Goldreich-Goldwasser-Halevi (GGH) encryption scheme is lattice-based cryptography with its security based on the shortest vector problem (SVP) and closest vector problem (CVP) with immunity to almost all attacks, including Shor's quantum algorithm and Nguyen's attack of higher lattice dimension. To improve the efficiency and security of the GGH Scheme by reducing the size of the public basis to be transmitted, we use an hourglass matrix obtained from quadrant interlocking factorization as a public key. The technique of quadrant interlocking factorization to yield a nonsingular hourglass matrix compensates the encryption scheme with better efficiency and security.

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This paper reviews the theory of matrices and determinants. Matrix and determinant are nowadays considered inseparable to some extent, but the determinant was discovered over two centuries before the term matrix was coined. Our review associate determinant with the matrix as part of linear systems but not with polynomials. Thus, the paper first gives the background on matrix with vast applications in all fields of study and then reviews the history of determinants which is based on its major contributors in chronological order from the sixteenth century to the twenty-first century