Orthogonal matrix and its application in Bloom’s threshold scheme

Abstract

Applying the Gram–Schmidt process (also called Gram–Schmidt orthogonalization) to a matrix $M\in GL(n, {\mathbb {R}})$, set of $n\times n$ invertible matrices over the field of real numbers, with the usual inner product gives easily an orthogonal matrix. However, the orthogonality in the vector space ${\mathbb {F}}_{q}^k$, where ${\mathbb {F}}_{q}$ is a binary finite field, is quite tricky as there are non-zero vectors which are orthogonal to themselves. For this reason the computational variants of Gram–Schmidt orthogonalization can fail. This paper presents an algorithm for constructing random orthogonal matrices over binary finite fields. The approach is inspired from the Gram–Schmidt procedure. Since the inverse of orthogonal matrix is easy to compute, the orthogonal matrices are used to construct a proactive variant of Bloom’s threshold secret sharing scheme.