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1. CMB 2011 (vol 55 pp. 850)

Shparlinski, Igor E.; Stange, Katherine E.
 Character Sums with Division Polynomials We obtain nontrivial estimates of quadratic character sums of division polynomials $\Psi_n(P)$, $n=1,2, \dots$, 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 $q^{1/2 + \varepsilon}$ for some fixed $\varepsilon > 0$. This work is motivated by an open question about statistical indistinguishability of some cryptographically relevant sequences that was recently brought up by K. Lauter and the second author. Keywords:division polynomial, character sumCategories:11L40, 14H52

2. CMB 2011 (vol 54 pp. 748)

Shparlinski, Igor E.
 On the Distribution of Irreducible Trinomials We obtain new results about the number of trinomials $t^n + at + b$ with integer coefficients in a box $(a,b) \in [C, C+A] \times [D, D+B]$ that are irreducible modulo a prime $p$. As a by-product we show that for any $p$ there are irreducible polynomials of height at most $p^{1/2+o(1)}$, improving on the previous estimate of $p^{2/3+o(1)}$ obtained by the author in 1989. Keywords:irreducible trinomials, character sumsCategories:11L40, 11T06