Constructions of vectorial Boolean functions with maximum component algebraic immunity

A. V. Miloserdov

Prikladnaya Diskretnaya Matematika. Supplement, no. 11 (2018), pp. 47-48

Résumé

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 $A$.

Détail

Keywords: vectorial Boolean functions, algebraic immunity, component algebraic immunity.

Bibliographie

[1] Courtois N. T., Meier W., “Algebraic attacks on stream ciphers with linear feedback”, LNCS, 2656, 2003, 345–359 | MR | Zbl

[2] Tokareva N., Gorodilova A., Agievich S., et al., “Mathematical methods in solutions of the problems from the Third International Students' Olympiad in Cryptography”, Prikladnaya diskretnaya matematika, 2018, no. 40, 34–58

[3] Dalai D. K., Maitra S., Sarkar S., “Basic theory in construction of Boolean functions with maximum possible annihilator immunity”, Designs, Codes and Cryptography, 40 (2006), 41–58 | DOI | MR | Zbl