Fast algebraic immunity of Boolean functions and LCD codes

TL;DR

This paper introduces a new measure called fast immunity profile to evaluate Boolean functions' resistance to fast algebraic attacks and links this resistance to LCD codes, enhancing cryptographic security analysis.

Contribution

It proposes a novel parameter for resistance to fast algebraic attacks, introduces the fast immunity profile, and connects perfect algebraic immune functions to LCD codes.

Findings

Defined the fast immunity profile for Boolean functions.

Evaluated the parameter on two Boolean function constructions.

Constructed infinite families of LCD codes from perfect algebraic immune functions.

Abstract

Nowadays, the resistance against algebraic attacks and fast algebraic attacks are considered as an important cryptographic property for Boolean functions used in stream ciphers. Both attacks are very powerful analysis concepts and can be applied to symmetric cryptographic algorithms used in stream ciphers. The notion of algebraic immunity has received wide attention since it is a powerful tool to measure the resistance of a Boolean function to standard algebraic attacks. Nevertheless, an algebraic tool to handle the resistance to fast algebraic attacks is not clearly identified in the literature. In the current paper, we propose a new parameter to measure the resistance of a Boolean function to fast algebraic attack. We also introduce the notion of fast immunity profile and show that it informs both on the resistance to standard and fast algebraic attacks. Further, we evaluate our parameter for two secondary constructions of Boolean functions. Moreover, A coding-theory approach to the characterization of perfect algebraic immune functions is presented. Via this characterization, infinite families of binary linear complementary dual codes (or LCD codes for short) are obtained from perfect algebraic immune functions. The binary LCD codes presented in this paper have applications in armoring implementations against so-called side-channel attacks (SCA) and fault non-invasive attacks, in addition to their applications in communication and data storage systems.