The main contribution of this brief is the introduction and the characterization of a novel Chebyshev-like rational map over finite fields. The referred map is identified as $t$ -Chebyshev map and depends on the definition of a finite field tangent function, which is also proposed. Among other new and interesting results, we demonstrate that the semigroup property holds for $t$ -Chebyshev maps and give a necessary and sufficient condition under which one of such maps induces a permutation on the set of elements of a field $\mathbb {F}_{q}$ . This makes these maps suitable for practical use in both public- and secret-key cryptography.

中文翻译：

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有限域上的正切函数和类切比雪夫有理映射

本简介的主要贡献是介绍和表征有限域上的新型切比雪夫有理映射。引用的地图被标识为 $t$ -Chebyshev 映射和依赖于有限域正切函数的定义，这也是被提出的。在其他新的有趣的结果中，我们证明了半群性质适用于 $t$ - 切比雪夫映射并给出一个充分必要条件，在该条件下，这样的映射中的一个在域的元素集合上引起排列 $\mathbb {F}_{q}$ . 这使得这些映射适用于公共和秘密密钥密码术的实际使用。