Abstract
In 2005, Guillot published a method for the construction of Boolean functions using linear codes through the Maiorana–McFarland construction of Boolean functions. In this work, we present a construction using Hermitian codes, starting from the classic Maiorana–McFarland construction. This new construction describes how the set of variables is divided into two complementary subspaces, one of these subspaces being a Hermitian Code. The ideal theoretical parameters of the Hermitian code are proposed to reach desirable values of the cryptographic properties of the constructed Boolean functions such as nonlinearity, resiliency order, and order of propagation. An extension of Guillot’s work is also made regarding parameters selection using algebraic geometric tools, including explicit examples.
Highlights
One of the major challenges and problems today is in confusion symmetric key algorithms, which depend heavily on good cryptographic properties of Boolean functions such as nonlinearity
We calculate the minimum weight of E0 and E1, we will denote it by w(E0) and w(E1) For each u ∈ E⊥ we determine w(E0) and w(E1), if we want to construct a Boolean function (l − 1)-resilient, we need max{w(E0), w(E1)} ≥ l and w(E0) ≥ l and w(E1) ≥ l
We have established the optimal parameters based on the dimension of the Hermitian code and, on this, adjust the value of resilience to build cryptographically strong Boolean functions
Summary
One of the major challenges and problems today is in confusion symmetric key algorithms, which depend heavily on good cryptographic properties of Boolean functions such as nonlinearity. The classical construction splits the set of variables into two separate subsets, with a decomposition of the complete working space into two complementary vector spaces being proposed One of these spaces is considered as a linear code and its parameters assign cryptographic properties to the constructed Boolean function. The cryptographical properties we are interested in are nonlinearity, resiliency, and propagation Building on such an idea, in [4], a methodology to construct Boolean functions from Guillot’s ideas was presented but, in this case, using Reed–Solomon codes. We follow on the same ideas of [4], a new methodology is proposed for the construction of Boolean functions using Hermitian codes and extends the construction of π four to one giving the appropriate parameters for it It theoretically bases the desirable values of the Hermitian code parameters to balance cryptographic properties such as nonlinearity, resiliency order, and propagation order
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