Plateaued functions play a significant role in cryptography, sequences for communications, and the related combinatorics and designs. Comparing to their importance, those functions have not been studied in detail in a general framework. Our motivation is to bring further results on the characterizations of bent and plateaued functions, and to introduce new tools which allow us firstly a better understanding of their structure and secondly to get methods for handling and designing such functions. We first characterize bent functions in terms of all even moments of the Walsh transform, and then plateaued (vectorial) functions in terms of the value distribution of the second-order derivatives. Moreover, we devote to cubic functions the characterization of plateaued functions in terms of the value distribution of the second-order derivatives, and hence this reveals non-existence of homogeneous cubic bent (and also (homogeneous) cubic plateaued for some cases) functions in odd characteristic. We use a rank notion which generalizes the rank notion of quadratic functions. This rank notion reveals new results about (homogeneous) cubic plateaued functions. Furthermore, we observe non-existence of a function whose absolute Walsh transform takes exactly 3 distinct values (one being zero). We finally provide a new class of functions whose absolute Walsh transform takes exactly 4 distinct values (one being zero).
Designs, Codes and Cryptography – Springer Journals
Published: Oct 23, 2017
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