2023/10/16 by Polujan, Alexandr, Pasalic, Enes, Kudin, Sadmir +1 · 1 citation
#Combinatorics (math.CO) #FOS: Computer and information sciences #FOS: Mathematics #Information Theory (cs.IT)
paper · doi:10.48550/arxiv.2310.10162
The concatenation of four Boolean bent functions f=f1||f2||f3||f4 is bent if and only if the dual bent condition f1^* + f2^* + f3^* + f4^* =1 is satisfied. However, to specify four bent functions satisfying this duality condition is in general quite a difficult task. Commonly, to simplify this problem, certain connections between fi are assumed, as well as functions fi of a special shape are considered, e.g., fi(x,y)=x⋅πi(y)+hi(y) are Maiorana-McFarland bent functions. In the case when permutations πi of \mathbbF2m have the (Am) property and Maiorana-McFarland bent functions fi satisfy the additional condition f1+f2+f3+f4=0, the dual bent condition is known to have a relatively simple shape allowing to specify the functions fi explicitly. In this paper, we generalize this result for the case when Maiorana-McFarland bent functions fi satisfy the condition f1(x,y)+f2(x,y)+f3(x,y)+f4(x,y)=s(y) and provide a construction of new permutations with the (Am) property from the old ones. Combining these two results, we obtain a recursive construction method of bent functions satisfying the dual bent condition. Moreover, we provide a generic condition on the Maiorana-McFarland bent functions stemming from the permutations of \mathbbF2m with the (Am) property, such that their concatenation does not belong, up to equivalence, to the Maiorana-McFarland class. Using monomial permutations πi of \mathbbF2m with the (Am) property and monomial functions hi on \mathbbF2m, we provide explicit constructions of such bent functions. Finally, with our construction method, we explain how one can construct homogeneous cubic bent functions, noticing that only very few design methods of these objects are known.