An investigation of even ordered magic squares (4, 6, and 8): characteristic polynomials, eigenvalues, and encryption

Abstract

In this paper, we discuss and mathematically compute the eigenvalues and the characteristic polynomials of special even square matrices of orders 4x4 and 8x8. Also, we introduce two 8th order compound magic squares. The computed values are verified using Maple software. First, for the 4th order square matrix, the characteristic polynomial was derived to be: λ(λ-2s)( λ²+4Θ) with the eigenvalues: 0,2 s, and two other conjugates. In further analysis, we performed numerical classification of the squares for the matrices of order 4. Second, for the 8th order magic square, the characteristic polynomial was obtained in the form: λ³(λ-4s)(λ⁴+Ωλ²+θ) where Ω,Θ are constants; the eigenvalues are 0,4 s, ∓√λ₁, ∓√λ₂; where λ₁, λ₂ are the roots of the quadratic equation: λ²+Ωλ+Θ=0. Third, for the franklin square, we obtained the eigenvalues 0,4 s, and the roots of the equation: λ²+aλ+b. Finally, we suggested a hybrid image encryption technique based on Franklin magic square matrices and improved substitution technique. The proposed a grayscale image encryption/ decryption algorithm uses circular rotation of bits and Franklin magic squares’ properties in conjunction with substitution techniques to obtain a very secure algorithm against attacks.