On the construction of affine extractors

In this paper, we address the problem of explicit construction of so-called 'affine extractors' of delta-entropy ratio, when delta <= 1/2 (for delta > 1/2, there is a well-known and easy example based on the Hadamard function). We first give examples for delta slightly below 1/2 and produce then a construction for arbitrary given delta > 0. All these examples are again of a simple algebraic nature. The mathematics involved includes recent developments in the sum-product theory in finite fields and certain new bounds on multilinear exponential sums.