The applications of generalized inverse systematic non-square binary
matrices span many domains including mathematics, error-correction
coding, machine learning, data storage, navigation signals, and
cryptography. In particular, they are employed in the McEliece and
Niederreiter public key cryptosystems. For a systematic non-square
matrix H of size (n-k) x n, n > k,
there exist 2 distinct inverse
matrices. This paper presents an algorithm to generate these matrices as
well as a method to construct a random inverse for systematic and
non-systematic binary matrices. The proposed approach is shown to have
lower computational complexity than the well-known Gauss-Jordan
techniques. The application to public key cryptography (PKC) is also
discussed.