IEEE Consumer Electronics Magazine - September/October 2021 - 43

The authors have observed that the Cubic
congruence could not work for the prime number
in the form of 6kþ1, and " n " is not 3. Therefore,
the prime numbers are not boundless. The
attacker may not face much difficulty getting the
plain text from cipher if the sender encrypts the
plain text using limited primes. Hence, the
authors move toward Polynomial congruence
for receiving the boundless prime numbers.
2) Polynomial congruence concept: As we have
discussed in Section II.A.1, the Cubic congruence
equation has a limited nature. The
authors thought about the equation xn a
(modp)where " a " and " n " are integers, and " p "
is a prime. Here, the range of primes is boundless,
so to find the " x, " the value of " p, " " a, " and
" n " should be known. The implementation
of the Polynomial congruence concept is done
in Algorithm 1, where the worst time complexity
is O(p). Authors also noted that if the value
of " n " is even, xn modp
ðp xÞn modp;
Else, xn modpðp xÞn modp and
Pp1
m¼1
nm mod p 0 ði:e:; 1 n 726989 ¼ > 316689 ¼ > USBY and
Decryption: USBY ¼ > 316689 ¼ > 726989 ¼
> HEY.
2) Encryption and Decryption for Image: Multimedia
content is being used in every field,
such as hospitality, education, banking,
insurance, science, and technology, etc. The
authors have also used the proposed technique
in sharing the image as implemented
in Algorithm 2. Initially, the sender chooses
a prime and value of " n, " and checks the
solution for " n " with the help of Algorithm 1.
Unfortunately, suppose the solution is not
possible; in that case, the sender should
choose another " n " value. If the solution is
possible with the new value of " n, " then
using a prime number and " n " value, the
sender encrypts every pixel of the image.
But there is a possibility of crossing the
pixel boundary value (i.e., 255) after applying
the polynomial congruence on the pixel
value. To overcome this issue, repeated iterations
are performed on the obtained value
until the desired result is achieved. Before
applying Polynomial congruence, decimation-by-2-permutation,
and 2-PHT (2-point
Pseudo Hadamard Transformation) are
applied to create the diffusion in the image
encryption between levels of linear layers,
as shown in Fig. 1. If the two bytes are given
to the 2-PHT as an input (i.e., a1, a2), then
the output bytes are (b1;b2):b1 ¼ð2a1þ
a2Þ % 256 and b2 ¼ða1 þ a2Þ % 256:
Finally, the sender shares the prime, the
value of " n, " and cipher with the receiver. After
receiving the cipher from the sender, the
receiver may able to generate the corresponding
key using Algorithm 1. The decryption mechanism
is done for every pixel using the same
43

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