IMAGE ENCRYPTION USING COLLATZ CONJECTURE
Example systems, methods, and apparatus are disclosed herein for image encryption using Collatz Conjecture. The proposed technology combines Chaos theory and modified Collatz Conjecture to develop a hashing function that is used to encrypt digital images. The Collatz Conjecture, when combined with chaos theory, has the potential to produce high-quality encryption. The Collatz Conjecture is a tremendously complex mathematical problem, and its application in image encryption enhances the unpredictability and complexity of the process. The proposed technology is compared with various previously documented encryption techniques. The evaluation is conducted using a range of metrics, including histogram analysis, correlation coefficient, mean square error (MSE), number of pixels changes rate (NPCR), unified average change intensity (UACI), entropy, and time complexity.
1 . A system for image encryption based on the Collatz Conjecture comprising:
a server;
a processor; and
a memory storing instructions, which when executed by the processor, cause the processor to
apply a cryptographic algorithm based on the Collatz Conjecture.
2 . The system of claim 1 , wherein the cryptographic algorithm includes a pixel-to-pixel encryption and a pixel-to-pixel decryption phase.
3 . The system of claim 2 , wherein the pixel-to-pixel encryption phase uses a first hash function:
x
n
+
1
=
10
M
·
{
R
K
(
f
(
x
n
)
sin
(
gx
n
)
)
+
x
n
)
,
if
x
n
ϵ
(
-
)
(
x
n
2
j
)
×
P
2
j
+
1
,
if
mod
(
x
n
,
2
j
)
=
0
,
3
≤
j
≤
5
(
R
(
f
(
x
n
)
sin
(
gx
n
)
)
+
x
n
,
if
mod
(
x
n
,
2
2
)
=
0
(
x
n
2
)
×
P
2
+
1
,
if
mod
(
x
n
,
2
)
=
0
x
n
×
P
m
o
d
(
x
n
,
N
P
)
+
1
,
if
mod
(
x
n
,
2
)
=
1
In this equation, two variable parameters, K and M, have been used, where,
R
K
(
x
)
=
⌊
❘
"\[LeftBracketingBar]"
x
×
1
0
K
❘
"\[RightBracketingBar]"
⌋
The function f is a sinusoidal function with increasing period n ∈ [0, 1].
f
(
x
n
)
=
sin
(
2
π
e
P
m
o
d
(
n
N
,
|
N
P
|
)
+
3
)
+
α
X
n
+
mod
(
n
,
5
)
-
2
where α = 1 or α = 2.
The modulus function mod (x, y) being used in the CollatzHash function is
the remainder of division of x by y, that is, mod (10, 3) = 1, mod (8, 2) = 0.
The function g is defined as follows:
g
(
n
)
=
❘
"\[LeftBracketingBar]"
(
(
mod
(
n
,
7
)
+
1
n
7
+
n
3
+
n
+
1
0
0
+
mod
(
n
,
1
)
+
1
n
5
+
1
0
0
)
-
1
)
❘
"\[RightBracketingBar]"
where N P is a list of primes used and P 1 = 3, P 2 = 3, P 3 = 5, . . . where the
number Pi = ith prime for i ≥ 2.
4 . The system of claim 2 , wherein the pixel-to-pixel decryption phase uses a second hash function:
5 . A method for image encryption based on the Collatz Conjecture comprising:
receiving an image;
applying a pixel-to-pixel encryption hash function;
applying a pixel-to-pixel decryption hash function; and
reconstructing the image.
6 . The method of claim 5 , wherein the pixel-to-pixel encryption hash function is:
x
n
+
1
=
1
0
M
.
{
R
K
(
f
(
x
n
)
sin
(
gx
n
)
)
+
x
n
)
,
if
x
n
ϵ
(
-
)
(
x
n
2
j
)
×
P
2
j
+
1
,
if
mod
(
x
n
,
2
j
)
=
0
,
3
≤
j
≤
5
(
R
(
f
(
x
n
)
sin
(
gx
n
)
)
+
x
n
,
if
mod
(
x
n
,
2
2
)
=
0
(
x
n
2
)
×
P
2
+
1
,
if
mod
(
x
n
,
2
)
=
0
x
n
×
P
mod
(
x
n
,
N
P
)
+
1
,
if
mod
(
x
n
,
2
)
=
1
In this equation, two variable parameters, K and M, have been used, where,
R
K
(
x
)
=
⌊
❘
"\[LeftBracketingBar]"
x
×
10
K
❘
"\[RightBracketingBar]"
⌋
The function ƒ is a sinusoidal function with increasing period n∈[0, 1].
f
(
x
n
)
=
sin
(
2
π
e
P
m
o
d
(
n
N
,
|
N
P
|
)
+
3
)
+
α
X
n
+
mod
(
n
,
5
)
-
2
where α=1 or α=2.
The modulus function mod (x, y) being used in the CollatzHash function is the remainder of division of x by y, that is, mod (10, 3)=1, mod (8, 2)=0.
The function g is defined as follows:
g
(
n
)
=
❘
"\[LeftBracketingBar]"
(
(
mod
(
n
,
7
)
+
1
n
7
+
n
3
+
n
+
1
0
0
+
mod
(
n
,
1
)
+
1
n
5
+
1
0
0
)
-
1
)
❘
"\[RightBracketingBar]"
where N P is a list of primes used and P 1 =3, P 2 =3, P 3 =5, . . . where the number
Pi
=
ith
prime
for
i
≥
2
.
7 . The method of claim 5 , wherein the pixel-to-pixel decryption hash function is: