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Published in:
2011 | OriginalPaper | Chapter
On Equivalence Classes of Boolean Functions
Authors : Qichun Wang, Thomas Johansson
Published in: Information Security and Cryptology - ICISC 2010
Publisher: Springer Berlin Heidelberg
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In FSE 2010, Rønjom and Cid put forward a nonlinear equivalence for Boolean functions and demonstrated that many cryptographic properties are not invariant among functions within the same equivalence class by providing some special examples. Their paper presented the idea and many problems were left open.
In this paper, we investigate equivalence of Boolean functions more deeply using a new method and discuss the number of Boolean functions in each equivalence class. We investigate further the cryptographic properties including algebraic immunity, algebraic degree and nonlinearity of equivalence classes, and deduce tight bounds on them. We find that there are many equivalence classes of Boolean functions with optimum algebraic immunity, optimum algebraic degree and a good nonlinearity. Moreover, we discuss how to construct equivalence classes with desired properties and show that it is possible to construct practical Boolean functions such that their equivalence classes have guaranteed cryptographic properties.