Permutation polynomials have important applications in cryptography, coding theory and combinatorial designs. In this letter, we construct four classes of permutation polynomials over 𝔽2n × 𝔽2n, where 𝔽2n is the finite field with 2n elements.

The differential uniformity, the boomerang uniformity, and the extended Walsh spectrum etc are important parameters to evaluate the security of S (substitution)-box. In this paper, we introduce efficient formulas to compute these cryptographic parameters of permutation polynomials of the form xrh(x(2n-1)/d) over a finite field of q=2n elements, where r is a positive integer and d is a positive divisor of 2n-1. The computational cost of those formulas is proportional to d. We investigate differentially 4-uniform permutation polynomials of the form xrh(x(2n-1)/3) and compute the boomerang spectrum and the extended Walsh spectrum of them using the suggested formulas when 6≤n≤12 is even, where d=3 is the smallest nontrivial d for even n. We also investigate the differential uniformity of some permutation polynomials introduced in some recent papers for the case d=2n/2+1.

Permutation polynomials over finite fields have been widely studied due to their important applications in mathematics and cryptography. In recent years, 2-to-1 mappings over finite fields were proposed to build almost perfect nonlinear functions, bent functions, and the semi-bent functions. In this paper, we generalize the 2-to-1 mappings to m-to-1 mappings, including their construction methods. Some applications of m-to-1 mappings are also discussed.

Permutation polynomials and their compositional inverses are crucial for construction of Maiorana-McFarland bent functions and their dual functions, which have the optimal nonlinearity for resisting against the linear attack on block ciphers and on stream ciphers. In this letter, we give the explicit compositional inverse of the permutation binomial $f(z)=z^{2^{r}+2}+alpha zinmathbb{F}_{2^{2r}}[z]$. Based on that, we obtain the dual of monomial bent function $f(x)={ m Tr}_1^{4r}(x^{2^{2r}+2^{r+1}+1})$. Our result suggests that the dual of f is not a monomial any more, and it is not always EA-equivalent to f.

In this paper, we propose a notion for high-dimensional generalizations of mutually orthogonal Latin squares (MOLS) and mutually orthogonal diagonal Latin squares (MODLS), called mutually dimensionally orthogonal d-cubes (MOC) and mutually dimensionally orthogonal diagonal d-cubes (MODC). Systematic constructions for MOC and MODC by using polynomials over finite fields are investigated. In particular, for 3-dimensional cubes, the results for the maximum possible number of MODC are improved by adopting the proposed construction.

Many linear codes with two or three weights have recently been constructed due to their applications in consumer electronics, communication, data storage system, secret sharing, authentication codes, association schemes, and strongly regular graphs. In this paper, two classes of p-ary linear codes with two or three weights are presented. The first class of linear codes with two or three weights is obtained from a certain non-quadratic function. The second class of linear codes with two weights is obtained from the images of a certain function on $mathbb{F}_{p^m}$. In some cases, the resulted linear codes are optimal in the sense that they meet the Griesmer bound.

Research on permutation polynomials over the finite field F22k with significant cryptographical properties such as possibly low differential uniformity, possibly high nonlinearity and algebraic degree has attracted a lot of attention and made considerable progress in recent years. Once used as the substitution boxes (S-boxes) in the block ciphers with Substitution Permutation Network (SPN) structure, this kind of polynomials can have a good performance against the classical cryptographic analysis such as linear attacks, differential attacks and the higher order differential attacks. In this paper we put forward a new construction of differentially 4-uniformity permutations over F22k by modifying the inverse function on some specific subsets of the finite field. Compared with the previous similar works, there are several advantages of our new construction. One is that it can provide a very large number of Carlet-Charpin-Zinoviev equivalent classes of functions (increasing exponentially). Another advantage is that all the functions are explicitly constructed, and the polynomial forms are obtained for three subclasses. The third advantage is that the chosen subsets are very large, hence all the new functions are not close to the inverse function. Therefore, our construction may provide more choices for designing of S-boxes. Moreover, it has been checked by a software programm for k=3 that except for one special function, all the other functions in our construction are Carlet-Charpin-Zinoviev equivalent to the existing ones.

Permutation polynomials over Zpn are useful in the design of cryptographic algorithms. In this paper, we obtain an equivalent condition for polynomial functions over Zpn to be permutations, and this equivalent condition can help us to analysis the randomness of such functions. Our results provide a method to distinguish permutation polynomials from random functions. We also introduce how to improve the randomness of permutation polynomials over Zpn.

Permutation polynomial based interleavers over integer rings, in particular quadratic permutation polynomials have been widely studied. In this letter, higher degree permutation polynomials for interleavers are considered for interleavers and permutation polynomials superior to quadratic permutation polynomials are found for some lengths.

Permutation polynomial based interleavers over integer rings have recently received attention for their excellent channel coding performance, elegant algebraic properties and simplicity of implementation. In this letter, it is shown that permutation polynomial based interleavers of practical interest is decomposed into linear permutation polynomials. Based on this observation, it is shown that permutation polynomial based interleavers as well as their inverses can be efficiently implemented.

Polynomials are called permutation polynomials if they induce bijective functions. This paper investigates algebraic properties of permutation polynomials over a finite field, especially properties associated with permutation cycles. A permutation polynomial has a simple structure but good randomness properties suitable for applications. The cycle structure of permutations are considered to be related to randomness. We investigate the algebraic structure from the viewpoint of randomness. First we show the relationship between polynomials and permutations using a matrix equation. Then, we give a general form of a permutation polynomial corresponding to a product C1C2Ck of pairwise disjoint cycles. Finally, permutation polynomials with fixed points -or with 2, 3 and 4-cycles -and their compositions are given together with distribution of degree of the permutation polynomials.