Trace of products in finite fields

Abstract : Let p be a prime number and let q = p(r). If C and D are large subsets of F-q(*) we study the trace of products cd with c is an element of C and d is an element of D and show that it is well distributed in F-p. We give an optimal condition (up to an absolute constant factor) on the size of the subsets C and D to ensure that the trace of products cd takes any given value in Fp. We also give a condition (optimal up to an absolute constant factor in most cases) on the size of the subsets C and D to ensure that the trace of cd meets the set of k-th powers for k >= 1, respectively the set of generators. Our method will enable us to take sets C and D whose size is substantially below root q. Character sums and Gaussian sums over F-p and F-q will play an important role in the proofs. Some estimates lead to interesting combinatorial questions in finite fields.
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https://hal.archives-ouvertes.fr/hal-02057370
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Submitted on : Tuesday, March 5, 2019 - 11:46:00 AM
Last modification on : Wednesday, March 6, 2019 - 1:27:42 AM

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Cathy Swaenepoel. Trace of products in finite fields. Finite Fields and Their Applications, Elsevier, 2018, 51, pp.93-129. ⟨10.1016/j.ffa.2018.01.005⟩. ⟨hal-02057370⟩

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