Evolution in multidimensional spaces

In this paper, we study evolutions in multidimensional spaces. By reducing multidimensional cases to one-dimensional, the properties of random evolutions in some semi-Markov media are studied

Ree geometries

We introduce a rank 3 geometry for any Ree group over a not necessarily perfect field and show that its full collineation group is the automorphism group of the corresponding Ree group. A similar result holds for two rank 2 geometries obtained as a truncation of this rank 3 geometry. As an application, we show that a polarity in any Moufang generalized hexagon is unambiguously determined by its set of absolute points, or equivalently, its set of absolute lines

EXPONENTIAL SUMS OVER POINTS OF ELLIPTIC CURVES WITH RECIPROCALS OF PRIMES

We consider exponential sums with x-coordinates of points qG and q−1G where G is a point of order T on an elliptic curve modulo a prime p and q runs through all primes up toN (with gcd (q,T)=1 in the case of the points q−1G). We obtain a new bound on exponential sums with q−1G and correct an imprecision in the work of W.D. Banks, J.B. Friedlander, M.Z. Garaev and I.E.Shparlinski on exponential sums with qG. We also note that similar sums with g1/q for an integer g with gcd (g,p)=1 have been estimated by J.Bourgain and I.E.Shparlinsk

The quadratic extension extractor for (hyper)elliptic curves in odd characteristic

We propose a simple and efficient deterministic extractor for the (hyper)elliptic curve C, defined over Fq2, where q is some power of an odd prime. Our extractor, for a given point P on C, outputs the first Fq-coefficient of the abscissa of the point P. We show that if a point P is chosen uniformly at random in C, the element extracted from the point P is indistinguishable from a uniformly random variable in Fq

Distribution of some sequences of points on elliptic curves

We estimate character sums over points on elliptic curves over a finite field of q elements. Pseudorandom sequences can be constructed by taking linear combinations with small coefficients (for example, from the set {-1, 0, 1}) of a fixed vector of points, which forms the seed of the generator. We consider several particular cases of this general approach which are of special practical interest and have occurred in the literature. For each of them we show that the resulting sequence has good uniformity of distribution properties