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Nowadays, data scrambling remains a vital technique to protect sensitive information by shuffling it in a way that makes it difficult to decipher or reverse-engineer while still maintaining its usability for legitimate purposes. As manipulating the usability of the scrambled data remains a challenge...
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| author | Romanuke, Vadim Yaremko, Svitlana Kuzmina, Olena Yehoshyna, Hanna |
| author_facet | Romanuke, Vadim Yaremko, Svitlana Kuzmina, Olena Yehoshyna, Hanna |
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| description | Nowadays, data scrambling remains a vital technique to protect sensitive information by shuffling it in a way that makes it difficult to decipher or reverse-engineer while still maintaining its usability for legitimate purposes. As manipulating the usability of the scrambled data remains a challenge on the background of risking losing data and getting them re-identified by attackers, scrambling and descrambling should be accomplished faster by not increasing data loss and re-identification risks. A scrambling algorithm must have a linear time complexity, still shuffling the data to minimize the risks further. A promising approach is based on the knight open tour problem, whose solutions appear like a random series of knight positions. Hence, a knight open tour algorithm is formalized, by which the knight seems to move chaotically across the chessboard. The formalization is presented as an indented pseudocode to implement it efficiently, whichever programming language is used. The output is a square matrix representing the knight open tour. Based on the knight tour matrix, data scrambler and descrambler algorithms are presented in the same manner. The algorithms have a linear time complexity. The knight-tour scrambling has a sufficiently low guess probability if an appropriate depth of scrambling is used, where the data is re-scrambled repetitively. The scrambling depth is determined by repetitive application of the chessboard matrix, whose size usually increases as the scrambling is deepened. Compared to the pseudorandom shuffling of the data along with storing the shuffled indices, the knight-tour descrambling key is stored and sent far simpler yet ensures proper data security. |
| doi_str_mv | 10.20535/SRIT.2308-8893.2024.3.03 |
| first_indexed | 2025-07-17T10:28:36Z |
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Publisher IASA at the Igor Sikorsky Kyiv Polytechnic Institute, 2024
44 ISSN 1681–6048 System Research & Information Technologies, 2024, № 3
UDC 004.056.55+004.421.5
DOI: 10.20535/SRIT.2308-8893.2024.3.03
DATA SCRAMBLER KNIGHT TOUR ALGORITHM
V.V. ROMANUKE, S.A. YAREMKO, O.M. KUZMINA, H.A. YEHOSHYNA
Abstract. Nowadays, data scrambling remains a vital technique to protect sensitive
information by shuffling it in a way that makes it difficult to decipher or reverse-
engineer while still maintaining its usability for legitimate purposes. As manipulat-
ing the usability of the scrambled data remains a challenge on the background of
risking losing data and getting them re-identified by attackers, scrambling and de-
scrambling should be accomplished faster by not increasing data loss and re-
identification risks. A scrambling algorithm must have a linear time complexity, still
shuffling the data to minimize the risks further. A promising approach is based on
the knight open tour problem, whose solutions appear like a random series of knight
positions. Hence, a knight open tour algorithm is formalized, by which the knight
seems to move chaotically across the chessboard. The formalization is presented as
an indented pseudocode to implement it efficiently, whichever programming lan-
guage is used. The output is a square matrix representing the knight open tour.
Based on the knight tour matrix, data scrambler and descrambler algorithms are pre-
sented in the same manner. The algorithms have a linear time complexity. The
knight-tour scrambling has a sufficiently low guess probability if an appropriate
depth of scrambling is used, where the data is re-scrambled repetitively. The scram-
bling depth is determined by repetitive application of the chessboard matrix, whose
size usually increases as the scrambling is deepened. Compared to the pseudoran-
dom shuffling of the data along with storing the shuffled indices, the knight-tour de-
scrambling key is stored and sent far simpler yet ensures proper data security.
Keywords: data scrambling, knight open tour problem, linear time complexity,
guess probability, scrambling depth.
INTRODUCTION
Data scrambling, also known as data obfuscation or data anonymization, is a
technique used to protect sensitive information by altering or shuffling it in a way
that makes it difficult to decipher or reverse-engineer while still maintaining its
usability for legitimate purposes [1; 2]. Data scrambling is an essential component
of data protection strategies, helping organizations safeguard sensitive informa-
tion while still benefiting from its utility [3; 4]. Scrambling is similar to encryp-
tion and ciphering, but these techniques differ in their fundamental approaches
and purposes [5; 6].
Scrambling is a data protection technique that aims to balance data privacy
with usability by partially obscuring data, making it reversible in most cases [7].
Encryption, on the other hand, focuses on data confidentiality by converting it
into ciphertext, which is typically not usable without decryption [8]. Ciphering is
a broader term that encompasses both scrambling and encryption, as it refers to
the process of transforming data to protect its confidentiality or privacy [9]. While
scrambling may involve various transformations, such as shuffling, substitution,
or masking, to make the data less readable [10], encryption is primarily used to
Data scrambler knight tour algorithm
Системні дослідження та інформаційні технології, 2024, № 3 45
secure data by converting it into a ciphertext using cryptographic algorithms
[8; 11; 12]. Encrypted data is typically not usable or meaningful without the
corresponding decryption key, as it appears as random ciphertext [5; 6; 9; 11].
The choice between scrambling and encryption depends on the specific use
case and requirements regarding data protection and usability. The main purposes
of data scrambling are data privacy and compliance. It protects, for example, per-
sonal identification data, financial records, proprietary business data, from unau-
thorized access or disclosure. Data scrambling helps organizations comply with
data protection regulations and privacy laws, such as GDPR or HIPAA [13; 14],
which require the safeguarding of sensitive data.
One common method of data scrambling is shuffling data using an appropri-
ate algorithm [15]. Knowing this particular algorithm allows descrambling the
scrambled data. Other methods are masking that replaces parts of sensitive data
with placeholders or pseudonyms, tokenization that replaces sensitive data with
tokens or references, which are meaningless without the associated mapping, and
data perturbation [16; 17]. The latter is a technique that adds random noise or pertur-
bation to numerical data to protect its privacy while preserving statistical properties.
Data scrambling is successfully used in secure storage, where data at rest is
protected from unauthorized access in databases, file systems, or backups. Data
sharing is another use case, where organizations share data with third parties for
analysis or collaboration without revealing sensitive details. In addition, scram-
bled data can be used in non-production and test environments to simulate real
data without exposing sensitive information [1; 2; 4; 7; 10].
The development of data scrambling includes understanding data sensitivity
priority, strengthening encryption, selecting appropriate anonymization tech-
niques based on the specific data and use case, and continuously monitoring and
auditing data scrambling processes to ensure their effectiveness and compliance.
However, determining a balance between data protection and data usability is
challenging [18]. The other two main challenges are data loss and re-
identification risks [19; 20]. Thus, improper implementation of data scrambling
techniques can lead to data loss or degradation of data quality. Besides, which is
the most important and where ones must be the most cautious, attackers may still
re-identify individuals or sensitive data if not properly scrambled [21].
While scrambling is frequently used for images due to its visual nature, it is
not limited to images and can be applied effectively to text and numerical data as
well to protect sensitive information while maintaining data usability [22]. This is
often seen in scenarios like redacting personally identifiable information in documents
or anonymizing user-generated content in social media moderation [8; 9; 13; 23], aca-
demic performance [24], and recommender system profiles [25]. Numerical data,
such as financial records, health records, or scientific research data, may also re-
quire protection through scrambling techniques [26; 27]. This is essential for com-
pliance with data privacy regulations like GDPR or HIPAA [13; 14].
PROBLEM STATEMENT
As manipulating the usability of the scrambled data remains a challenge on the
background of risking to lose data and get them re-identified by attackers, scram-
bling and descrambling should be accomplished faster by not increasing data loss
V.V. Romanuke, S.A. Yaremko, O.M. Kuzmina, H.A. Yehoshyna
ISSN 1681–6048 System Research & Information Technologies, 2024, № 3 46
and re-identification risks. A scrambling algorithm must be of a linear time com-
plexity still shuffling the data so that to further minimize the risks. A promising
approach is based on the knight open tour problem [28; 29] whose solutions ap-
pear like a random series of knight positions. The goal of the research is to apply
this property of the solutions to data scrambling. For achieving the goal, the fol-
lowing five tasks are to be fulfilled:
1. To formalize a knight open tour algorithm, by which the knight is seemed
to move chaotically across the chessboard. The formalization is to be presented as
an indented pseudocode to efficiently implement it, whichever programming lan-
guage is used. The output is a square matrix representing the knight open tour.
2. To algorithmize a data scrambler and descrambler based on the knight
tour matrix. Both the algorithms must be given as indented pseudocodes.
3. To estimate the time complexity of the algorithms. In addition, to com-
pare their performance to other approach of scrambling by shuffling the data, in-
cluding the probability of illegitimately descrambling by attackers.
4. To discuss the significance and practical applicability of the suggested
knight open tour algorithm. The proper contribution to the field of data scram-
bling should be emphasized.
5. To conclude on the suggestion and findings along with mentioning a pos-
sibility to extend and advance the research.
KNIGHT TOUR MATRIX
The directions the knight can move on the chessboard are completely described
by the horizontal and vertical sets
}2,1,1,2,2,1,1,2{}{ )hor(
hor lsS (1)
and
}1,2,2,1,1,2,2,1{}{ )vert(
vert lsS . (2)
Sets (1) and (2) are such that
)hor(
4
)hor(
ll ss 4,1 l ,
and
)vert(
4
)vert(
ll ss 4,1l .
If the knight starts its open tour at horizontal position x and vertical position y
on a chessboard of size MM , the knight open tour algorithm finds a sequence
of the remaining 12 M chessboard positions that constitute the tour. The se-
quence is written by the set of integers from 1 to 2M , where 1 corresponds to the
starting position of the knight. These integers are put on the chessboard, forming
thus an MM matrix
MMrtM byxMK ][),,(B , (3)
where },1{ 2Mbrt and ),,( yxMK is the algorithm mapping the size of the
chessboard and the knight starting position into matrix MB by (3).
While the knight tour problem is NP-hard in general [28], there is a number
of heuristic algorithms that allow finding a solution in linear time — that is, in a
Data scrambler knight tour algorithm
Системні дослідження та інформаційні технології, 2024, № 3 47
time amount proportional to number 2M . One of such heuristics is the Warns-
dorff’s rule [30; 31]. This rule finds a single solution. According to the Warns-
dorff’s rule, the knight is moved so that it always proceeds to the position from
which the knight will have the fewest onward moves. The priority queue of avail-
able neighbors is stored as a set
W
www qqQ 121 ]}[{ , }0{N W . (4)
The algorithm is presented as an indented pseudocode (Algorithm 1), where set
(4) dynamically changes its size inside the outer loop.
Algorithm 1 . The Warnsdorff’s rule indented pseudocode
for 1k with step 1 to 2Mk do
kbyx , Q
for 1l with step 1 to Ml do
)hor(
hor lsxm , )vert(
vert lsym
if 1horm and 1hor Mm and 1vertm and 1vert Mm
if 0
horvert
mmb
0c
for 1u with step 1 to Mu do
)hor(
horhor usmg , )vert(
vertvert usmg
if 1horg and 1hor Mg and 1vertg and
1vert Mg
if 0
horvert
ggb then cc (obs) ,
1(obs) cc
if Q then ]}{[ lcQ
else
if cqw 1 then QQ )obs( , }],[{ )obs(QlcQ
else
if cqw 1
if lqw 2 then QQ )obs( ,
}],[{ )obs(QlcQ
else QQ )obs( ,
}][,{ )obs( lcQQ
if cqw 1 then QQ )obs( ,
}][,{ )obs( lcQQ
if Q then 12
* ql , xx )obs( , )hor()obs(
*l
sxx , yy )obs( ,
)vert()obs(
*l
syy
else break
The knight open tour resulting from Algorithm 1 mostly seems to be rather
chaotic. An example of the tour for 1616 chessboard is shown in Fig. 1, where
the entries of the knight tour matrix 16B are also given (odd and even numbers
V.V. Romanuke, S.A. Yaremko, O.M. Kuzmina, H.A. Yehoshyna
ISSN 1681–6048 System Research & Information Technologies, 2024, № 3 48
differ in their color). Some traceries are still noticeable, though. Especially near
the margins — where the knight has fewer possibilities to move onward.
When we have two-dimensional data of N points in each dimension, it can
be formally presented as a square matrix
NNijd ][D , (5)
where ijd is either real or complex number. In particular, if NM , the data in
matrix (5) can be scrambled using the pattern of the knight tour matrix (3) by just
shuffling the entries of D in accordance with MB . When NM , we can shuffle
not an entry but an aa square of 2a entries, where
M
N
a and it obviously
must be integer. This is the core of the algorithm for data scrambling and de-
scrambling based on the knight tour matrix (3).
DATA SCRAMBLER AND DESCRAMBLER
A data scrambler is an operator that maps data matrix (5) using the knight tour
matrix (3) into an NN matrix
NNijM hF ][),( BDH . (6)
Fig. 1. A knight open tour built by the Warnsdorff’s rule algorithm for 16 16 chessboard
Data scrambler knight tour algorithm
Системні дослідження та інформаційні технології, 2024, № 3 49
Operator ),( MF BD in (6) is realized by a data scrambler algorithm (Algorithm 2),
which, using matrices D and MB , dynamically builds an 12M array
1
)square(
2
)square(
1square 2][
Mkk qqQ (7)
containing coordinates of successive positions to shuffle.
Algorithm 2 . A data scrambler by (6) using the knight open tour matrix (3)
for 1l with step 1 to Ml do
for 1u with step 1 to Mu do
uMlk )1( , lqk )square(
1 , uqk )square(
2
for 1l with step 1 to Ml do
for 1u with step 1 to Mu do
lubk , )square(
1scr kql , )square(
2scr kqu
****** jiji
dh for mali )1(* , mauj )1(* , am ,1
mali )1( scr
** , mauj )1( scr
**
It is worth noting that the data can be a three-dimensional matrix as well.
Then the data scrambler algorithm processes the third-dimension layers in paral-
lel, using, for instance, the data vectorization approach [32]. Thus, the color im-
age in Fig. 2 is scrambled by using an ordinary chessboard matrix 8B with its
Fig. 2. The image of data of a 4096 4096 3 array (the image is taken from [33])
V.V. Romanuke, S.A. Yaremko, O.M. Kuzmina, H.A. Yehoshyna
ISSN 1681–6048 System Research & Information Technologies, 2024, № 3 50
starting position }6,7{ ( 6x , 7y ). The scrambled image is shown in Fig. 3.
In this scientific-data-like example, the scrambling result may not seem that con-
vincing, but the mosaic in Fig. 3 is nonetheless pretty hard to assemble it back to
the original image in Fig. 2. Nevertheless, many mosaic squares contain parts of
consistent information. This is due to every mosaic square is a 512 512 color
image.
To decrease the data readability, the size of the chessboard should be in-
creased. On the other hand, the scrambler can be applied once again to the scram-
bled data. Thus, Fig. 4 presents a result of scrambling the scrambled image in
Fig. 3 by using matrix 16B with its starting position }12,7{ ( 12x , 7y ). Now
the readability of the image parts is lower, although many parts and their local
information are still distinguishable. However, assembling the mosaic in Fig. 4
back to the original image is much more difficult compared to the mosaic in Fig. 3.
Indeed, the mapping of data (5) by (6) and (3) is a very simple approach
whose guess probability is 2M . To go deeper in scrambling, consider its depth
, where }0{N , and an additional sequence of integers
1}{M , where
usually
1 MMM by 1,1 . (8)
Fig. 3. The scrambled image in Fig. 2 by using 8B with its starting position 7, 6
Data scrambler knight tour algorithm
Системні дослідження та інформаційні технології, 2024, № 3 51
Then the scrambling of depth is recursively fulfilled as
),(1 MF BHH for ,1 (9)
by HH 1 . The case 0 is the simplest possible scrambling here. Overall,
there are 1 transformations by operator F . They result in a set 1
1}{
H of
scrambled data matrices. The last matrix in this set, 1H , is the final result of
the data scrambling of depth . The guess probability herein becomes equal to
1
1
22
MM . (10)
So, the image in Fig. 4 is the scrambling result of depth 1, and its guess probabil-
ity is
61035156250000.02)168()( 1412212
1
2 MM .
Going deeper by using matrix 32B with its starting position }21,6{ ( 21x ,
6y ) further strengthens the encryption (Fig. 5) decreasing the guess probability
by 1024 times.
Fig. 4. The scrambled image in Fig. 3 by using 16B with its starting position }12,7{
V.V. Romanuke, S.A. Yaremko, O.M. Kuzmina, H.A. Yehoshyna
ISSN 1681–6048 System Research & Information Technologies, 2024, № 3 52
Herein, while obeying inequality (8), conditions
MM 21 by 1,1 and MM 21 (11)
are followed. This is convenient to establish the guess probability such as it is de-
sired to be. Scrambling to depth 3 (Fig. 6), where matrix 64B has its starting posi-
tion }46,3{ ( 46x , 3y ), makes the guess practically impossible within a rea-
sonable amount of time (unless the latest advanced computational techniques are
applied, like, e. g., quantum computing):
361222212
3
2
2
2
1
2 2)6432168()( MMMM
11105.1
66871947673
1 .
Besides, while the usability is maintained, the data in Fig. 6 is not readable even
partially — every mosaic square is just a 6464 color image.
Fig. 7 shows that at depth 4 for this particular example the scrambled image
is perceived as noise. Every mosaic square is just a 3232 color image. The
guess probability is 16384 times lower than that for the scrambled image in Fig. 6:
5012222212
4
2
3
2
2
2
1
2 2)1286432168()( MMMMM
16109.8
8426241125899906
1 .
Fig. 5. The scrambled image in Fig. 4 (depth 2) by using 32B with its starting position
}21,6{ , where the guess probability is 062504644775390000000596.02 24
Data scrambler knight tour algorithm
Системні дослідження та інформаційні технології, 2024, № 3 53
The data readability is nearly at naught. Therefore, Fig. 7 is the final result of
scrambling the data in Fig. 2.
Fig. 6. The scrambled image in Fig. 5 (depth 3) by using 64B with its starting position
}46,3{ , where the guess probability is 362
Fig. 7. The scrambled image in Fig. 6 (depth 4) by using 128B with its starting position
6, 36 , where the guess probability is 502
V.V. Romanuke, S.A. Yaremko, O.M. Kuzmina, H.A. Yehoshyna
ISSN 1681–6048 System Research & Information Technologies, 2024, № 3 54
A data descrambler is an operator that maps scrambled data matrix (6) using
the knight tour matrix (3) back into matrix (5):
),( MG BHD . (12)
Operator ),( MG BH in (12) is realized by a data descrambler algorithm (Algo-
rithm 3), which, using matrices H and MB , dynamically builds array (7).
Algorithm 3. A data descrambler by (7) using the knight open tour matrix (3)
for 1l with step 1 to l M do
for 1u with step 1 to u M do
1k l M u , (square)
1kq l , (square)
2kq u
for 1l with step 1 to l M do
for 1u with step 1 to u M do
luk b , (square)
scr 1kl q , (square)
scr 2ku q
** ** * *i j i j
d h for * 1i l a m , * 1j u a m , 1,m a
**
scr 1i l a m , **
scr 1j u a m
The descrambling of depth is recursively fulfilled as
),(
121 MG BHH for ,1 . (13)
Obviously, there are 1 transformations by operator G restoring the
backward set of scrambled data matrices
11}{H . The last matrix in this set,
HH 1 , is put into operator (12) and then the final result of the data descram-
bling is obtained.
Henceforth, the data scrambler by (6) and (9) requires knowing 1 chess-
board matrices
}}{,{ 1
MM BB (14)
whose sizes obey inequality (8) or, in particular, the doubling by (11). This en-
sures the guess probability equal to (10) or, if the doubling by (11) is used, the
guess probability equal to
1
2
2
1
22
1
2
2
1
22 44)2(2 MMMMMM
1
)1(214
M . (15)
Each of matrices (14) is defined by its size and the starting position of the knight.
Therefore, the data scrambler is defined by the sizes and starting positions
}}{,{ 1
MM and }},{},,{{ 1
xyxy , (16)
respectively. Consequently, the data descrambler by (12) and (13) is defined by
sets (16). Knowing (16) allows descrambling data in matrix 1H without any
data losses.
Data scrambler knight tour algorithm
Системні дослідження та інформаційні технології, 2024, № 3 55
KNIGHT-TOUR SCRAMBLING VERSUS SHUFFLER SCRAMBLING
It is obvious that, despite a linear time complexity, too deep scrambling may take
significant amounts of time. However, the deep scrambling is really needful only
for decreasing the guess probability as much as possible. In other situations, the
data scrambler can be applied just once to produce a scrambled data matrix of
sufficiently low readability. For example, the data image in Fig. 2 scrambled with
a matrix 128B looks blurred (Fig. 8) and its scientific data is not readable almost
much as the image in Fig. 7, although the guess probability is just 142 (rela-
tively, it is pretty high).
Fig. 8. The scrambled image in Fig. 2 by using the chessboard matrix of size 128 (the zero
depth of scrambling, 0 ), where the guess probability is the same as for the scrambled
image in Fig. 4 with the scrambling depth 1
Could the same effect be when, instead of the data scrambler knight tour al-
gorithm, a more random and less sophisticated approach is used? For instance,
such an approach is the pseudorandom shuffling of the data along with storing the
shuffled indices. The shuffler scrambling is defined just by an NN matrix
NNij ][S (17)
of randomly permutated unique 2N integers from 1 to 2N . Then the shuffler
scrambling and descrambling algorithms are those Algorithms 2 and 3, respec-
tively, where only 1a and matrix (17) is used instead of matrix (3). The guess
probability in this case is 2N as the shuffler algorithm is presumed to be known
and it can be defined by the position of an integer value from 1 to 2N .
Obviously, if the knight-tour scrambling is of depth 0, then the shuffler
scrambling results in a lower guess probability unless M N :
V.V. Romanuke, S.A. Yaremko, O.M. Kuzmina, H.A. Yehoshyna
ISSN 1681–6048 System Research & Information Technologies, 2024, № 3 56
2 2M N for M N .
However, as the scrambling is developed deeper, the guess probability may sig-
nificantly drop, as it has been demonstrated above by Figs. 4–7.
Consider a range of the data set size as follows:
}256,128,64,32,16,8{}2,2,2,2,2,2{ 876543 N . (18)
Then data matrix (5) with integers
Z ]255;0[ijd
is re-generated for 100 repetitions for each of the six values of N in (18). The
depth of scrambling is
3log2 N .
So, the knight-tour data scrambler is applied as deep as possible, i. e.
},}2{,8{}}2{,8{}}{,{ 1
1
3
1
3
1 NMM
.
That is, the knight tour matrix NM BB
, whichever N is. This is expectedly
the worst-case scenario with respect to the computation time of the knight-tour
data scrambler compared to the shuffler. Table 1 presents statistics after averaging
over the 100 repetitions of the knight-tour data scrambler versus shuffler. The
similarity index ratio is calculated as the ratio of the averaged similarity rate by
the knight-tour data scrambler to the averaged similarity rate by the shuffler,
where the averaged similarity rate is calculated as the element-wise number of
coincidences in matrices D and the scrambled data matrix divided by 2N . The
knight-tour-scrambling guess probability herein is calculated by (15). The statis-
tics remain almost the same for any other 100-repetition generations. Although
the knight-tour-to-shuffler time ratios reveal some favorability of the shuffler
computation time, the shuffler is a far less reliable scrambler due to its guess
probability is too high compared to the knight-tour-scrambling guess probability.
Besides, the knight-tour-to-shuffler similarity index ratio confirms that the knight-
tour data scrambler produces a scrambled object which resembles the original less
than a scrambled object by the shuffler resembles it.
T a b l e 1 . Knight-tour scrambling versus shuffler scrambling
N
Knight-tour-
to-shuffler
scrambling
time ratio
Knight-tour-
to-shuffler
descrambling
time ratio
Similarity
index
ratio
Knight-tour-scrambling
guess probability
Shuffler-scrambling
guess probability
8 0.5797 1.1134 0.2308 2 68 2 0.015625 0.015625
16 1.3375 1.3913 1.0055
1422 2168
510103515625.6
0.00390625
32 1.3805 1.416 0.7135 222 32168
824 1062
4109.765625
64 1.3392 1.3928 0.9337 2222 6432168
1136 105.12
4102.44140625
128 1.2306 1.2232 0.995 1650 109.82 51056.10351562
256 1.1258 1.1157 1.0079 2066 104.12 5106251.52587890
Data scrambler knight tour algorithm
Системні дослідження та інформаційні технології, 2024, № 3 57
Consider another series of 100 simulations for the knight-tour data scram-
bler. Let
}1024,512,256,128,64,32,16,8{}2,2,2,2,2,2,2,2{ 109876543 N (19)
and the remaining simulation parameters be the same. Let the depth be varied
from 0 to 5. Then, for each repetition, there is an instance of some size in (19) and
}5,0{ . The averaged interrelation between scrambling and descrambling
computation times is shown in Table 2. The interrelation varies within 20 % on
average for any other 100-repetition generations. The knight-tour data descram-
bler seemingly operates faster owing to the memory allocation and code compila-
tion that are done following the scrambling.
T a b l e 2 . Knight-tour scrambling-to-descrambling time ratio
Depth of scrambling N
0 1 2 3 4 5
8 0.9248
16 1.0638 1.0302
32 1.0756 1.0231 1.056
64 1.0765 1.0578 1.039 1.0206
128 1.151 1.179 1.1399 1.0888 1.0859
256 1.1518 1.361 1.3882 1.3129 1.102 1.0409
512 1.2119 1.5472 1.6715 1.732 1.42 1.1119
1024 1.1883 1.6135 1.8532 2.0819 1.9851 1.3881
Obviously, as the size of the data matrix increases, the computation time
grows. Table 3 shows the computation time relative growth with respect to the
computation time taken to scramble with a chessboard matrix of size N . At the
first glance, there is a quadratic time complexity with respect to size N along
each column, but the real size of the algorithm input is 2N , so Table 3 confirms a
linear time complexity of Algorithm 2. So does Table 4 showing the computation
time relative growth with respect to the computation time taken to descramble
with a chessboard matrix of size N . However, it is worth noting that Algorithm 3
being of a linear time complexity seems to be faster (this is clearly seen when Ta-
ble 4 is cell-by-cell compared to Table 3). This effect has been already explained
above — the data descrambler by Algorithm 3 virtually utilizes the memory pre-
allocation and code pre-compilation, done for Algorithm 2 in this case.
T a b l e 3 . Knight-tour scrambling time relative growth as N increases
Depth of scrambling N
0 1 2 3 4 5
8 1
16 0.5562 1
32 0.4834 1.0351 1
64 0.5661 1.1575 1.1084 1
128 1.0355 1.832 1.5465 1.2523 1
256 2.7697 4.2452 2.86 1.8061 1.1855 1
512 9.1934 12.8017 7.7231 3.8901 1.8098 1.1376
1024 35.2239 42.9207 23.9562 11.245 4.1936 1.5951
V.V. Romanuke, S.A. Yaremko, O.M. Kuzmina, H.A. Yehoshyna
ISSN 1681–6048 System Research & Information Technologies, 2024, № 3 58
T a b l e 4 . Knight-tour descrambling time relative growth as N increases
Depth of scrambling N
0 1 2 3 4 5
8 1
16 0.4835 1
32 0.4156 1.0423 1
64 0.4863 1.1273 1.1265 1
128 0.832 1.6007 1.4327 1.174 1
256 2.2238 3.2135 2.1756 1.4041 1.1681 1
512 7.0152 8.524 4.8793 2.2924 1.3839 1.0649
1024 27.412 27.4043 13.6509 5.5127 2.2939 1.1961
Another important property is how the computation time grows as the
scrambling depth is increased. Table 5 shows the computation time relative
growth with respect to the scrambling depth. It appears that the data scrambler has
a quadratic time complexity with respect to the scrambling depth. Roughly the
same time complexity is observed for the data descrambler (Table 6). This is par-
ticularly explained with that the chessboard matrix size is increased twice along a
dimension, i. e. it is increased fourfold if to count its entries.
T a b l e 5 . Knight-tour scrambling time relative growth as the depth is increased
Depth of scrambling N
0 1 2 3 4 5
8 1
16 1 3.57
32 1 4.2513 15.4973
64 1 4.0595 14.6675 59.8536
128 1 3.5128 11.1889 40.9824 202.9467
256 1 3.0433 7.7362 22.0973 89.947 782.9858
512 1 2.7648 6.2936 14.3384 41.3688 268.3466
1024 1 2.4194 5.0952 10.8178 25.0193 98.2041
T a b l e 6 . Knight-tour descrambling time relative growth as the depth is increased
Depth of scrambling
N
0 1 2 3 4 5
8 1
16 1 3.6864
32 1 4.4696 15.785
64 1 4.1312 15.1969 63.1311
128 1 3.4293 11.2978 43.3244 215.123
256 1 2.5756 6.4187 19.3856 94.0116 866.4042
512 1 2.1657 4.5632 10.0329 35.3074 292.477
1024 1 1.7818 3.2672 6.1745 14.9772 84.0675
The knight-tour scrambling similarity index showing the unit-normalized
part of the number of coincidences in the data matrix and the scrambled data ma-
trix appears to be quite stable regardless of the data set size and the scrambling
Data scrambler knight tour algorithm
Системні дослідження та інформаційні технології, 2024, № 3 59
depth (Table 7). The scrambling similarity index scarcely exceeds 0.5 %. The av-
erage similarity index varies within 20 % on for any other 100-repetition genera-
tions. A maximum variation is spotted at a 53 % rate.
T a b l e 7 . Knight-tour scrambling similarity index
Depth of scrambling N
0 1 2 3 4 5
8 0.0033
16 0.0037 0.0043
32 0.0041 0.0041 0.0051
64 0.0038 0.0038 0.0048 0.0041
128 0.0039 0.0039 0.0048 0.0041 0.004
256 0.0039 0.0039 0.0048 0.0042 0.0039 0.004
512 0.0039 0.0039 0.0049 0.0041 0.0039 0.0039
1024 0.0039 0.0039 0.0049 0.0041 0.0039 0.0039
The final remark of the knight-tour scrambling versus shuffler scrambling
relates to the descrambling key. While the knight-tour data descrambler must hold
the 1 sizes and starting positions (16), its memory size requirement is very
low. In the double precision format, it is just
)1(24)1(38 bytes,
and it is
)1(12)1(34 bytes
in the single precision format. In practice, nevertheless, values (16) are integers
(strictly speaking, naturals), so they are written with the unsigned 16-bit precision
at most, when they are allowed to vary between 1 and 65535. In this case the data
descrambler key occupies just
)1(6)1(32 bytes.
If it is known beforehand that
256}},{max},,{max,}{max,{max 11
xyxyMM
then the data descrambler key occupies half of the latter by being written with the
unsigned 8-bit precision — it is )1(3 bytes. The shuffler descrambling, on
the other hand, needs to know matrix (17) of unique 2N integers from 1 to 2N ,
which would occupy 28N or 24N bytes if the double or single precision format
was used, respectively. Inasmuch as using the unsigned 8-bit precision is unlikely
here, then either 22N or 24N bytes are occupied depending on whether the un-
signed 16-bit or 32-bit precision is respectively used. Therefore, the memory oc-
cupation by the shuffler descrambling key may seriously affect the data transfer
rate. For instance, if 10241024 images are shuffling-scrambled and transferred,
then an additional amount of 4 megabytes of information should be transferred
along or afterwards for descrambling. This is quite unacceptable due to a
10241024 color image represented in a 8-bit scale itself is 3 megabytes of in-
formation, whereas a 10241024 grayscale image is just 1 megabyte. Contrari-
wise, the knight-tour scrambling at the depth of, say, 5, is accompanied with only
36 bytes ensuring an acceptably low guess probability.
V.V. Romanuke, S.A. Yaremko, O.M. Kuzmina, H.A. Yehoshyna
ISSN 1681–6048 System Research & Information Technologies, 2024, № 3 60
DISCUSSION OF THE CONTRIBUTION
In addition to the knight-tour scrambling advantages, a convention for scrambling
keys can be applied to avoid sending values (16) for descrambling. Such conven-
tions are really plausible and effective when a small group of data chunks is sent
over. Theoretically, a convention for the entries of matrix (17) might be applied,
too. But once the shuffler scrambling key must be unexpectedly changed (e. g., by
the reason of an oncoming cyberthreat), it will be required nonetheless at the de-
scrambling side, unless there is another convention for the case of changing the
key. Anyway, an information signal about the change should be sent. Such incon-
venience makes the shuffler scrambling too bulky and far inefficient.
However, a deeper shuffling still can be applied by the analogy to the
scrambling of some depth. For such a scrambling, matrix (17) may be used re-
peatedly, or a series of such matrices is used. A twofold application of the shuf-
fling decreases the guess probability down to 4N , which becomes acceptable for
images or data sets comprising at least a few hundred (numeric or symbolic) ele-
ments.
The suggested knight open tour algorithm is a significant contribution to the
field of data scrambling by three reasons. First, its implementation by Algorithm
2 for scrambling and Algorithm 3 for descrambling, with Algorithm 1 for generat-
ing a knight open tour, is very simple and easy to understand. Second, the knight-
tour scrambling maintains a linear time complexity. Third, its guess probability is
sufficiently low if an appropriate depth of scrambling is used. Eventually, the data
scrambler knight tour algorithm can be combined with other linear-time-
complexity scrambling algorithms to further strengthen the data protection.
In addition to the simplicity, the data scrambler and descrambler by the
knight open tour algorithm are practically symmetric. Hence, they are easy appli-
cable yet ensuring proper security of data. Despite the knight open tour algorithm
is presented for the square matrix, it is scalable to any size. After all, the chess-
board matrix can be used partially for non-square data matrices. Moreover, the
data can be reshaped into a vector, either row or column, comprising an odd num-
ber of entries, whereupon a part of the chessboard matrix is still used to scramble
the data.
CONCLUSION
A data scrambling algorithm has been suggested based on a knight open tour
problem solution, whose structure seems chaotic enough to substitute ordinary
shuffling. The linear time complexity of the algorithm inspires its practical em-
bedding mainly owing to the scrambling non-sophisticated subtlety and low guess
probability. The latter is regulated by changing the scrambling depth determined
by repetitive application of the chessboard matrix. The size of this matrix is usu-
ally increased as the scrambling is deepened.
The research is possible to extend and advance in the way of studying more
favorable knight open tours for the given chessboard matrix. As the computation
time grows in a quadratic manner by deepening the scrambling, but the guess
probability is lowered far stronger, a tradeoff between the depth and re-
identification risk might be ascertained. Another open question is which strategy
of increasing the chessboard matrix size would be optimal to attain the tradeoff at
the shallowest possible depth.
Data scrambler knight tour algorithm
Системні дослідження та інформаційні технології, 2024, № 3 61
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INFORMATION ON THE ARTICLE
Vadim V. Romanuke, ORCID: 0000-0001-9638-9572, Vinnytsia Institute of Trade and
Economics of State University of Trade and Economics, Ukraine, e-mail:
romanukevadimv@gmail.com
Svitlana A. Yaremko, ORCID: 0000-0002-0605-9324, Vinnytsia Institute of Trade and
Economics of State University of Trade and Economics, Ukraine, e-mail:
s.yaremko@vtei.edu.ua
Olena M. Kuzmina, ORCID: 0000-0002-0061-9933, Vinnytsia Institute of Trade and
Economics of State University of Trade and Economics, Ukraine, e-mail:
o.kuzmina@vtei.edu.ua
Hanna A. Yehoshyna, ORCID: 0000-0002-2381-1231, Northwestern Polytechnic,
Grande Prairie, Canada, e-mail: hyehoshyna@nwpolytech.ca
АЛГОРИТМ ЦИКЛУ ШАХОВОГО КОНЯ ДЛЯ СКРЕМБЛЮВАННЯ ДАНИХ
/ В.В. Романюк, С.А. Яремко, О.М. Кузьміна, Г.А. Єгошина
Анотація. Скремблювання даних у наш час залишається важливою методикою
для захисту конфіденційної інформації за допомогою певного перемішування,
після якого розшифрування є надважким, однак зберігається можливість легі-
тимних дій з даними після скремблювання. Оскільки повноцінне маніпулю-
вання діями з даними після скремблювання залишається проблемою на тлі ри-
зику втрати даних та їх несанкціонованого розшифрування, скремблювання та
дескремблювання мають виконуватись ще швидше, не збільшуючи ризики
втрати та розшифрування. Алгоритм скремблювання повинен мати лінійну ча-
сову складність та ще більше мінімізувати зазначені ризики. Перспективним
підходом є задача побудови відкритого циклу шахового коня, розв’язки якої
виглядають наче випадкові послідовності позицій коня. Відтак формалізується
алгоритм відкритого циклу шахового коня, за яким кінь наче хаотично пересу-
вається по шаховій дошці. Ця формалізація представлена у формі відформато-
ваного псевдокоду для подальшого його ефективного впровадження незалеж-
но від мови програмування. На виході отримано квадратну матрицю, котра
відображає відкритий цикл шахового коня. На основі матриці циклу шахового
коня подано алгоритми скремблера та дескремблера даних у тому ж стилі. Ці
алгоритми мають лінійну часову складність. Скремблювання на основі циклу
шахового коня має досить низьку імовірність випадкової дешифрації за умови,
якщо використана прийнятна глибина скремблювання, де дані повторно скре-
мблюються. Глибина скремблювання визначається багаторазовим застосуван-
ням матриці шахової дошки, розмір якої зазвичай збільшується з поглиблен-
ням скремблювання. Порівняно з псевдовипадковим перемішуванням даних
разом зі збереженням індексів перемішування, ключ дескремблювання на ос-
нові циклу шахового коня зберігається і пересилається набагато простіше, за-
безпечуючи в той же час прийнятний захист даних.
Ключові слова: скремблювання даних, задача побудови відкритого циклу ша-
хового коня, лінійна часова складність, імовірність випадкової дешифрації,
глибина скремблювання.
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| id | journaliasakpiua-article-315142 |
| institution | System research and information technologies |
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| language | English |
| last_indexed | 2025-07-17T10:28:36Z |
| publishDate | 2024 |
| publisher | The National Technical University of Ukraine "Igor Sikorsky Kyiv Polytechnic Institute" |
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| spelling | journaliasakpiua-article-3151422024-11-16T18:06:34Z Data scrambler knight tour algorithm Алгоритм циклу шахового коня для скремблювання даних Romanuke, Vadim Yaremko, Svitlana Kuzmina, Olena Yehoshyna, Hanna data scrambling knight open tour problem linear time complexity guess probability scrambling depth скремблювання даних задача побудови відкритого циклу шахового коня лінійна часова складність імовірність випадкової дешифрації глибина скремблювання Nowadays, data scrambling remains a vital technique to protect sensitive information by shuffling it in a way that makes it difficult to decipher or reverse-engineer while still maintaining its usability for legitimate purposes. As manipulating the usability of the scrambled data remains a challenge on the background of risking losing data and getting them re-identified by attackers, scrambling and descrambling should be accomplished faster by not increasing data loss and re-identification risks. A scrambling algorithm must have a linear time complexity, still shuffling the data to minimize the risks further. A promising approach is based on the knight open tour problem, whose solutions appear like a random series of knight positions. Hence, a knight open tour algorithm is formalized, by which the knight seems to move chaotically across the chessboard. The formalization is presented as an indented pseudocode to implement it efficiently, whichever programming language is used. The output is a square matrix representing the knight open tour. Based on the knight tour matrix, data scrambler and descrambler algorithms are presented in the same manner. The algorithms have a linear time complexity. The knight-tour scrambling has a sufficiently low guess probability if an appropriate depth of scrambling is used, where the data is re-scrambled repetitively. The scrambling depth is determined by repetitive application of the chessboard matrix, whose size usually increases as the scrambling is deepened. Compared to the pseudorandom shuffling of the data along with storing the shuffled indices, the knight-tour descrambling key is stored and sent far simpler yet ensures proper data security. Скремблювання даних у наш час залишається важливою методикою для захисту конфіденційної інформації за допомогою певного перемішування, після якого розшифрування є надважким, однак зберігається можливість легітимних дій з даними після скремблювання. Оскільки повноцінне маніпулювання діями з даними після скремблювання залишається проблемою на тлі ризику втрати даних та їх несанкціонованого розшифрування, скремблювання та дескремблювання мають виконуватись ще швидше, не збільшуючи ризики втрати та розшифрування. Алгоритм скремблювання повинен мати лінійну часову складність та ще більше мінімізувати зазначені ризики. Перспективним підходом є задача побудови відкритого циклу шахового коня, розв’язки якої виглядають наче випадкові послідовності позицій коня. Відтак формалізується алгоритм відкритого циклу шахового коня, за яким кінь наче хаотично пересувається по шаховій дошці. Ця формалізація представлена у формі відформатованого псевдокоду для подальшого його ефективного впровадження незалежно від мови програмування. На виході отримано квадратну матрицю, котра відображає відкритий цикл шахового коня. На основі матриці циклу шахового коня подано алгоритми скремблера та дескремблера даних у тому ж стилі. Ці алгоритми мають лінійну часову складність. Скремблювання на основі циклу шахового коня має досить низьку імовірність випадкової дешифрації за умови, якщо використана прийнятна глибина скремблювання, де дані повторно скремблюються. Глибина скремблювання визначається багаторазовим застосуванням матриці шахової дошки, розмір якої зазвичай збільшується з поглибленням скремблювання. Порівняно з псевдовипадковим перемішуванням даних разом зі збереженням індексів перемішування, ключ дескремблювання на основі циклу шахового коня зберігається і пересилається набагато простіше, забезпечуючи в той же час прийнятний захист даних. The National Technical University of Ukraine "Igor Sikorsky Kyiv Polytechnic Institute" 2024-09-28 Article Article application/pdf https://journal.iasa.kpi.ua/article/view/315142 10.20535/SRIT.2308-8893.2024.3.03 System research and information technologies; No. 3 (2024); 44-63 Системные исследования и информационные технологии; № 3 (2024); 44-63 Системні дослідження та інформаційні технології; № 3 (2024); 44-63 2308-8893 1681-6048 en https://journal.iasa.kpi.ua/article/view/315142/306031 |
| spellingShingle | скремблювання даних задача побудови відкритого циклу шахового коня лінійна часова складність імовірність випадкової дешифрації глибина скремблювання Romanuke, Vadim Yaremko, Svitlana Kuzmina, Olena Yehoshyna, Hanna Алгоритм циклу шахового коня для скремблювання даних |
| title | Алгоритм циклу шахового коня для скремблювання даних |
| title_alt | Data scrambler knight tour algorithm |
| title_full | Алгоритм циклу шахового коня для скремблювання даних |
| title_fullStr | Алгоритм циклу шахового коня для скремблювання даних |
| title_full_unstemmed | Алгоритм циклу шахового коня для скремблювання даних |
| title_short | Алгоритм циклу шахового коня для скремблювання даних |
| title_sort | алгоритм циклу шахового коня для скремблювання даних |
| topic | скремблювання даних задача побудови відкритого циклу шахового коня лінійна часова складність імовірність випадкової дешифрації глибина скремблювання |
| topic_facet | data scrambling knight open tour problem linear time complexity guess probability scrambling depth скремблювання даних задача побудови відкритого циклу шахового коня лінійна часова складність імовірність випадкової дешифрації глибина скремблювання |
| url | https://journal.iasa.kpi.ua/article/view/315142 |
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