2 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