Abstract
The p-ary function f(x) mapping GF(p 4k) to GF(p) and given by \(f(x)={\rm Tr}_{4k}\big(ax^d+bx^2\big)\) with a,b ∈ GF(p 4k) and d = p 3k + p 2k − p k + 1 is studied with the respect to its exponential sum. In the case when either \(a^{p^k(p^k+1)}\neq b^{p^k+1}\) or a 2 = b d with b ≠ 0, this sum is shown to be three-valued and the values are determined. For the remaining cases, the value of the exponential sum is expressed using Jacobsthal sums of order p k + 1. Finding the values and the distribution of those sums is a long-lasting open problem.
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Helleseth, T., Kholosha, A. (2010). Sequences, Bent Functions and Jacobsthal Sums. In: Carlet, C., Pott, A. (eds) Sequences and Their Applications – SETA 2010. SETA 2010. Lecture Notes in Computer Science, vol 6338. Springer, Berlin, Heidelberg. https://doi.org/10.1007/978-3-642-15874-2_35
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DOI: https://doi.org/10.1007/978-3-642-15874-2_35
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