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We show that the reductions modulo primes $p\le x$ of the elliptic curve $$ Y^2 = X^3 + f(a)X + g(b), $$ behave as predicted by the Lang-Trotter and Sato-Tate conjectures, on average over integers $a \in [-A,A]$ and $b \in [-B,B]$ for $A$ and $B$ reasonably small compared to $x$, provided that $f(T), g(T) \in \Z[T]$ are not powers of another polynomial over $\Q$. For $f(T) = g(T) = T$ first results of this kind are due to E. Fouvry and M. R. Murty and have been further extended by other autho...
Following Kraitchik and Lehmer, we say that a positive integer $n\equiv1\pmod 8$ is an $x$-pseudosquare if it is a quadratic residue for each odd prime $p\le x$, yet is not a square. We extend this defintion to algebraic curves and say that $n$ is an $x$-pseudopoint of a curve $f(u,v) = 0$ (where $f \in \Z[U,V]$) if for all sufficiently large primes $p \le x$ the congruence $f(n,m)\equiv 0 \pmod p$ is satisfied for some $m$. We use the Bombieri bound of exponential sums along a curve to estim...
We study the collection of group structures that can be realized as a group of rational points on an elliptic curve over a finite field (such groups are well known to be of rank at most two). We also study various subsets of this collection which correspond to curves over prime fields or to curves with a prescribed torsion. Some of our results are rigorous and are based on recent advances in analytic number theory, some are conditional under certain widely believed conjectures, and others are...
Let $\E$ be an elliptic curve over a finite field $\F_{q}$ of $q$ elements, with $\gcd(q,6)=1$, given by an affine Weierstra\ss\ equation. We also use $x(P)$ to denote the $x$-component of a point $P = (x(P),y(P))\in \E$. We estimate character sums of the form $$ \sum_{n=1}^N \chi\(x(nP)x(nQ)\) \quad \text{and}\quad \sum_{n_1, \ldots, n_k=1}^N \psi\(\sum_{j=1}^k c_j x\(\(\prod_{i =1}^j n_i\) R\)\) $$ on average over all $\F_q$ rational points $P$, $Q$ and $R$ on $\E$, where $\chi$ is a quadra...
We obtain nontrivial estimates of quadratic character sums of division polynomials $\Psi_n(P)$, $n=1,2, ...$, evaluated at a given point $P$ on an elliptic curve over a finite field of $q$ elements. Our bounds are nontrivial if the order of $P$ is at least $q^{1/2 + \epsilon}$ for some fixed $\epsilon > 0$. This work is motivated by an open question about statistical indistinguishability of some cryptographically relevant sequences which has recently been brought up by K. Lauter and the secon...
For a prime $p$ and an integer $a \in \Z$ we obtain nontrivial upper bounds on the number of solutions to the congruence $x^x \equiv a \pmod p$, $1 \le x \le p-1$. We use these estimates to estimate the number of solutions to the congruence $x^x \equiv y^y \pmod p$, $1 \le x,y \le p-1$, which is of cryptographic relevance.
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