Security enhancement and analysis of images using a novel Sudoku-based encryption algorithm

Figures & data

Algorithm 1 Read Image, Store Keys

1: read the image

2: store the height and width of the image as the first two keys in ‘keys.txt’

3: SudokuSize = input size of the sudoku

4: if user wants custom keyspace then

5: keyspace = input the custom keyspace.

6: convert the custom keyspace into the equivalent interger keyspace.

7: end if

8: n = input number of encryption rounds.

9: store n in ‘keys.txt’

10: input n Sudoku keys or generate randomly using seed values from keys.txt.

11: store the n keys in ‘keys.txt’.

Algorithm 2 Generate Pseudo Random & Perform Thresholding

1. read in the first Sudoku puzzle from ‘keys.txt’, set counter = 0

2. for each element of row of Sudoku do

3. for each element of column of Sudoku do

4.  if row number = column number then

5.   counter $+=1$

6.  end if

7. end for

8. end for

9. for each pixel of image do

10. if image is color then

11.  if $pixel[red]+randomNumber\le 255$ then

12.   set pixel[red] + = randomNumber

13.  else

14.   set pixel[red] -= (255 - randomNumber)

15.  end if

16.  if $pixel[green]+randomNumber\le 255$ then

17.   set pixel[green] += randomNumber

18.  else

19.   set pixel[green] -= (255 - randomNumber)

20.  end if

21.  if $pixel[blue]+randomNumber\le 255$ then

22.   set pixel[blue] += randomNumber

23.  else

24.   set pixel[blue] -= (255 - randomNumber)

25.  end if

26. else if image is greyscale then

27.  if $pixel+randomNumber\le 255$ then

28.   set pixel += randomNumber

29.  else

30.   set pixel -= (255 - randomNumber)

31.  end if

32. end if

33. end for

34. store the resulting image.

Algorithm 3 Image Padding

1. if height<SudokuSize then

2. add padding SudokuSize - height

3. else if width<SudokuSize then

4. add padding SudokuSize - width

5. else if $height\text{\%}SudokuSize!=0$ then

6. add padding (height//SudokuSize+1)*SudokuSize - height

7. else if $width\text{\%}SudokuSize!=0$ then

8. add padding (width//SudokuSize + 1)*SudokuSize - width

9. end if

10. randomly shuffle each array of image according to seed value

11. store the padded & shuffled image.

12. store the height of padded image as seed value in ‘keys.txt’ as the final key

Algorithm 4 Sudoku Based Encryption

1. shuffle the rows of paddedImage with the last key of ‘keys.txt’ the seed value.

2. for $(i=1i<n+1i++)$ do

3. read Sudokus from the key.txt file.

4. shuffle image pixels in the row in blocks of SudokuSize according to Sudoku $K_n$.

5. shuffle the image rows in batches of SudokuSize according to Sudoku $K_n$.

6. rotate image clockwise by 90 degrees.

7. end for

8. store the encrypted image.

Algorithm 5 Sudoku Based Decryption of Encrypted Image

1. im = read image to be decrypted.

2. reverse shuffle rows of paddedImage with the last key of ‘keys.txt’ as the seed value.

3. for $(i=1i<n+1i++)$ do

4. rotate image anti-clockwise by 90 degrees.

5. read Sudokus from the key.txt file in reverse order.

6. reverse shuffle image pixels in each row in blocks of sudokuSize according to Sudoku $K_n$.

7. reverse shuffle image rows in batches of sudokuSize according to Sudoku $K_n$.

8. end for

9. reverse shuffle image rows using the last key of ‘keys.txt’.

10. obtain padded, decrypted image with thresholding.

Algorithm 6 Remove Image Padding

1. im = decrypted, padded image

2. $h_{im}$, $w_{im}$ = height & width of im

3. read height and width of image from key space as $h_{key}$ & $w_{key}$

4. remove $(h_{im}-h_{key})/2$ padding from bottom and top.

5. remove $(w_{im}-w_{key})/2$ padding from left and right.

6. store the image obtained after removal of padding.

Algorithm 7 Restore original image by reversing thresholding

im = unpadded image

for each pixel of image do

if image is color then

if $pixel[red]+randomNumber\ge 0$ then

set pixel[red] −= randomNumber

else

set pixel[red] += (255 - randomNumber)

end if

if $pixel[green]+randomNumber\ge 0$ then

set pixel[green] −= randomNumber

else

set pixel[green] += (255 - randomNumber)

end if

if $pixel[blue]+randomNumber\ge 0$ then

set pixel[blue] −= randomNumber

else

set pixel[blue] += (255 - randomNumber)

end if

else if image is greyscale then

if $pixel+randomNumber\ge 0$ then

set pixel −= randomNumber

else

set pixel += (255 - randomNumber)

end if

end if

end for

obtain decrypted image.

Figure 1. Flow diagram of the encryption process.

Figure 2. Flow diagram of the decryption process.

Figure 3. Image 1 before encryption.

Figure 4. Image 1 after modified thresholding.

Figure 5. Image 1 after 50 iterations.

Figure 6. Image 1 after 75 iterations.

Figure 7. Image 1 after 100 iterations.

Figure 8. Image 1 after decryption.

Figure 9. Color Histogram before Encryption.

Figure 10. Color Histogram after Encryption.

Figure 11. Image 2 before encryption.

Figure 12. Image 2 after modified thresholding.

Figure 13. Image 2 after 50 iterations.

Figure 14. Image 2 after 75 iterations.

Figure 15. Image 2 after 100 iterations.

Figure 16. Image 2 after decryption.

Figure 17. Color Histogram before Encryption.

Figure 18. Color Histogram after Encryption.

Figure 19. Image 3 before encryption.

Figure 20. Image 3 after modified thresholding.

Figure 21. Image 3 after 50 iterations.

Figure 22. Image 3 after 75 iterations.

Figure 23. Image 3 after 100 iterations.

Figure 24. Image 3 after decryption.

Figure 25. Color Histogram before Encryption.

Figure 26. Color Histogram after Encryption.

Figure 27. Image 4 before encryption.

Figure 28. Image 4 after modified thresholding.

Figure 29. Image 4 after 50 iterations.

Figure 30. Image 4 after 75 iterations.

Figure 31. Image 4 after 100 iterations.

Figure 32. Image 4 after decryption.

Figure 33. Color Histogram before Encryption.

Figure 34. Color Histogram after Encryption.

Figure 35. Image 5 before encryption.

Figure 36. Image 5 after modified thresholding.

Figure 37. Image 5 after 50 iterations.

Figure 38. Image 5 after 75 iterations.

Figure 39. Image 5 after 100 iterations.

Figure 40. Image 5 after decryption.

Figure 41. Color Histogram before Encryption.

Figure 42. Color Histogram after Encryption.

Figure 43. Image 6 before encryption.

Figure 44. Image 6 after modified thresholding.

Figure 45. Image 6 after 50 iterations.

Figure 46. Image 6 after 75 iterations.

Figure 47. Image 6 after 100 iterations.

Figure 48. Image 6 after decryption.

Figure 49. Color Histogram before Encryption.

Figure 50. Color Histogram after Encryption.

Figure 51. Image 1 before encryption.

Figure 52. Image 1 after modified thresholding.

Figure 53. Image 1 after 50 iterations.

Figure 54. Image 1 after 75 iterations.

Figure 55. Image 1 after 100 iterations.

Figure 56. Image 1 after decryption.

Figure 57. Color Histogram before Encryption.

Figure 58. Color Histogram after Encryption.

Figure 59. Image 2 before encryption.

Figure 60. Image 2 after modified thresholding.

Figure 61. Image 2 after 50 iterations.

Figure 62. Image 2 after 75 iterations.

Figure 63. Image 2 after 100 iterations.

Figure 64. Image 2 after decryption.

Figure 65. Color Histogram before Encryption.

Figure 66. Color Histogram after Encryption.

Figure 67. Image 3 before encryption.

Figure 68. Image 3 after modified thresholding.

Figure 69. Image 3 after 50 iterations.

Figure 70. Image 3 after 75 iterations.

Figure 71. Image 3 after 100 iterations.

Figure 72. Image 3 after decryption.

Figure 73. Color Histogram before Encryption.

Figure 74. Color Histogram after Encryption.

Figure 75. Image 4 before encryption.

Figure 76. Image 4 after modified thresholding.

Figure 77. Image 4 after 50 iterations.

Figure 78. Image 4 after 75 iterations.

Figure 79. Image 4 after 100 iterations.

Figure 80. Image 4 after decryption.

Figure 81. Color Histogram before Encryption.

Figure 82. Color Histogram after Encryption.

Figure 83. Image 5 before encryption.

Figure 84. Image 5 after modified thresholding.

Figure 85. Image 5 after 50 iterations.

Figure 86. Image 5 after 75 iterations.

Figure 87. Image 5 after 100 iterations.

Figure 88. Image 5 after decryption.

Figure 89. Color Histogram before Encryption.

Figure 90. Color Histogram after Encryption.

Figure 91. Image 6 before encryption.

Figure 92. Image 6 after modified thresholding.

Figure 93. Image 6 after 50 iterations.

Figure 94. Image 6 after 75 iterations.

Figure 95. Image 6 after 100 iterations.

Figure 96. Image 6 after decryption.

Figure 97. Color Histogram before Encryption.

Figure 98. Color Histogram after Encryption.

Figure 99. Sample Sudoku.

Figure 100. Sudoku defined on arbitrary key space.

Figure 101. Time analysis in seconds.

Figure 102. Time analysis of key generation in seconds.

Table 1. Comparison of iterations for Lena.

Number of iterations	Time Required (s)
25	5.792709500000001
50	11.773323399999981
75	17.823201600000004
100	24.863347800000156

Table 2. Time required for key generation.

Number of keys (Standard $9\times 9$ Sudoku)	Time Required (s)
25	10.219445300000189
50	23.623966499999824
75	37.47240509999983
100	49.08801289999974

Figure 103. Time analysis for the complete encryption process in seconds.

Table 3. Total time per set of iterations for Lena for a Sudoku of size $9\times 9$ in seconds.

Total iterations	Time Required (s)
25	16.01215480000019
50	35.397289899999805
75	55.29560669999984
100	73.9513606999999

Table 4. Total time required for 100 iterations of all images using $9\times 9$ Sudoku and $16\times 16$ Sudoku.

Image	Image Dimensions (In Pixels)	Sudoku Used	Time Required (s)
Lena	$512\times 512$	$9\times 9$	24.863347800000156
		$16\times 16$	41.441451500000085
Orca	$2047\times 1235$	$9\times 9$	44.64921710000004
		$16\times 16$	56.51922779999995
Towers	$1054\times 1600$	$9\times 9$	34.06326979999994
		$16\times 16$	58.09327759999996
Color Square	$417\times 417$	$9\times 9$	17.11347049999995
		$16\times 16$	17.371556599999958
Eiffel	$556\times 367$	$9\times 9$	16.330925800000387
		$16\times 16$	26.45713459999979
B/W Forest	$800\times 531$	$9\times 9$	34.238081500000135
		$16\times 16$	53.50442720000001

Table 5. Average pixel intensities of Images before and after encryption using $9\times 9$ Sudoku and $16\times 16$ Sudoku.

		Original Image Pixel Intensities			Encrypted Image Pixel Intensities
Image	Sudoku Used	Red	Green	Blue	Red	Green	Blue
Lena	$9\times 9$	180.24	99.03	105.44	121.58	141.24	170.42
	$16\times 16$				183.12	183.34	109.59
Orca	$9\times 9$	105.19	109.06	109.31	118.25	142.49	144.44
	$16\times 16$				144.33	120.64	127.94
Towers	$9\times 9$	93.99	125.06	154.02	146.32	163.63	145.57
	$16\times 16$				110.05	86.58	116.89
Color Square	$9\times 9$	128.99	147.78	90.33	67.11	113.55	76.16
	$16\times 16$				171.92	145.50	182.58
Eiffel	$9\times 9$	164.69	161.18	157.03	129.77	141.66	137.09
	$16\times 16$				176.02	83.68	131.37
B/W Forest	$9\times 9$	69.68	69.68	69.68	78.33	78.33	78.33
	$16\times 16$				111.41	111.41	111.41

Table 6. Comparison of Shannon entropies of original images and encrypted Images.

Image	Entropy of Original Image	Entropy of Encrypted Image
Lena	7.76088	7.99174
Orca	7.85013	7.98612
Towers	7.79760	7.98530
Color Square	4.82313	7.96223
Eiffel	7.66564	7.99613
B/W Forest	7.35455	7.96326

Table 7. Information Entropies comparison for different algorithms on Lena.

Algorithm	Entropy Value for Lena
Proposed Algorithm	7.99174
X. Wu et al. (2018)	7.9895
Baptista's algorithm (Zhang & Liu, 2011)	7.9260
Wong's algorithm (Zhang & Liu, 2011)	7.9690
RC5 (Hanchinamani & Kulakarni, 2014)	7.9812
RC6 (Hanchinamani & Kulakarni, 2014)	7.9829
The MSK (Roy et al., 2021)	7.9983

Table 8. Obtained NPCR and UACI values for encrypted images using different Sudokus.

Image Name	Soduku Used	NPCR Value	UACI Value
Lena	$9\times 9$	99.60034	35.65403
	$16\times 16$	99.64510	28.00500
Orca	$9\times 9$	99.59824	29.13991
	$16\times 16$	99.61983	34.83915
Towers	$9\times 9$	99.56582	29.05051
	$16\times 16$	99.58515	27.19965
Colors Square	$9\times 9$	99.39061	49.77233
	$16\times 16$	99.89322	47.20972
Eiffel	$9\times 9$	99.58817	33.91635
	$16\times 16$	99.62297	26.14911
B/W Forest	$9\times 9$	99.37311	26.89946
	$16\times 16$	99.56968	32.29829

Table 9. Comparison of NPCR and UACI values between different algorithms on Lena.

Algorithm	NPCR Value (%)	UACI Value (%)
Proposed Algorithm	99.60	35.65
Norouzi et al. (2014)	99.65	33.55
Arpaci et al. (2020)	99.62	33.44
D. Mehta, Jha, et al. (2022)	99.61	33.58
Seyedzadeh and Mirzakuchaki (2012)	99.67	33.49

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