457 research outputs found
Hadamard Matrices, -Linearly Independent Sets and Correlation-Immune Boolean Functions with Minimum Hamming Weights
It is known that correlation-immune (CI) Boolean functions used in the framework of side channel attacks need to have low Hamming weights. In 2013, Bhasin et al. studied the minimum Hamming weight of -CI Boolean functions, and presented an open problem: the minimal weight of a -CI function in variables might not increase with . Very recently, Carlet and Chen proposed some constructions of low-weight CI functions, and gave a conjecture on the minimum Hamming weight of -CI functions in variables.
In this paper, we determine the values of the minimum Hamming weights of -CI Boolean functions in variables for infinitely many \u27s and give a negative answer to the open problem proposed by Bhasin et al. We then present a method to construct minimum-weight 2-CI functions through Hadamard matrices, which can provide all minimum-weight 2-CI functions in variables. Furthermore, we prove that the Carlet-Chen conjecture is equivalent to the famous Hadamard conjecture. Most notably, we propose an efficient method to construct low-weight -variable CI functions through -linearly independent sets, which can provide numerous minimum-weight -CI functions. Particularly, we obtain some new values of the minimum Hamming weights of -CI functions in variables for . We conjecture that the functions constructed by us are of the minimum Hamming weights if the sets are of absolute maximum -linearly independent. If our conjecture holds, then all the values for and most values for general are determined
Simplicity conditions for binary orthogonal arrays
It is known that correlation-immune (CI) Boolean functions used in the
framework of side-channel attacks need to have low Hamming weights. The
supports of CI functions are (equivalently) simple orthogonal arrays when their
elements are written as rows of an array. The minimum Hamming weight of a CI
function is then the same as the minimum number of rows in a simple orthogonal
array. In this paper, we use Rao's Bound to give a sufficient condition on the
number of rows, for a binary orthogonal array (OA) to be simple. We apply this
result for determining the minimum number of rows in all simple binary
orthogonal arrays of strengths 2 and 3; we show that this minimum is the same
in such case as for all OA, and we extend this observation to some OA of
strengths and . This allows us to reply positively, in the case of
strengths 2 and 3, to a question raised by the first author and X. Chen on the
monotonicity of the minimum Hamming weight of 2-CI Boolean functions, and to
partially reply positively to the same question in the case of strengths 4 and
5
Simplicity conditions for binary orthogonal arrays
It is known that correlation-immune (CI) Boolean functions used in the framework of side channel attacks need to have low Hamming weights. The supports of CI functions are (equivalently) simple orthogonal arrays, when their elements are written as rows of an array. The minimum Hamming weight of a CI function is then the same as the minimum number of rows in a simple orthogonal array. In this paper, we use Rao's Bound to give a sufficient condition on the number of rows, for a binary orthogonal array (OA) to be simple. We apply this result for determining the minimum number of rows in all simple binary orthogonal arrays of strengths 2 and 3; we show that this minimum is the same in such case as for all OA, and we extend this observation to some OA of strengths 4 and 5. This allows us to reply positively, in the case of strengths 2 and 3, to a question raised by the first author and X. Chen on the monotonicity of the minimum Hamming weight of 2-CI Boolean functions, and to partially reply positively to the same question in the case of strengths 4 and 5
Boolean Dynamics with Random Couplings
This paper reviews a class of generic dissipative dynamical systems called
N-K models. In these models, the dynamics of N elements, defined as Boolean
variables, develop step by step, clocked by a discrete time variable. Each of
the N Boolean elements at a given time is given a value which depends upon K
elements in the previous time step.
We review the work of many authors on the behavior of the models, looking
particularly at the structure and lengths of their cycles, the sizes of their
basins of attraction, and the flow of information through the systems. In the
limit of infinite N, there is a phase transition between a chaotic and an
ordered phase, with a critical phase in between.
We argue that the behavior of this system depends significantly on the
topology of the network connections. If the elements are placed upon a lattice
with dimension d, the system shows correlations related to the standard
percolation or directed percolation phase transition on such a lattice. On the
other hand, a very different behavior is seen in the Kauffman net in which all
spins are equally likely to be coupled to a given spin. In this situation,
coupling loops are mostly suppressed, and the behavior of the system is much
more like that of a mean field theory.
We also describe possible applications of the models to, for example, genetic
networks, cell differentiation, evolution, democracy in social systems and
neural networks.Comment: 69 pages, 16 figures, Submitted to Springer Applied Mathematical
Sciences Serie
A new class of codes for Boolean masking of cryptographic computations
We introduce a new class of rate one-half binary codes: {\bf complementary
information set codes.} A binary linear code of length and dimension
is called a complementary information set code (CIS code for short) if it has
two disjoint information sets. This class of codes contains self-dual codes as
a subclass. It is connected to graph correlation immune Boolean functions of
use in the security of hardware implementations of cryptographic primitives.
Such codes permit to improve the cost of masking cryptographic algorithms
against side channel attacks. In this paper we investigate this new class of
codes: we give optimal or best known CIS codes of length We derive
general constructions based on cyclic codes and on double circulant codes. We
derive a Varshamov-Gilbert bound for long CIS codes, and show that they can all
be classified in small lengths by the building up construction. Some
nonlinear permutations are constructed by using -codes, based on the
notion of dual distance of an unrestricted code.Comment: 19 pages. IEEE Trans. on Information Theory, to appea
- …