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

In this paper, we discuss and mathematically compute the eigenvalues characteristic polynomials of special even square matrices orders 4x4 8x8. Also, introduce two 8th order compound magic squares. The computed values are verified using Maple software. First, for 4th matrix, polynomial was derived to be: λ(λ-2s)( λ²+4Θ) with eigenvalues: 0,2 s, other conjugates. further analysis, performed numerical classification squares 4. Second, square, obtained in form: λ<sup>3</sup>(λ-4s)(λ<sup>4</sup>+Ωλ<sup>2</sup>+θ) where Ω,Θ constants; 0,4 ∓√λ<sub>1</sub>, ∓√λ<sub>2</sub>; λ<sub>1</sub>, λ<sub>2</sub> roots quadratic equation: λ<sup>2</sup>+Ωλ+Θ=0. Third, franklin λ<sup>2</sup>+aλ+b. Finally, suggested a hybrid image encryption technique based on Franklin improved substitution technique. proposed grayscale encryption/ decryption algorithm uses circular rotation bits squares’ properties conjunction techniques obtain very secure against attacks.