Constructions of Involutions Over Finite Fields

• Published in: IEEE transactions on information theory Vol. 65; no. 12; pp. 7876 - 7883
• Main Authors: Zheng, Dabin, Yuan, Mu, Li, Nian, Hu, Lei, Zeng, Xiangyong
• Format: Journal Article
• Language: English
• Published: New York IEEE 01.12.2019; The Institute of Electrical and Electronics Engineers, Inc. (IEEE)
• Subjects: block cipher; Ciphers; Criteria; Cryptography; Encryption; Fields (mathematics); Indexes; Information theory; involution; Linear equations; Mathematics; Numbers; Permutation polynomial; Permutations; Polynomials; Subgroups; Task analysis
• ISSN: 0018-9448; 1557-9654
• DOI: 10.1109/TIT.2019.2919511

An involution over finite fields is a permutation polynomial whose inverse is itself. Owing to this property, involutions over finite fields have been widely used in applications, such as cryptography and coding theory. Following the idea by Wang to characterize the involutory behavior of the generalized cyclotomic mappings, this paper gives a more concise criterion for <inline-formula> <tex-math notation="LaTeX">x^{r}h(x^{s})\in {\mathbb F} _{q}[x] </tex-math></inline-formula> being involutions over the finite field <inline-formula> <tex-math notation="LaTeX">{\mathbb F}_{q} </tex-math></inline-formula>, where <inline-formula> <tex-math notation="LaTeX">r\geq 1 </tex-math></inline-formula> and <inline-formula> <tex-math notation="LaTeX">s\,|\, (q-1) </tex-math></inline-formula>. By using this criterion, we propose a general method to construct involutions of the form <inline-formula> <tex-math notation="LaTeX">x^{r}h(x^{s}) </tex-math></inline-formula> over <inline-formula> <tex-math notation="LaTeX">{\mathbb F}_{q} </tex-math></inline-formula> from given involutions over some subgroups of <inline-formula> <tex-math notation="LaTeX">{\mathbb F}_{q}^{*} </tex-math></inline-formula> by solving congruent and linear equations over finite fields. Then, many classes of explicit involutions of the form <inline-formula> <tex-math notation="LaTeX">x^{r}h(x^{s}) </tex-math></inline-formula> over <inline-formula> <tex-math notation="LaTeX">{\mathbb F}_{q} </tex-math></inline-formula> are obtained.