=Paper=
{{Paper
|id=Vol-2300/Paper27
|storemode=property
|title=Methods of Crypto Protection of Color Image Pixels in Different Code Systems
|pdfUrl=https://ceur-ws.org/Vol-2300/Paper27.pdf
|volume=Vol-2300
|authors=Nataliia Vozna,Yaroslav Nykolaichuk,Orest Volynskyi,Petro Humennyi,Andrij Sydor
|dblpUrl=https://dblp.org/rec/conf/acit4/VoznaNVHS18
}}
==Methods of Crypto Protection of Color Image Pixels in Different Code Systems==
https://ceur-ws.org/Vol-2300/Paper27.pdf
110
Methods of Crypto Protection of Color Image Pixels in
Different Code Systems
Nataliia Vozna1, Yaroslav Nykolaichuk1, Orest Volynskyi2, Petro Humennyi1, Andrij Sydor1
1. Department of Specialized Computer Systems, Ternopil National Economic University, UKRAINE, Ternopil, 8 Chekhova str., email:
nvozna@ukr.net
2. Department of Cyber Security, Ternopil National Economic University, UKRAINE, Ternopil, 8 Chekhova str., email: orestsks@ukr.net
Abstract: The relevance of the development of recognition was carried out by segmentation methods on the
theoretical foundations, methods and algorithms for basis of histogram thresholding and cumulative histograms. It
encoding color image pixels by the problem-oriented is analysis of the statistic estimates of the mean value,
multifunctional data structuring and the representation dispersion, asymmetry and the degree of contrast of the
of color image code pixels in Rademacher (R), Krestenson intensity histograms homogeneity taking into account the
(K), Rademacher-Krestenson (RK), Haar-Krestenson dispersion of pixels coordinates of image fragments and
(HK) and Galois (G) Systems is substantiated in this silhouettes, as well as image clustering methods [1-5].
article. The purpose of the research is to increase the As a result, it was found that the main components of the
efficiency of the algorithms for digital image transforms, algorithms of the above-mentioned methods for image
processing and recognition using modular arithmetic of processing are the following arithmetic operations:
extended Galois fields on the basis of mathematics of
arithmetic operations of a non-positional residue number
summarizing ( ∑ xi ), division ( P(i ) = ni / n0 ), absolute
system. difference ( xi − x j ), square ( xi2 ), multiplication ( xi × x j ),
[ ]
Keywords: crypto protection, color image pixels,
Rademacher and Krestenson Systems, Residue Number square difference ( xi − x j 2 ), sum of multiplication
System.
( ∑ x x ), which are commonly performed due to the low-
i j
I. INTRODUCTION speed arithmetic of the binary number system.
Successful development of modern computer technology, III. THE METHOD FOR ENCODING RGB PIXELS
microelectronics and telecommunication systems promotes IN THE RADEMACHER AND KRESTENSON SYSTEMS
designing and mass production of color TV displays as well
as personal computers, mobile devices, camcorders, tablet PC According to the international RGB color model, colors
screens, industrial and large format color displays. are presented as a combination of three main colors: red (R),
The large-scale application of various types of video green (G) and blue (B) [4].
equipment in all branches of industry and their wide-spread In this case in the computer RGB system, the main color
personal use determines a high level of importance of the has 256 gradations. Thus, the color code of the RGB system
solutions to theoretical and applied problems of increasing is made up of three bytes, that is, 24 bits in the Rademacher
and optimizing the efficiency of video image structuring system.
during the processes of creation, encoding, transformation, The colors of the Hamming distance pixels on a monitor,
crypto protection, transmission, archiving and access given in Cartesian coordinates, can be coded in the Residue
receiving to color images as well as their use. Number System (K). This is implemented by introducing
The examples of setting and successful solving the three relatively simple modules ( P1 , P2 , P3 ), which allow
problems referring to this issue on the basis of the encoding each pixel of the RGB system in the binary system
mathematical foundations development, the implementation by forward integer transform of the residue number system
of the algorithms and hardware and software tools for image (RNS) according to the expression [6]:
processing and recognition were thoroughly highlighted in 3
the works of scientific researches [1-5]. N k = res ∑ b ⋅ B (mod P )
i =1
i i 0 (1)
Considerable attention is paid to solving research
problems in this field and creating algorithms of the image where B i - the orthogonal bases of RNS, which are
structural properties and features. calculated according to diophantine equations:
II. METHODS OF MULTIFUNCTIONAL B1 = P2 ⋅ P3 ⋅ m1 ≡ 1(mod P1 ) ; (2)
STRUCTURING OF COLOR IMAGE PIXELS IN THE B2 = P1 ⋅ P3 ⋅ m2 ≡ 1(mod P2 ) ; (3)
SYSTEM OF EXTENDED GALOIS FIELDS B3 = P1 ⋅ P2 ⋅ m3 ≡ 1(mod P3 ) , (4)
The analysis of the mathematical foundations of the where m1 , m 2 , m3 - inverse elements of the RNS [8];
existing algorithms for color image processing and
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P0 = P1 ⋅ P2 ⋅ P3 - color image pixel encoding range with IV. THE METHOD FOR COLOR IMAGE PIXELS
color depth K = Eˆ [log P ] , Eˆ [•] - integer function with
0 2 0
ENCODING IN THE RADEMACHER-KRESTENSON AND
rounding to a larger integer. THE HAAR-KRESTENSON SYSTEMS
RGB pixels encoding in the Rademacher-Krestenson The encoding of color image pixels according to the RGB
system is provided by selecting the following values of the color model is carried out by the 24-bit binary code, when the
encoding range of bi remainders in the Rademacher system: intensity of each of the colors is represented by the 8-bit
b1 = bR ; 0 ≤ bR ≤ 255 ; ( 00000000 ÷ 11111111 ); binary code of the Rademacher System:
b2 = bG ; 0 ≤ bG ≤ 255 ; ( 00000000 ÷ 11111111 ); r8−1 g 8−1 b8−1
b3 = bB ; 0 ≤ bB ≤ 255 ; ( 00000000 ÷ 11111111 ). ... ... ...
In addition, taking into account the coefficients m = 1.0 , R ri ; Gg i ; B bi
n = 4.5907 , p = 0.0601 , in order to achieve the most ... ... ...
r0 g 0 b0
saturated green color, the range of its change can be set as
0 ≤ bG ≤ 254 that provides relevant simplicity of the
0 ≤ ri ≤ 255 ; 0 ≤ g i ≤ 255 0 ≤ bi ≤ 255 .
following modules: P1 = 256 , P2 = 255 , P3 = 257 . Encoding of the color image RGB pixels in the
To verify the relevant simplicity of the selected modules Rademacher-Krestenson (RK) and Haar-Krestenson (HK)
system, they are factorized into Systems is carried out by selecting relatively simple modules
multipliers: 256 = 2 , 255 = 5 * 51 , 257 - a prime number,
8
system ( P1 , P2 , P3 ), whose product exceeds the range of
i.e. P0 = 16776960 , where P0 < 2 24 = 16777216 . That is, the quantization of the brightness values ( ri , g i , bi ).
condition for creating a 24-bit pixel code in the Rademacher- Such a condition can be satisfied by a different set of the
Krestenson System is satisfied. RNS discrete transformer modules, for
In binary system module codes are represented as: example, P1 = 5, P2 = 7, P3 = 8 , which provide encoding of ri ,
P1 = 100000000 ( 2) , P2 = 11111111( 2) , P3 = 100000001( 2) .
g i and bi brightness in P0 = 5 * 7 * 8 = 280 > 255 range. The
Then: P0 = 111111111111111100000001( 2) .
following code structure is created in the R-K System, which
As a module P1 = 28 is among the modules P1 , P2 , P3 , unambiguously represents the corresponding RGB-pixel
then, according to the inverse RNS transform, the remainder code:
of N k (G – color features) will be presented without a 2 c 2 d 2
decoding it by eight low orders of N k , which is in the R ∨ G ∨ B a1 ; c1 d 1
a c d
Rademacher system. 0 0 0
According to the Diophantine equations solution (2-4), the P1 = 5 P2 = 7 ; P3 = 8 ,
following values of the inverse elements mi and basic
where ai ∈ 0,1 ; ci ∈ 0,1 ; d i ∈ 0,1 ; i ∈ 0,2 .
numbers Bi are received:
In this case, each value ai , ci , d i is calculated as the
m1 = 255 , B1 = 16711425 ; m 2 = 128 , B 2 = 8421376 ;
remainder according to the expressions: ai = res(ri mod P1 ) ;
m3 = 129 , B3 = 8421120
ci = res( g i mod P2 ) , d i = res(bi mod P3 ) .
The verification of the calculation accuracy of the RNS
transform is performed according to the equation: For a given set of modules, the inverse elements mi and
N k = (bR ⋅ B1 + bG ⋅ B2 + bB ⋅ B3 ) ⋅ (mod P0 ) = 1 when the basic numbers Bi are determined according to the
bR = 1 , bG = 1 , bB = 1 . Diophantine equations solutions (2-4):
That is, m1 = 1 , B1 = 56 , m2 = 3 , B2 = 120 , m3 = 3 ,
N k = (1 ⋅16711425 + 1 ⋅ 8421376 + 1 ⋅ 8421120) ⋅ (mod P0 ) = 1 . B3 = 105 .
For example, R = 10 , G = 200 , B = 100 . Accuracy of the obtained mi and Bi values is verified
Then
according to the expression (1):
N k = (10 ⋅16711425 + 200 ⋅ 8421376 + 100 ⋅ 8421120) ⋅ N1 = (1 ⋅ 56 + 1 ⋅ 120 + 1 ⋅ 105) mod 280 = 1
,
⋅ (mod 16776960) = 9187850 For example, the following values of color intensity of the
which corresponds to the binary representation of the RGB-pixel are set as: ri = 10 , g i = 100 , bi = 37 .
RGB pixel in the Krestenson System Then, RGB-pixel codes are received in the Rademacher
(100011000011001000001010 2 ). System:
Decoding of such representation is as follows:
ri = 00001010 ( 2) ; g i = 01100100 ( 2) ; bi = 00100101( 2) ;
ri = resN k (mod P1 ) ; g i = resN k (mod P2 ) ;
in the Rademacher-Krestenson system:
bi = resN k (mod P3 ) .
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112
P1 P2 P3
P1 P2 P3
The representation of ri , g i and bi color brightness
ri = (000011101) (5,7,8) ; g i = (000010010) (5,7,8) ;
digital values in different systems leads, correspondingly, to
P1 P2 P3 different code length according to the expressions:
bi = (010010101) (5,7,8) . 1. K R = log 2 2 8 = 8 bits in the Rademacher System (R).
Representation of the RGB pixel code for each ri , g i and 3
bi intensity value in the Haar-Krestenson System is made
2. K R −C = ∑ [ Eˆ (log P − 1)] = 3 + 3 + 3 = 9 bits in
i −1
2 i
according to the structure: the Rademacher-Krestenson system (R-K).
a P1 −1 c P2 −1 d P3 −1 n
....
...
...
3. K H −C = ∑ P = 5 + 7 + 8 = 20 bits in the Haar-
i
i =1
R ∨ G ∨ B a i ; i
c d i Krestenson System (H-K).
.... ... ...
V. STRUCTURE DEVELOPMENT AND
a 0 c 0 d 0
EXPERIMENTAL STUDIES OF STRUCTURAL, TIME
P1 = 5 P2 = 7 ; P3 = 8 , AND HARDWARE COMPLEXITY OF ADC WITH THE
where i ∈ 0, Pi − 1 R AND H-K OUTPUT CODES.
For the specified color intensity values of the RGB pixel It is expedient to make multifunctional encoding of RGB
ri = 10 , g i = 100 , bi = 37 , the following code structure in pixels in the R-K and H-K systems at the level of analog-to-
the H-K system is obtained: digital conversion of the analog signals intensity of the RGB
ri = (10000..0001000..00000100) ; sensors. Such a principle of multifunctional data structuring
in color formation is implemented by parallel ADC, the
g i = (10000..0010000..00100000) ;
structure of which is shown in Fig 1.
bi = (00100..0010000..00000100) .
Fig.1. The structure of a multi-purpose parallel ADC with output codes in the Haar-Krestenson System.
ADC consists of 1 – input analogue bus; 2 – paraphase τ ADC 2 = τ k 2 + τ LE2 + τ LE3 ,
comparators; 3 – input reference bus; 4– exemplary
resistors; 5 – the first logic elements "AND-NOT"; 6 – the where τ k 2 = 2υ - switching time for paraphase
second logic elements "AND-NOT", 7 – output ADC bus. comparator;
ADC efficiency is determined according to the τ LE2 = 1υ - switching time for two-input logic element
expression:
"AND-NOT";
ACIT 2018, June 1-3, 2018, Ceske Budejovice, Czech Republic
113
τ LE3 = 1υ - switching time for multi-input logic substantiated. This allows to increase the efficiency of
algorithms for digital image transform, processing and
element (LE) "AND-NOT";
recognition on the basis of the mathematics of arithmetic
That is, the efficiency of ADC is determined by the total
operations of the non-positional Residue Number System.
delay of signals:
The analysis of the mathematical foundations of
τ ADC2 = (2 + 1 + 1)υ = 4 micro cycles. existing algorithms for color image processing and
When calculating the time complexity of the ADC recognition was carried out by segmentation methods on
components, it is taken into account that the switching the basis of histogram thresholding and cumulative
time of the paraphase comparator is 2.5 times less in histograms, statistic estimates of the mean value,
comparison with the single-phase comparator due to dispersion, asymmetry and the degree of contrast of the
positive trigger feedback between the direct and inverse intensity of histograms. This is exemple homogeneity
outputs. taking into account the dispersion of pixels coordinates of
image fragments and silhouettes, as well as image
VI. THE METHOD OF CRYPTO PROTECTION OF clustering methods.
COLOR IMAGE RGB PIXELS. It is proposed to carry out structured encoding of color
Crypto protection of the RGB image pixels is image pixels by the codes of non-positional number
performed in order to restrict unauthorized access to color systems of R-K, H-K and G. This allows to increase the
images that are generated in real time. It's encoded in efficiency of algorithms for image processing by 2-3
different number systems, transmitted via communication orders.
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ACIT 2018, June 1-3, 2018, Ceske Budejovice, Czech Republic