves the underlying problem for a quaternion analog of the Charles–Goren–Lauter hash function, and has security implications for the original CGL construction in terms of supersingular elliptic curves.
Using triality, we define a relative Arason invariant for orthogonal involutions on a -possibly division- central simple algebra of degree 8. This invariant detects hyperbolicity, but it does not detect isomorphism. We produce explicit examples, in index 4 and 8, of non isomorphic involutions with trivial relative Arason invariant.
Like an idea whose time has come, nonabelian Galois cohomology burst into the world in the mid 50's. There had been harbingers, of course. Châtelet's méthode galoisienne for genus 1 curves and Weil's observations on homogeneous spaces had opened the way, and it was a small step to write down the basic operations so that they make sense in a noncommutative situation. Within a few years, several pioneers realized almost simultaneously that the formalism of Galois cohomology could be used to classify various algebraic structures and to illuminate the definition of some of their invariants. This simple and remarkably penetrating idea, soon popularized by Serre's famous monograph Cohomologie galoisienne, immediately took hold. Galois cohomology is indeed algebra at its best: a few formal basic operations with a broad spectrum of far-reaching applications.
Grégory Berhuy's monograph provides a very welcome introduction to Galois descent techniques and nonabelian Galois cohomology, aimed at people who are new to the subject. Beginners will find here a thorough discussion of the technical details that are usually left to the reader. Together with advanced readers, they will appreciate a tasteful tour of applications, including some to which the author, himself an avid cocyclist, has contributed. (Incidentally, the title of Section III.8.1 also offers a glimpse into his taste in movies.) As may be expected, the list of applications discussed here is far from exhaustive, and in the last chapters the exposition is more demanding.
The aim of this paper is to give an overview of the various facets of psychiatry in France. The selection of topics discussed, as well as the views expressed in this paper, are our own, and some of them might not reflect the views of other French psychiatrists.