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We estimate character sums with n!, on average, and individually. These bounds are used to derive new results about various congruences modulo a prime p and obtain new information about the spacings between quadratic nonresidues modulo p. In particular, we show that there exists a positive integer n ≪ p 1/2+ε , such that n! is a primitive root modulo p. We(More)
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 ∈ F q returns (x + s) e for a given divisor e | 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(More)
We estimate the number of solutions of certain diagonal congru-ences involving factorials. We use these results to bound exponential sums with products of two factorials n!m! and also derive asymp-totic formulas for the number of solutions of various congruences with factorials. For example, we prove that the products of two factori-als n!m! with max{n, m}(More)
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