New constructions of permutation polynomials of the form xrhxq-1 over Fq2

Permutation polynomials over finite fields have been studied extensively recently due to their wide applications in cryptography, coding theory, communication theory, among others. Recently, several authors have studied permutation trinomials of the form (Formula presented.) over (Formula presented.), where (Formula presented.), (Formula presented.) and (Formula presented.) are integers. Their methods are essentially usage of a multiplicative version of AGW Criterion because they all transformed the problem of proving permutation polynomials over (Formula presented.) into that of showing the corresponding fractional polynomials permute a smaller set (Formula presented.), where (Formula presented.). Motivated by these results, we characterize the permutation polynomials of the form (Formula presented.) over (Formula presented.) such that (Formula presented.) is arbitrary and q is also an arbitrary prime power. Using AGW Criterion twice, one is multiplicative and the other is additive, we reduce the problem of proving permutation polynomials over (Formula presented.) into that of showing permutations over a small subset S of a proper subfield (Formula presented.), which is significantly different from previously known methods. In particular, we demonstrate our method by constructing many new explicit classes of permutation polynomials of the form (Formula presented.) over (Formula presented.). Moreover, we can explain most of the known permutation trinomials, which are in Ding et al. (SIAM J Discret Math 29:79–92, 2015), Gupta and Sharma (Finite Fields Appl 41:89–96, 2016), Li and Helles

New constructions of involutions over finite fields

Involutions over finite fields are permutations whose compositional inverses are themselves. Involutions especially over Fq 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) = xrh(xs) over Fq to be involutory on Fq, where s∣(q − 1). Based on this, we provide three explicit classes of involutions of the form xrh(xq− 1) over Fq2. Recently, Zheng et al. (Finite Fields Appl., 56, 1–16 2019) found an equivalent relationship between permutation polynomials of g(x)qi−g(x)+cx+(1−c)δ and g(xqi−x+δ)+cx. The other part work of this paper is to consider the involutory property of these two classes of permutation polynomials, which fall into the additiv

Application of K/Sr co-doped calcium polyphosphate bioceramic as scaffolds for bone substitutes

Ion doping is one of the most important methods to modify the properties of bioceramics for better biodegrade abilities, biomechanical properties, and biocompatibilities. This paper presents a novel ion doping method applied in calcium polyphosphate (CPP)-based bioceramic scaffolds substituted by potassium and strontium ions (K/Sr) to form (K/Sr–CPP) scaffolds for bone tissue regeneration. The microstructure and crystallization of the scaffolds were detected by scanning electron microscopy and X-ray diffraction. Compressive strength and degradation tests were assessed to evaluate the mechanical and chemical stabilities of K/Sr–CPP in vitro. The cell biocompatibility was measured with respect to the cytotoxicity of the extractions of scaffolds. Muscle pouches and bone implantation were performed to evaluate the biodegradability and osteoconductivity of the scaffolds. The results indicated that the obtained K/Sr–CPP scaffolds had a single beta-CPP phase. The unit cell volume and average grain size increased but the crystallization decreased after the ions were doped into the CPP structure. The K/Sr–CPP scaffolds yielded a higher compressive strength and a better degradation property than the pure CPP scaffold. The MTT assay and in vivo results reveal that the K/Sr–CPP scaffolds exhibited a better cell biocompatibility and a tissue biocompatibility than CPP and hydroxyapatite scaffolds. This study proves the potential applications of K/Sr–CPP scaffolds in bone repair