We study a subclass of p-ary functions in n variables, denoted by A n , which is a collection of p-ary functions in n variables satisfying a certain condition on the exponents of its monomial terms. Firstly, we completely classify all p-ary (n − 1)-plateaued functions in n variables by proving that every (n − 1)-plateaued function should be contained in A n . Secondly, we prove that if f is a p-ary r-plateaued function contained in A n with deg f > 1 + n−r 4 (p −1), then the highest degree term of f is only a single term. Furthermore, we prove that there is no p-ary r-plateaued function in A n with maximum degree (p − 1) n−r 2 + 1. As application, we partially classify all (n − 2)-plateaued functions in A n when p = 3, 5, and 7, and p-ary bent functions in A 2 are completely classified for the cases p = 3 and 5.
Bibliographical noteFunding Information:
The first author was supported by the National Research Foundation of Korea(NRF) grant funded by the Korea government(MEST) (NRF-2017R1A2B2004574), the second and third named authors were supported by Basic Science Research Program through the National Re-search Foundation of Korea(NRF) funded by the Ministry of Education(2009-0093827), and the second named author is supported by National Research Foundation of Korea (NRF) grant founded by the Korea government(MEST)(NRF-2017R1A6A3A11030486) and a research grant of Kangwon National University in 2018, and the third named author also by the National Research Foundation of Korea(NRF) grant funded by the Korea government(MEST)(NRF-2017R1A2B2004574).
© 2018 Korean Mathematial Soiety.
- Bent function
- Cryptographic function
- Plateaued function