Algorithm 3: ECC with AES

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

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

AES symmetric key and initialization vector (IV).

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);

2.  Key Generation for AES

(a) Sender randomly generates 16 bytes long AES symmetric key and the

Initialization Vector;

(b) These parameters must be securely transmitted to receiver;

3.  AES Encryption

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

channels R, G, and B.

(b) Convert the pixels in each channel to bytes using the function (PL) where PL

denotes the list of pixels. The function returns an immutable array of bytes.

(c) Perform AES encryption for each of the channel bytes. This generates bytes in the

encrypted form.

4.  ECC Encryption

(a) Choose a base of representing the numbers. We have chosen 256 to represent the

range of 8-bit pixel values;

(b) Represent the prime number p in base 256 as a list of integers from 0 to 255.

Measure the length of this representation as L. Initialize group size as $L-1$

(c) Group the cipher bytes from each channel separately with group size initialized

as above. For each of the channels, convert each group of cipher bytes to big

integers treating base of representation as 256. If numbers of big integers are odd,

append with a random value (possibly less than 256);

(d) Pair up the big integers to represent it as a point on the XY-plane. If the

X co-ordinate of the pair and that of the shared secret key are identical, then we

apply point doubling formula over the pair co-ordinates. Otherwise apply point

addition formula with SSK co-ordinates.

(e) From Step 4, we get a point on XY plane not at all necessary to be on the

curve chosen. Represent its co-ordinates in base 256 digits. Each of these.

representations must be of size equal to $L=groupsize+1$. Append 0 zeros

otherwise to make the required length.

(f) The base 256 representations of cipher points are treated as image pixels and cipher

image is constructed correspondingly. The width of cipher image is kept same as

width of input image and height is computed based on the total number of cipher

pixels collected. We observe that size of cipher image is generally more than the

input image.

(g) Send the cipher image, original image size and ECC public key to receiver.

5.  ECC Decryption

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

SSK = nBPA = nBnAG + nAnBG = (x, y);

(b) Represent the prime number p in base 256 as a list of integers from 0 to 255.

Measure he length of this representation as L. Initialize group size as:

$groupsize=L-1$

(c) Collect the cipher pixels for each channel and group them with each group size

equal to groupsize + 1;

(d) Convert each group in respective channels to corresponding big integers taking

base to be 256. Pair up the big integers treating them as points on the XY-plane.

(e) Perform reflection of $SSK$ point with respect x-axis: $(x,-ymodp)$;

(f) Perform point doubling with reflected $SSK$ co-ordinates if the x co-ordinates are

same or perform point addition formula otherwise. The point obtained for each

pair lies on XY-plane but need not be a point the elliptic curve

(g) Represent each of the points obtained in base 256 with each representation of

length equal to groupsize. Append zeros if required.

(h) Obtained values are bytes obtained from AES encryption.

6.  AES Decryption

(a) Obtain the AES key and IV from secure communication.

(b) Perform AES decryption for the bytes obtained from above for each of the respective

image channels.

(c) Represent the bytes as the plain text image of size as mentioned by the sender.