Abstract
Based on a criterion of permutation polynomials of the form \(x^rf(x^{\frac{q-1}{m}})\) by Wan and Lidl (1991) and some very elementary techniques we show existence of permutation binomials of the following forms
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1
\(x(x^{\frac{2^n-1}{3}}+a) \in \mathbb{F}_{2^n}[x]\), for n > 4
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2
\(x^{\frac{2^{2n}-1}{2^{n}-1} + 1}+ax = x^{2^n+2} + ax \in \mathbb{F}_{2^{2n}}[x]\), for n ≥ 3.
In (i), we extend a result of Carlitz (1962) for even characteristic. Moreover we present the count of such permutation binomials when a is in a certain subfield of \(\mathbb{F}_{2^n}\). In (ii), we reprove, using much simpler technique, a recent result of Charpin and Kyureghyan (2008) and give the number of permutation binomials of this form. Finally, we discuss some cryptographic relevance of these results.
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Sarkar, S., Bhattacharya, S., Çeşmelioğlu, A. (2012). On Some Permutation Binomials of the Form \(x^{\frac{2^n-1}{k}+1} +ax\) over \(\mathbb{F}_{2^n}\) : Existence and Count. In: Özbudak, F., Rodríguez-Henríquez, F. (eds) Arithmetic of Finite Fields. WAIFI 2012. Lecture Notes in Computer Science, vol 7369. Springer, Berlin, Heidelberg. https://doi.org/10.1007/978-3-642-31662-3_17
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DOI: https://doi.org/10.1007/978-3-642-31662-3_17
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