Character Sums and Deterministic Polynomial Root Finding in Finite Fields

We obtain a new bound of certain double multiplicative character sums. We use this bound together with some other previously obtained results to obtain new algorithms for finding roots of polynomials modulo a prime $p$

Counting Additive Decompositions of Quadratic Residues in Finite Fields

We say that a set $S$ is additively decomposed into two sets $A$ and $B$ if $S = \{a+b : a\in A, \ b \in B\}$. A. S\'ark\"ozy has recently conjectured that the set $Q$ of quadratic residues modulo a prime $p$ does not have nontrivial decompositions. Although various partial results towards this conjecture have been obtained, it is still open. Here we obtain a nontrivial upper bound on the number of such decompositions

On Congruences with Products of Variables from Short Intervals and Applications

We obtain upper bounds on the number of solutions to congruences of the type $$(x_1+s)...(x_{\nu}+s)\equiv (y_1+s)...(y_{\nu}+s)\not\equiv0 \pmod p$$ modulo a prime $p$ with variables from some short intervals. We give some applications of our results and in particular improve several recent estimates of J. Cilleruelo and M. Z. Garaev on exponential congruences and on cardinalities of products of short intervals, some double character sum estimates of J. B. Friedlander and H. Iwaniec and some results of M.-C. Chang and A. A. Karatsuba on character sums twisted with the divisor function

Multiplicative Congruences with Variables from Short Intervals

Recently, several bounds have been obtained on the number of solutions to congruences of the type $$(x_1+s)...(x_{\nu}+s)\equiv (y_1+s)...(y_{\nu}+s)\not\equiv0 \pmod p$$ modulo a prime $p$ with variables from some short intervals. Here, for almost all $p$ and all $s$ and also for a fixed $p$ and almost all $s$, we derive stronger bounds. We also use similar ideas to show that for almost all primes, one can always find an element of a large order in any rather short interval

On the Hidden Shifted Power Problem

We consider the problem of recovering a hidden element $s$ of a finite field \F_q of $q$ elements from queries to an oracle that for a given x\in \F_q returns $(x+s)^e$ for a given divisor $e\mid q-1$. We use some techniques from additive combinatorics and analytic number theory that lead to more efficient algorithms than the naive interpolation algorithm, for example, they use substantially fewer queries to the oracle.Comment: Moubariz Garaev (who has now become a co-author) has introduced some new ideas that have led to stronger results. Several imprecision of the previous version have been corrected to

Prime chains and Pratt trees

We study the distribution of prime chains, which are sequences p_1,...,p_k of primes for which p_{j+1}\equiv 1\pmod{p_j} for each j. We give estimates for the number of chains with p_k\le x (k variable), and the number of chains with p_1=p and p_k \le px. The majority of the paper concerns the distribution of H(p), the length of the longest chain with p_k=p, which is also the height of the Pratt tree for p. We show H(p)\ge c\log\log p and H(p)\le (\log p)^{1-c'} for almost all p, with c,c' explicit positive constants. We can take, for any \epsilon>0, c=e-\epsilon assuming the Elliott-Halberstam conjecture. A stochastic model of the Pratt tree, based on a branching random walk, is introduced and analyzed. The model suggests that for most p, H(p) stays very close to e \log\log p.Comment: v4. Very minor revision. Small corrections, e.g. sentence preceding Theorem 6, last sentence in the proof of Lemma 5.2. Updated reference [22