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Frobenius linear translators giving rise to new infinite classes of permutations and bent functionsCepak, Nastja ; Pašalić, Enes ; Muratović-Ribić, AmelaWe show the existence of many infinite classes of permutations over finite fields and bent functions by extending the notion of linear translators, introduced by Kyureghyan (J. Combin. Theory Ser. A ... 118(3), 1052-1061, 2011). We call these translators Frobenius translators since the derivatives of ▫$f : F_{p^n} \rightarrow F_{p^k}$▫, where ▫$n = rk$▫, are of the form ▫$f(x + u\phi) - f(x) = u^{p^i}b$▫, for a fixed ▫$b \in F_{p^k}$▫ and all ▫$u \in F_{p^k}$▫, rather than considering the standard case corresponding to ▫$i = 0$▫. This considerably extends a rather rare family ▫$\{f\}$▫ admitting linear translators of the above form. Furthermore, we solve a few open problems in the recent article (Cepak et al., Finite Fields Appl. 45, 19-42, 2017) concerning the existence and an exact specification of ▫$f$▫ admitting classical linear translators, and an open problem introduced in Hodžić et al. (2018), of finding a triple of bent functions ▫$f_1, f_2, f_3$▫ such that their sum ▫$f_4$▫ is bent and that the sum of their duals ▫$f_1^\ast +f_2^\ast +f_3^\ast +f_4^\ast = 1$▫. Finally, we also specify two huge families of permutations over ▫$F_{p^n}$▫ related to the condition that ▫$G(y) = -L(y)+(y+\delta)^s -(y+\delta)^{p^ks}$▫ permutes the set ▫$S =\{\beta \in F_{p^n} : Tr^n_k(\beta) = 0\}$▫, where ▫$n = 2k$▫ and ▫$p > 2$▫. Finally, we offer generalizations of constructions of bent functions in Mesnager et al. (2017) and described some new bent families using the permutations found in Cepak et al. (Finite Fields Appl. 45, 19-42, 2017).Source: Cryptography and communications. - ISSN 1936-2447 (Vol. 11, iss. 6, Nov. 2019, str. 1275-1295)Type of material - article, component part ; adult, seriousPublish date - 2019Language - englishCOBISS.SI-ID - 18709849
source: Cryptography and communications. - ISSN 1936-2447 (Vol. 11, iss. 6, Nov. 2019, str. 1275-1295)
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Cepak, Nastja | 37552 |
Pašalić, Enes | 27777 |
Muratović-Ribić, Amela |
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