Nilpotent linearized polynomials over finite fields and applications

TL;DR

This paper introduces nilpotent linearized polynomials over finite fields, explores their properties, and demonstrates their applications in constructing permutations, involutions, and cryptographic functions.

Contribution

It defines and studies nilpotent linearized polynomials, providing methods for their construction and applications in permutation and involution design over finite fields.

Findings

Existence and construction methods for NLP's

Explicit permutation and inverse constructions from NLP's

Construction of fixed-point-free involutions for cryptography

Abstract

Let $q$ be a prime power and $\mathbb F_{q^n}$ be the finite field with $q^n$ elements, where $n>1$. We introduce the class of the linearized polynomials $L(x)$ over $\mathbb F_{q^n}$ such that $$L^{(t)}(x):=\underbrace{L(L(\cdots(x)\cdots))}_{t \quad\text{times}}\equiv 0\pmod {x^{q^n}-x}$$ for some $t\ge 2$, called nilpotent linearized polynomials (NLP's). We discuss the existence and construction of NLP's and, as an application, we show how to construct permutations of $\mathbb F_{q^n}$ from these polynomials. For some of those permutations, we can explicitly give the compositional inverse map and the cycle structure. This paper also contains a method for constructing involutions over binary fields with no fixed points, which are useful in block ciphers.