Abstract
In this paper, we study blended representations of Boolean functions, and construct the following two classes of Boolean functions. Two bounds on the r-order nonlinearity were given by Carlet in the IEEE TRANSACTIONS ON INFORMATION THEORY, vol. 54. In general, the second bound is better than the first bound. But it was unknown whether it is always better. Recently, Mesnager et al. constructed a class of Boolean functions where the second bound is strictly worse than the first bound, for r = 2. However, it is still an open problem for r ≥ 3. Using the blended representation, we construct a class of Boolean functions based on the trace function and show that the second bound can also be strictly worse than the first bound, for r = 3. The second class is based on the hidden weighted bit function, which seems to have the best cryptographic properties among all currently known functions.
Highlights
Boolean functions have many applications in logic, electrical engineering, reliability theory, game theory, combinatorics, computational complexity, coding theory, cryptography, etc [14]
In this paper, we study blended representations of Boolean functions, and construct the following two classes of Boolean functions
An n-variable Boolean function f is a function from Fn2 into F2, and it can be represented by the output column of its truth table, i.e., a binary string of length 2n f (0, . . . , 0), f (1, . . . , 0), f (0, 1, . . . , 0), f (1, 1, . . . , 0), . . . , f (1, . . . , 1)
Summary
Boolean functions have many applications in logic, electrical engineering, reliability theory, game theory, combinatorics, computational complexity, coding theory, cryptography, etc [14]. In 2018, Wang et al proved that the maximum 3-order nonlinearity of 7-variable Boolean functions with degree at most 4 is 20 [46]. In [31], Mesnager et al constructed a class of Boolean functions where the first bound is tight and the second bound is strictly worse than the first bound, for r = 2. It is still an open problem for r ≥ 3. We construct a class of Boolean functions based on the trace function and show that the second bound can be strictly worse than the first bound, for r = 3.
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