Proof of a conjectured three-valued family of Weil sums of binomials

Abstract : We consider Weil sums of binomials of the form W F,d (a) = x∈F ψ(x d − ax), where F is a finite field, ψ : F → C is the canonical additive character, gcd(d, |F × |) = 1, and a ∈ F ×. If we fix F and d and examine the values of W F,d (a) as a runs through F × , we always obtain at least three distinct values unless d is degenerate (a power of the characteristic of F modulo F ×). Choices of F and d for which we obtain only three values are quite rare and desirable in a wide variety of applications. We show that if F is a field of order 3 n with n odd, and d = 3 r + 2 with 4r ≡ 1 (mod n), then W _{F,d} (a) assumes only the three values 0 and ±3^{(n+1)/2}. This proves the 2001 conjecture of Dobbertin, Helleseth, Kumar, and Martinsen.
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Pascale Charpin, Nicolas Sendrier, Jean-Pierre Tillich. The 9th International Workshop on Coding and Cryptography 2015 WCC2015, Apr 2015, Paris, France. 2015, Proceedings of the 9th International Workshop on Coding and Cryptography 2015 WCC2015. 〈wcc2015.inria.fr〉
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Daniel J. Katz, Philippe Langevin. Proof of a conjectured three-valued family of Weil sums of binomials. Pascale Charpin, Nicolas Sendrier, Jean-Pierre Tillich. The 9th International Workshop on Coding and Cryptography 2015 WCC2015, Apr 2015, Paris, France. 2015, Proceedings of the 9th International Workshop on Coding and Cryptography 2015 WCC2015. 〈wcc2015.inria.fr〉. 〈hal-01276424〉

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