Closed formula for exponential sums of cubic $ k $-cycles over Galois fields

José E. Calderón-Gómez; Rodrigo A. León-Prato; Luis A. Medina; Eiver Rodríguez

doi:10.3934/amc.2026038

Article Contents

2026, Volume 25: 1-20. Doi: 10.3934/amc.2026038

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Closed formula for exponential sums of cubic $ k $-cycles over Galois fields

1.
Department of Mathematical Sciences, University of Puerto Rico, Mayagüez, PR 00681
2.
Department of Mathematics, University of Puerto Rico, San Juan, PR 00931

^*Corresponding author: Eiver Rodríguez
^*Corresponding author: Eiver Rodríguez

Received: March 19, 2026

Revised: April 17, 2026

Early access: April 28, 2026

Published online: April 28, 2026

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Abstract

Determining whether a Boolean function is balanced remains computationally demanding, especially as the number of variables increases. A classical approach involves computing the function's exponential sum, which vanishes if and only if the function is balanced. In this article, we investigate a class of cubic Boolean functions that are invariant under the action of the cyclic subgroup $ \langle \sigma_n^k \rangle $ and derive a closed form expression for their exponential sums. For the subclass of mixed cubic functions, we establish an explicit characterization of the exponential sum in terms of Lucas sequences, leading to an effective criterion for balancedness.

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Mathematics Subject Classification: Primary: 06E30, 11L03.

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