On Inverses for Quadratic Permutation Polynomials over Integer Rings

Analysis of cubic permutation polynomials for turbo codes

Quadratic permutation polynomials (QPPs) have been widely studied and used as interleavers in turbo codes. However, less attention has been given to cubic permutation polynomials (CPPs). This paper proves a theorem which states sufficient and necessary conditions for a cubic permutation polynomial to be a null permutation polynomial. The result is used to reduce the search complexity of CPP interleavers for short lengths (multiples of 8, between 40 and 352), by improving the distance spectrum over the set of polynomials with the largest spreading factor. The comparison with QPP interleavers is…

The Group Theoretic Roots of Information: permutations, symmetry, and entropy

We propose a new interpretation of measures of information and disorder by connecting these concepts to group theory in a new way. Entropy and group theory are connected here by their common relation to sets of permutations. A combinatorial measure of information and disorder is proposed, in terms of integers and discrete functions, that we call the integer entropy. The Shannon measure of information is the limiting case of a richer, more general conceptual structure that reveals relations among finite groups, information, and symmetries. It is shown that the integer entropy converges uniforml…

Five Constructions of Permutation Polynomials over $\gf(q^2)$

Four recursive constructions of permutation polynomials over $\gf(q^2)$ with those over $\gf(q)$ are developed and applied to a few famous classes of permutation polynomials. They produce infinitely many new permutation polynomials over $\gf(q^{2^\ell})$ for any positive integer $\ell$ with any given permutation polynomial over $\gf(q)$. A generic construction of permutation polynomials over $\gf(2^{2m})$ with o-polynomials over $\gf(2^m)$ is also presented, and a number of new classes of permutation polynomials over $\gf(2^{2m})$ are obtained.

On Two Conjectures about Permutation Trinomials over $\mathbb{F}_{3^{2k}}$

Permutation polynomials with few terms attracts researchers' interest in recent years due to their simple algebraic form and some additional extraordinary properties. In this paper, by analyzing the quadratic factors of a fifth-degree polynomial and a seventh-degree polynomial over the finite field $\mathbb{F}_{3^{2k}}$, two conjectures on permutation trinomials over $\mathbb{F}_{3^{2k}}$ proposed recently by Li, Qu, Li and Fu are settled, where $k$ is a positive integer.

Several Classes of Permutation Trinomials From Niho Exponents

Motivated by recent results on the constructions of permutation polynomials with few terms over the finite field $\mathbb{F}_{2^n}$, where $n$ is a positive even integer, we focus on the construction of permutation trinomials over $\mathbb{F}_{2^n}$ from Niho exponents. As a consequence, several new classes of permutation trinomials over $\mathbb{F}_{2^n}$ are constructed from Niho exponents based on some subtle manipulation of solving equations with low degrees over finite fields.

A general construction of regular complete permutation polynomials

Let $r\geq 3$ be a positive integer and $\mathbb{F}_q$ the finite field with $q$ elements. In this paper, we consider the $r$-regular complete permutation property of maps with the form $f=τ\circσ_M\circτ^{-1}$ where $τ$ is a PP over an extension field $\mathbb{F}_{q^d}$ and $σ_M$ is an invertible linear map over $\mathbb{F}_{q^d}$. We give a general construction of $r$-regular PPs for any positive integer $r$. When $τ$ is additive, we give a general construction of $r$-regular CPPs for any positive integer $r$. When $τ$ is not additive, we give many examples of regular CPPs over the extension…

关于 APN 置换的一个猜想

On a conjecture on APN permutations

本文利用有限域代数几何工具研究两个九维零星二次 APN 置换的三变量表示族，证明其在更高维数中不存在其他 APN 置换。

A new class of permutation trinomials constructed from Niho exponents

Permutation polynomials over finite fields are an interesting subject due to their important applications in the areas of mathematics and engineering. In this paper we investigate the trinomial $f(x)=x^{(p-1)q+1}+x^{pq}-x^{q+(p-1)}$ over the finite field $\mathbb{F}_{q^2}$, where $p$ is an odd prime and $q=p^k$ with $k$ being a positive integer. It is shown that when $p=3$ or $5$, $f(x)$ is a permutation trinomial of $\mathbb{F}_{q^2}$ if and only if $k$ is even. This property is also true for more general class of polynomials $g(x)=x^{(q+1)l+(p-1)q+1}+x^{(q+1)l+pq}-x^{(q+1)l+q+(p-1)}$, where …

A conjecture on permutation trinomials over finite fields of characteristic two

In this paper, by analyzing the quadratic factors of an $11$-th degree polynomial over the finite field $\ftwon$, a conjecture on permutation trinomials over $\ftwon[x]$ proposed very recently by Deng and Zheng is settled, where $n=2m$ and $m$ is a positive integer with $\gcd(m,5)=1$.

New Instances of Quadratic APN Functions

In a recent work, Beierle, Brinkmann and Leander presented a recursive tree search for finding APN permutations with linear self-equivalences in small dimensions. In this paper, we describe how this search can be adapted to find many new instances of quadratic APN functions. In particular, we found 12,921 new quadratic APN functions in dimension eight, 35 new quadratic APN functions in dimension nine and five new quadratic APN functions in dimension ten up to CCZ-equivalence. Remarkably, two of the 35 new APN functions in dimension nine are APN permutations. Among the 8-bit APN functions, ther…

奇维二次APN置换的最优简约代表元

Optimal Reduced Representatives of Quadratic APN Permutations in Odd Dimension

该研究针对奇维二次APN置换，证明奇数n≥5时存在特定可逆映射使简约代表元导数满足条件，相关结果在Lean 4中验证，为伽罗瓦环上最优微分均匀性置换多项式提供反例。