944 research outputs found
Constructions of Almost Optimal Resilient Boolean Functions on Large Even Number of Variables
In this paper, a technique on constructing nonlinear resilient Boolean
functions is described. By using several sets of disjoint spectra functions on
a small number of variables, an almost optimal resilient function on a large
even number of variables can be constructed. It is shown that given any ,
one can construct infinitely many -variable ( even), -resilient
functions with nonlinearity . A large class of highly
nonlinear resilient functions which were not known are obtained. Then one
method to optimize the degree of the constructed functions is proposed. Last,
an improved version of the main construction is given.Comment: 14 pages, 2 table
On the Algebraic Immunity - Resiliency trade-off, implications for Goldreich\u27s Pseudorandom Generator
Goldreich\u27s pseudorandom generator is a well-known building block for many theoretical cryptographic constructions from multi-party computation to indistinguishability obfuscation. Its unique efficiency comes from the use of random local functions: each bit of the output is computed by applying some fixed public -variable Boolean function to a random public size- tuple of distinct input bits.
The characteristics that a Boolean function must have to ensure pseudorandomness is a puzzling issue. It has been studied in several works and particularly by Applebaum and Lovett (STOC 2016) who showed that resiliency and algebraic immunity are key parameters in this purpose.
In this paper, we propose the first study on Boolean functions that reach together maximal algebraic immunity and high resiliency.
1) We assess the possible consequences of the asymptotic existence of such optimal functions. We show how they allow to build functions reaching all possible algebraic immunity-resiliency trade-offs (respecting the algebraic immunity and Siegenthaler bounds).
We provide a new bound on the minimal number of variables~, and thus on the minimal locality, necessary to ensure a secure Goldreich\u27s pseudorandom generator. Our results come with a granularity level depending on the strength of our assumptions, from none to the conjectured asymptotic existence of optimal functions.
2) We extensively analyze the possible existence and the properties of such optimal functions. Our results show two different trends. On the one hand, we were able to show some impossibility results concerning existing families of Boolean functions that are known to be optimal with respect to their algebraic immunity, starting by the promising XOR-MAJ functions. We show that they do not reach optimality and could be beaten by optimal functions if our conjecture is verified.
On the other hand, we prove the existence of optimal functions in low number of variables by experimentally exhibiting some of them up to variables. This directly provides better candidates for Goldreich\u27s pseudorandom generator than the existing XOR-MAJ candidates for polynomial stretches from to
Fast algebraic immunity of Boolean functions and LCD codes
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
New construction of Boolean functions with maximun algebraic immunity
Because of the algebraic attacks, a high
algebraic immunity is now an important criteria for Boolean
functions used in stream ciphers. In this paper, by using the
relationship between some flats and support of a n variables
Boolean function f, we introduce a general method to determine the
algebraic immunity of a Boolean function and finally construct some
balanced functions with optimum algebraic immunity
On the algebraic immunity - resiliency trade-off, implications for Goldreich's pseudorandom generator
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