Workshop on Generalized Almost Perfect Nonlinear Functions and Fermat Curves

Reference No.	2024a017
Type/Category	Grant for General Research-Short-term Joint Research
Title of Research Project	Workshop on Generalized Almost Perfect Nonlinear Functions and Fermat Curves
Principal Investigator	Masamichi Kuroda（Nippon Bunri University School of Engineering・Associate professor）
Research Period	September 20,2024. ～ September 24,2024.
Keyword(s) of Research Fields	almost perfect nonlinear function, generalized almost perfect nonlinear function, geometric irreducibility, Fermat curve
Abstract for Research Report	As highly nonlinear functions on finite fields, almost perfect nonlinear (APN) functions have been studied and applied to cryptography and coding theory in the case of characteristic 2. For example, in cryptography, it is one of the key components in constructing block cipher algorithms which have highly resistant to differential and linear cryptanalysis, and in coding theory, exceptional numbers were classified completely by giving the complete classification of exceptional APN functions. On the other hand, basic research of APN functions for odd characteristic has not been sufficient or applied to these areas. The reason for non-existence of applications to these areas may be that APN functions for odd characteristic do not satisfy an analogue of the algebraic property for characteristic 2. Recently, as a generalization of APN functions from even characteristic to odd characteristic preserving the algebraic property, generalized almost perfect nonlinear (GAPN) functions are defined. It is possible that polynomials with Integer coefficients which are not APN on finite fields of characteristic 2 are GAPN on finite fields of odd characteristic. In particular, since the existence of such generating functions is an advantage in applications, GAPN functions have potential for industrial applications. GAPN functions began to be studied both nationally and internationally, and an academic paper is published in Cryptography and Communications in 2023. The purpose of this study is to make progress on the classification of exceptional GAPN functions for odd characteristic. As mentioned above, exceptional APN functions for characteristic 2 have been applied to cryptography and coding theory. Hence, based on the results obtained in this study, it may be possible to apply GAPN functions to these areas by generalizing the already known applications. In addition, Fermat curves play an important role in this study. From the above, this study is an interdisciplinary study of mathematics in which various areas (cryptography, coding theory, number theory, arithmetic geometry, algebraic geometry, combinatorics, computer mathematics, etc.) are related.
Organizing Committee Members (Workshop) Participants (Short-term Joint Usage)	Kentaro Mitsui（University of the Ryukyus・Associate professor） Akinari Hoshi（Niigata University・Professor）