An Effective Color Image Encryption Based on Henon Map, Tent Chaotic Map, and Orthogonal Matrices

. 2022 Jun 8;22(12):4359. doi: 10.3390/s22124359

Algorithm 2. Key Generation of Permutation Process

Input:

Equation of plane a x + b y + c z = d, a, b, c, d \in ℜ .

Output:

Orthogonal matrix K_{2} .

1.
$Let the orthogonal line O be spanned by the unit vector t$
$t = \frac{(a, b, c)}{\sqrt{a^{2} + b^{2} + c^{2}}},$
$from the expression Z_{w} = w - 2 t 〈 w, t 〉$ $, where 〈 w, t 〉$ $is the inner product of w$ $and t$ .
2.
$For w = {w_{1}, w_{2}, w_{3}, \dots, w_{m}} \in R^{m}$ $be the basis of O$ $. The basis vectors for m = 3$ are
$w_{1} = [w_{11}, w_{12}, w_{13}] = [1, 0, 0] w_{2} = [w_{21}, w_{22}, w_{23}] = [0, 1, 0] w_{3} = [w_{31}, w_{32}, w_{33}] = [0, 0, 1] .$
3.
$The orthogonal key matrix K_{2}$ will be
$K_{2} = [\begin{matrix} z_{w 11} & z_{w 12} & z_{w 13} \\ z_{w 21} & z_{w 22} & z_{w 23} \\ z_{w 31} & z_{w 32} & z_{w 33} \end{matrix}] .$