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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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https://hal.inria.fr/hal-01276424
Contributor : Jean-Pierre Tillich <>
Submitted on : Friday, February 19, 2016 - 1:00:31 PM
Last modification on : Wednesday, July 24, 2019 - 4:02:06 PM
Long-term archiving on: : Saturday, November 12, 2016 - 11:41:57 PM

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Daniel J. Katz, Philippe Langevin. Proof of a conjectured three-valued family of Weil sums of binomials. The 9th International Workshop on Coding and Cryptography 2015 WCC2015, Anne Canteaut, Gaëtan Leurent, Maria Naya-Plasencia, Apr 2015, Paris, France. ⟨hal-01276424⟩

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