Involutions over finite fields are permutations whose compositional inverses are themselves. Involutions especially over \( \mathbb {F}_{q} \) with q is even have been used in many applications, including cryptography and coding theory. The explicit study of involutions (including their fixed points) has started with the paper (Charpin et al. IEEE Trans. Inf. Theory, 62(4), 2266–2276 2016) for binary fields and since then a lot of attention had been made in this direction following it; see for example, Charpin et al. (2016), Coulter and Mesnager (IEEE Trans. Inf. Theory, 64(4), 2979–2986, 2018), Fu and Feng (2017), Wang (Finite Fields Appl., 45, 422–427, 2017) and Zheng et al. (2019). In this paper, we study constructions of involutions over finite fields by proposing an involutory version of the AGW Criterion. We demonstrate our general construction method by considering polynomials of different forms. First, in the multiplicative case, we present some necessary conditions of f(x) = x^rh(x^s) over \(\mathbb {F}_{q}\) to be involutory on \(\mathbb {F}_{q}\), where s∣(q − 1). Based on this, we provide three explicit classes of involutions of the form x^rh(x^q− 1) over \(\mathbb {F}_{q^{2}}\). Recently, Zheng et al. (Finite Fields Appl., 56, 1–16 2019) found an equivalent relationship between permutation polynomials of \(g(x)^{q^{i}} - g(x) + cx +(1-c)\delta \) and \(g\left (x^{q^{i}} - x + \delta \right ) +c x\). The other part work of this paper is to consider the involutory property of these two classes of permutation polynomials, which fall into the additive case of the AGW criterion. On one hand, we reveal the relationship of being involutory between the form \( g(x)^{q^{i}} - g(x) + cx +(1-c)\delta \) and the form \( g\left (x^{q^{i}} - x + \delta \right ) +c x \) over \( \mathbb {F}_{q^{m}} \) ; on the other hand, the compositional inverses of permutation polynomials of the form \( g\left (x^{q^{i}} - x + \delta \right ) + cx \) over \( \mathbb {F}_{q^{m}} \) are computed, where \( \delta \in \mathbb {F}_{q^{m}} \), \( g(x) \in \mathbb {F}_{q^{m}}[x] \) and integers m, i satisfy 1 ≤ i ≤ m − 1. In addition, a class of involutions of the form \( g\left (x^{q^{i}} - x + \delta \right ) + cx \) is constructed. Finally, we study the fixed points of constructed involutions and compute the number of all involutions with any given number of fixed points over \( \mathbb {F}_{q} \).

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有限域对合的新构造

有限域上的对合是其反构成本身的置换。对合尤其是在\（\ mathbb {F} _ {Q} \）与q甚至已经在许多应用中，包括加密和编码理论使用。对内卷积（包括其固定点）的显式研究始于针对二进制字段的论文（Charpin等人，IEEE Trans。Inf。Theory，62（4），2266–2276 2016），此后引起了很多关注沿此方向进行；参见例如Charpin等。（2016），Coulter and Mesnager（IEEE Trans。Inf。Theory，64（4），2979-2986，2018），Fu and Feng（2017），Wang（Finite Fields Appl。，45，422–427，2017）和Zheng等。（2019）。在本文中，我们通过提出AGW标准的非强制版本来研究有限域上的对合构造。我们通过考虑不同形式的多项式来证明我们的一般构造方法。首先，在乘法情况下，我们给出\（\ mathbb {F} _ {q} \）上的f（x）= x ^r h（x ^s）的一些必要条件，使其不符合\（\ mathbb {F} _ {q} \），其中s ∣（q − 1）。基于此，我们提供了x ^r h（x ^{q− 1}）在\（\ mathbb {F} _ {q ^ {2}} \）上。最近，Zheng等。（有限域申请，56，1-16 2019）发现的置换多项式之间的等效关系\（G（X）^ {Q ^ {I}} - G（X）+ CX +（1-c）的\增量\）和\（g \ left（x ^ {q ^ {i}}-x + \ delta \ right）+ cx \）。本文的另一部分工作是考虑这两类置换多项式的对合性质，它们属于AGW准则的加法情形。一方面，我们揭示了形式\（g（x）^ {q ^ {i}}-g（x）+ cx +（1-c）\ delta \）与形式\（ g \ left（x ^ {q ^ {i}}-x + \ delta \ right）+ cx \）在\（\ mathbb {F} _ {q ^ {m}} \）上; 在另一方面，形式的置换多项式组成逆\（克\左右（x ^ {Q ^ {I}} - X + \增量\右）+ CX \）超过\（\ mathbb {F} _ {q ^ {m}} \），其中\（\ delta \ in \ mathbb {F} _ {q ^ {m}} \），\（g（x）\ in \ mathbb {F} _ { C 1-4 {M}} [X] \）和整数中号，我满足1≤我≤米- 1。此外，一类的形式的对合\（克\左右（x ^ {q ^ {I}} -x + \ delta \ right）+ cx \）被构造。最后，我们研究构造对合的不动点，并计算\（\ mathbb {F} _ {q} \）上具有给定数量的不动点的所有对合的数量。