Abstract
Linearized polynomials over finite fields have been intensively studied over the last several decades. Interesting new applications of linearized polynomials to coding theory and finite geometry have been also highlighted in recent years. Let p be any prime. Recently, preimages of the p −linearized polynomials \({\sum }_{i=0}^{\frac kl-1} X^{p^{li}}\) and \({\sum }_{i=0}^{\frac kl-1} (-1)^{i} X^{p^{li}}\) were explicitly computed over \({\mathbb {F}}_{p^{n}}\) for any n. This paper extends that study to p −linearized polynomials over \({\mathbb {F}}_{p}\), i.e., polynomials of the shape \(L(X)=\sum \limits _{i=0}^{t} \alpha _{i} X^{p^{i}}, \alpha _{i}\in {\mathbb {F}}_{p}.\) Given a k such that L(X) divides \(X-X^{p^{k}}\), the preimages of L(X) can be explicitly computed over \({\mathbb {F}}_{p^{n}}\) for any n.
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The authors are very grateful to the referees and the Associate Editor for several valuable comments, which improved the exposition of the results.
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Kim, K.H., Mesnager, S., Choe, J.H. et al. Preimages of p −Linearized Polynomials over \({\mathbb {F}}_{p}\). Cryptogr. Commun. 14, 75–86 (2022). https://doi.org/10.1007/s12095-021-00514-x
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DOI: https://doi.org/10.1007/s12095-021-00514-x