Kasami Power Functions, Permutation Polynomials and Cyclic Difference Sets

The study of computing compositional inverses of permutation polynomials over finite fields efficiently is motivated by an open problem proposed by G. L. Mullen (1991), as well as the potential applications of these permutation polynomials (Dillon 1974, Khachatrian and Kyureghyan, Discrete Appl. Math. 216, 622–626 2017, Lidl 1985, Lidl and Muller 1984, Rivest et al., ACM Commun. Comput. Algebra. 1978, 120–126 1976, Schwenk and Huber, Electron. Lett. 34, 759–760 1998). It is well known that every permutation polynomial over a finite field $\mathbb {F}_{q}$ can be reduced to a permutation polynomial of the form xrh(xs) with s∣(q − 1) and $h(x) \in \mathbb {F}_{q}[x]$ (Akbary et al., Finite Fields Appl. 15(2), 195–206 2009, Wang, Finite Fields Appl. 22, 57–69 2013). Recently, several explicit classes of permutation polynomials of the form xrh(xs) over ${\mathbb F}_{q}$ have been constructed. However, all the known methods to compute the compositional inverses of permutation polynomials of this form seem to be inadequately explicit, which could be a hurdle to potential applications. In this paper, for any prime power q, we introduce a new approach to explicitly compute the compositional inverse of a permutation polynomial of the form xrh(xs) over ${\mathbb F}_{q}$ , where s∣(q − 1) and $\gcd (r,q-1)= 1$ . The main idea relies on transforming the problem of computing the compositional inverses of permutation polynomials over ${\mathbb F}_{q}$ into computing the compositional inverses of two restricted permutation mappings, where one of them is a monomial over $\mathbb {F}_{q}$ and the other is the polynomial xrh(x)s over a particular subgroup of $\mathbb {F}_{q}^{*}$ with order (q − 1)/s. This is a multiplicative analog of Tuxanidy and Wang (Finite Fields Appl. 28, 244–281 2014), Wu and Liu (Finite Fields Appl. 24, 136–147 2013). We demonstrate that the inverses of these two restricted permutations can be explicitly obtained in many cases. As consequences, many explicit compositional inverses of permutation polynomials given in Zieve (Proc. Am. Math. Soc. 137, 2209–2216 2009), Zieve (arXiv: 1310.0776 , 2013), Zieve (arXiv: 1312.1325v3 , 2013) are obtained using this method.

The compositional inverse of a class of bilinear permutation polynomials over finite fields of characteristic 2

In this paper, by using linear translators, we characterize a new class of permutation polynomials of the form [Formula: see text], which has a more general form than [Formula: see text]. Then, we present the compositional inverses of such permutation polynomials. Furthermore, by specifying the functions [Formula: see text] and [Formula: see text], we can get some new permutation polynomials of the forms [Formula: see text] and [Formula: see text] where [Formula: see text] is the trace function.

Permutation polynomials over Zn form the basis of the RSA and Dickson public key schemes. Previously, however, only those permutation polynomials whose inverse permutation polynomial was easy to evaluate have been used in cryptography. The authors propose a way to avoid this restriction in public key cryptography by implementing secret key decryption and signature generation by computation of the gcd of two polynomials. This allows the implementation of new classes of public key scheme.

A link between two classes of permutation polynomials

On a class of permutation polynomials and their inverses

Cryptographic functions with low differential uni-formity have wide applications in cryptography and cyber security. In this paper, we further investigate c-differential uniformity proposed by Ellingsen et al. (IEEE Transactions on Information Theory. 2020, 66(9): 5781–5789). Specifically, two classes of power functions with low (-1)-differential uniformity over finite fields with odd characteristic are obtained. In addition, a class of permutation polynomials and <tex xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">$\mathrm{P}_{-1}\mathrm{N}$</tex> or <tex xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">$\text{AP}_{-1}\mathrm{N}$</tex> functions over <tex xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">$\mathbb{F}_{3^{n}}$</tex> are proposed. These functions with low <tex xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">$c$</tex>-differential uniformity are utilized to resist against <tex xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">$c$</tex>-differential attacks in cyberspace from those with the usual low differential uniformity.

International audience

Two types of permutation polynomials with special forms

Permutation polynomials of the form [formula omitted] over [formula omitted

Permutation polynomials with low boomerang uniformity have wide applications in cryptography. In this paper, by utilizing the Weil sums technique and solving some certain equations over [Formula: see text], we determine the boomerang uniformity of these permutation polynomials: (1) [Formula: see text], where [Formula: see text], [Formula: see text] with [Formula: see text]; (2) [Formula: see text], where [Formula: see text], [Formula: see text] with [Formula: see text]; (3) [Formula: see text], where [Formula: see text], [Formula: see text] with [Formula: see text]. The results show that the boomerang uniformity of [Formula: see text], [Formula: see text] and [Formula: see text] can attain [Formula: see text].

Let q = 2 m . In this paper, we propose several classes of permutation polynomials of the forms L 1 ( x d 1 ) k 1 + L 2 ( x d 2 ) k 2 , L 1 ( x d 1 ) k 1 + L 3 ( x d 2 + x d 3 ) k 2 over F q 2 and L 1 ( x d 1 ) + L 3 ( c 1 x d 2 + c 2 x d 3 ) over F q 3 from certain linearized polynomials L 1 ( x ) = x + a 0 x q , L 2 ( x ) = b 0 x + b 1 x q , L 3 ( x ) = x + x q . The numerical results suggest that the sufficient conditions we find in the first class of permutation polynomials (see Theorem 3.2) are also necessary. Some known results on permutation polynomials over F q 2 are covered by ours.

Complete permutation polynomials with the form [formula omitted] over [formula omitted

In this chapter we present upper bounds on the minimum distance of turbo codes using QPP interleavers obtained by Eirik Rosnes in (2012), as well as an upper bound on the minimum distance of turbo codes using PP-based interleaver of any degree given in Ryu et al. (2015). We note that most of these upper bounds are partial, meaning that they are valid only for certain interleaver lengths complying with certain constraints on their prime decomposition. A single upper bound is general (see Theorem 5.10, Eq. (5.112)), meaning that it holds for any interleaver length, but there is a constraint on the minimum degree of the inverse polynomials to be equal to two (i.e. the QPP interleaver must have an inverse QPP (Ryu and Takeshita 2006)). Because some of the above mentioned upper bounds use inverse polynomials of QPPs, in the first section of this chapter we present the necessary and sufficient condition for a QPP to admit at least one quadratic inverse, derived by Jonghoon Ryu and Oscar Y. Takeshita in Ryu and Takeshita (2006). At the end of the first section we give, without proof, a theorem from Ryu (2007), Ryu and Takeshita (2011) which allows us to determine all inverse permutation polynomials of a QPP of the smallest degree.

Some classes of monomial complete permutation polynomials over finite fields of characteristic two