Regular complete permutation polynomials over $${\mathbb {F}}_{2^{n}}$$ F 2 n

Applications of permutation polynomials in cryptography are closely related to their cycle structures. For example, many block ciphers use permutation polynomials defined over $${\mathbb {F}}_{2^{n}}$$ with few fixed points and nontrivial cycles of length 2 as their S-boxes. In addition, permutation polynomials with long cycles have been widely used to generate keystream sequences. Complete permutation polynomials over $${\mathbb {F}}_{2^{n}}$$ have a single fixed point and good bit independence, which can provide good choices for S-boxes in block ciphers. However, there are only a limited number of known infinite families of complete permutation polynomials with explicit cycle structures. In this paper, we construct two classes of r-regular complete permutation polynomials g(x) over $${\mathbb {F}}_{2^{n}}$$ , i.e., g(x) is a complete permutation polynomial over $${\mathbb {F}}_{2^n}$$ with a single fixed point and all the other cycles of the same length r. The first class of regular complete permutation polynomials over $${\mathbb {F}}_{2^{km}}$$ is based on some permutation polynomials over $${\mathbb {F}}_{2^{m}}$$ for positive integers k, m, and the second class of regular complete permutation polynomials over $${\mathbb {F}}_{2^{n}}$$ for a positive integer n is obtained from piecewise functions. In addition, when g(x) is a r-regular complete permutation polynomial, we also consider the r-regular complete permutation property of $$g(x)+x$$ .