Geodesic
Constructions of vectorial Boolean functions with maximum component algebraic immunity
Prikladnaya Diskretnaya Matematika. Supplement, no. 11 (2018), pp. 47-48.

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Matrices $A$ have been found so that the function $F\colon\mathbb F_2^n\to\mathbb F_2^n$ of the form $F(x)=(f(x),f(Ax),\dots,f(A^{n-1}x))$ where $f$ is the Dalai function in $n=3,4$ variables has the maximal component algebraic immunity. There are no vectorial Boolean functions $F\colon\mathbb F_2^5\to\mathbb F_2^5$ of the form $F(x)=(f(x),f(Ax),f(A^2x)),f(A^3x),f(A^4x))$ with the maximal component algebraic immunity where $f$ is the Dalai function in $5$ variables. Let $f$ be a Boolean function with the maximal algebraic immunity in an odd number $n$ of variables and $A$ be a non-degenerate matrix $n\times n$. Then the function $g(x)=f(x)+f(Ax)$ has the maximal algebraic immunity only if exactly half of the set supp$(f)$ remains in the set $\operatorname{supp}(f)$ after the action of the linear transformation $A$.
Keywords: vectorial Boolean functions, algebraic immunity, component algebraic immunity. 
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     author = {A. V. Miloserdov},
     title = {Constructions of vectorial {Boolean} functions with maximum component algebraic immunity},
     journal = {Prikladnaya Diskretnaya Matematika. Supplement},
     pages = {47--48},
     publisher = {mathdoc},
     number = {11},
     year = {2018},
     language = {ru},
     url = {http://geodesic.mathdoc.fr/item/PDMA_2018_11_a13/}
}
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A. V. Miloserdov. Constructions of vectorial Boolean functions with maximum component algebraic immunity. Prikladnaya Diskretnaya Matematika. Supplement, no. 11 (2018), pp. 47-48. http://geodesic.mathdoc.fr/item/PDMA_2018_11_a13/

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