Simplicity conditions for binary orthogonal arrays

TL;DR

This paper establishes conditions under which binary orthogonal arrays are simple, linking their structure to correlation-immune Boolean functions and their Hamming weights, and determines minimal sizes for certain strengths.

Contribution

It provides a Rao's Bound-based sufficient condition for simplicity in binary orthogonal arrays and determines minimal row counts for strengths 2 to 5.

Findings

Minimum row count for simple OA of strengths 2 and 3 matches that of all OA.

Extended the minimal row count result to some OA of strengths 4 and 5.

Confirmed the monotonicity of the minimum Hamming weight for 2-CI Boolean functions for strengths 2 and 3.

Abstract

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.