Algorithm 2: ECC with Hill Cipher

Input: The image of size 256 ×256 to be encrypted or decrypted.

Elliptic curve parameters: $\left(Fp\right)=\{a,b,p,G\}$.

Output: The corresponding cipher image or original image of size 256 × 256.

1.  Key Generation for ECC (The ECC keys are generated as shown in Section 3.1, Algorithm 1;

2.  Computing the self-invertible matrix

(a)  Compute: $K_1=xG=(k_{11},k_{12})$ and $K_2=yG=(k_{21}+k_{22})$;

(b)  Compute: $K_{12}=I_2-k_{11};K_{21}=I_2+k_{11}$; and $k_{11}+k_{22}=0$;

(c)  The $4\times 4$ self-invertible matrix is derived as: $$K_m=\left[\begin{array}{cc}{k}_{11}& k_{12}\\ k_{k21}& k_{22}\end{array}\right]$$

where, $K_{11}=\left[\begin{array}{cc}{k}_{11}& k_{12}\\ k_{21}& k_{22}\end{array}\right]$

3.  Encryption process:

(a)  Read the image to be encrypted and collect the image pixels separately for the

channels R, G, and B.

(b) Group every channel of pixels into 4 × 4 matrices and perform matrix multiplication

with computed self-invertible matrix.

(c) The encryption is done using the subsequent formula:         $\begin{array}{c}{C}_i=K_m·P_i=\left[\begin{array}{cccc}{k}_{11}& k_{12}& k_{13}& k_{14}\\ k_{21}& k_{22}& k_{23}& k_{24}\\ k_{31}& k_{32}& k_{33}& k_{34}\\ k_{41}& k_{42}& k_{43}& k_{44}\end{array}\right]\left[\begin{array}{cccc}{p}_{11}& p_{12}& p_{13}& p_{14}\\ p_{21}& p_{22}& p_{23}& p_{24}\\ p_{31}& p_{32}& p_{33}& p_{34}\\ p_{41}& p_{42}& p_{43}& p_{44}\end{array}\right]=\left[\begin{array}{cccc}{c}_{11}& c_{12}& c_{13}& c_{14}\\ c_{21}& c_{22}& c_{23}& c_{24}\\ c_{31}& c_{32}& c_{33}& c_{34}\\ c_{41}& c_{42}& c_{43}& c_{44}\end{array}\right]\hfill \end{array}$

where, $K_m$ is the self-invertible matrix and $P_i$ is the current input image block to be encrypted;

(d) Allocate the cipher pixels exactly to the same position as of the corresponding input

image pixels. A cipher image is formed of size identical to the size of input image.

(e) Send the cipher image and ECC public key to the receiver. The encryption process

can be visualized as in Figure 1.

4.  Decryption process

(a) Compute the shared secret key from the public key of sender as:

$SSK=nBPA=nBnAG=nAnBG=(x,y)$

(b) Compute the self-invertible matrix exactly as described above using the shared

secret key.

(c) Group the cipher pixels into $4\times 4$ matrices and perform matrix multiplication with

self-invertible matrix from Step 2.

(d) Allocate the plain text pixels to the same position as of the corresponding cipher

pixels. The original image is formed back and retrieved as without any intensity loss.