Square transposition: an approach to the transposition process in block cipher

Bulletin of Electrical Engineering and Informatics
Vol. 10, No. 6, December 2021, pp. 3385∼3392
ISSN: 2302-9285, DOI: 10.11591/eei.v10i6.3129 r 3385
Square transposition: an approach to the transposition
process in block cipher
Magdalena A. Ineke Pekereng, Alz Danny Wowor
Department of Informatic Engineering, Universitas Kristen Satya Wacana, Salatiga, Indonesia
Article Info
Article history:
Received Dec 26, 2020
Revised Mar 30, 2021
Accepted Oct 7, 2021
Keywords:
AES
DES
Input scheme
Retrieval scheme
Square transposotion
ABSTRACT
The transposition process is needed in cryptography to create a diffusion effect on
data encryption standard (DES) and advanced encryption standard (AES) algorithms
as standard information security algorithms by the National Institute of Standards and
Technology. The problem with DES and AES algorithms is that their transposition in-
dex values form patterns and do not form random values. This condition will certainly
make it easier for a cryptanalyst to look for a relationship between ciphertexts because
some processes are predictable. This research designs a transposition algorithm called
square transposition. Each process uses square 8 × 8 as a place to insert and retrieve
64-bits. The determination of the pairing of the input scheme and the retrieval scheme
that have unequal flow is an important factor in producing a good transposition. The
square transposition can generate random and non-pattern indices so that transposition
can be done better than DES and AES.
This is an open access article under the CC BY-SA license.
Corresponding Author:
Magdalena A. Ineke Pekereng
Department of Informatics Engineering
Universitas Kristen Satya Wacana
Jl. Notohamidjojo 1-10, Salatiga 50718, Indonesia
Email: ineke.pakereng@uksw.edu
1. INTRODUCTION
Diffusional transposition process is useful for the spread of plaintext redundancy in a ciphertext. Mod-
ern cryptography such as data encryption standard (DES)and advanced encryption standard (AES) as informa-
tion security standards used by National Institute of Standards and Technology (NIST) also contain transposi-
tion as one of the important processes in the algorithm [1], [2]. DES with initial permutation (IP) and inverse
initial permutation (IP)−1
are truly essential in the transposition process [3]. Whereas, AES with shift rows
capability is simpler in transposition [4]. The two algorithms use index values to determine the shift of each
object. The excess diffusion in the algorithm is one of the factors that make DES and AES still attractive and
feasible to use, which makes both of them are chosen by researchers as their information security methods
[5]-[21].
The DES transposition index value in Figure 1 shows patterned results. 64-bit outputs in DES always
form 8-bit groups. Each eight index values produce the same pattern, starting from the highest value that
gradually decreases. For example ai as the index value where (i = 1, 2, 3, · · · , 8) in the first group, the value
of the same position in the next group always becomes ai + 1 (mod 64).
The transposition on AES also has a patterned index value as shown in Figure 2. The AES index
value forms a group of 4 characters. The first group (4-0) consists of four upward histograms and 0 different
Journal homepage: http://beei.org

3386 r ISSN: 2302-9285
histograms. The second group: (3-1) consists of three upward histograms and 1 histogram that has different
values. The same condition applies to the third group (2-2) and the fourth group (1-3).
The index value is expected as a new position to make the sequence of each character to be more
irregular so that the diffusion factor will increasingly appear in the ciphertext. The moving of objects based
on index values in DES and AES indicates a problem because it forms a certain pattern. The problem in DES
and AES is that when the sequence or pattern of data {1, 2, 3, · · · , n} is known, the probability for finding the
{n+1, n+2, · · · } data is great. This will weaken the algorithm and the patterned condition will certainly make
it easier for cryptanalysts to find a plaintext-ciphertext relationship because part of the process is predictable.
A study related to the transposition process was also carried out by [22]-[25], who dismissed the shift
row operation as a transposition operation in AES cryptography. Thus, XOR as an additional operation can
be performed repeatedly up to three times. The research done by [26], [27] adds various processes to correct
the shortcomings of transposition in the algorithm. Although the improvement of the transposition process by
adding the algorithm in parallel will certainly obtain a good result, the adding of the algorithm takes more time,
and space. In terms of efficiency, the algorithm is less elegant to be used as information security.
Figure 1. Index values of DES transposition Figure 2. Index values of AES transposition
This research designs a transposition algorithm called Square Transposition. A square of n × n size
is used as the medium to hold (m = n × n)-bit. Each bit input is entered into a square using certain rules and
taking of bit is also done with certain rules. Determination of whether the designed algorithm is good or not is
seen based on its statistical testing. Statistical testing is done to determine the randomness of each index value.
In addition, correlation testing is used to measure the algorithm’s ability to disguise the relationship between
input and output. Finally, DES and AES are compared to find out the power of Square Transposition in the
algorithm for the transposition process.
2. PROPOSED RESEARCH
2.1. Square transposition
Square transposition consists of two processes namely bit-input into a square and bit-retrieval with a
certain predetermined size. Suppose T = text input, ti = i-th text character and ai = i-th binary character, then:
T = {t1, t2, · · · , tn}; n|8, n ∈ Z+
(1)
Where, t1 = {a01, a02, a03, · · · , a08}, t2 = {a09, a10, a11, · · · , a16}, t3 = {a17, a18, a19, · · · , a24}, · · · ,
tn = {a8n−7, a8n−6, a8n−5, · · · , a8n}. If n6 | 8, then padding is done as many as k, so that it will result in (2).
With (n + k)|8; k = 1, 2, · · · , 7.
T = {t1, t2, · · · , tn, tn+1, tn+2, · · · , tn+k} (2)
The square that is used as the transposition media can be adjusted to the bit size of the text input. This research
chooses 64-bit text input, so it will be a square size of 8 × 8 shown in Figure 3.
Figure 3. Square transposition 8 × 8
Bulletin of Electr Eng & Inf, Vol. 10, No. 6, December 2021 : 3385 – 3392

Bulletin of Electr Eng & Inf ISSN: 2302-9285 r 3387
The entry scheme is a way to place each bit of ai; i ∈ Z+
64 in the entry of square with certain rules.
For example, every bit after entering into a square is the order of bits given in (3).
Tsq = {a∗
1, a∗
2, a∗
3, · · · , a∗
64} (3)
A retrieval scheme is a way to take every bit of a∗
i , i ∈ Z+
64 from the square with a certain rule. Notation for
each bit taken from square (a∗
i (j)); ∃ i, j ∈ Z+
64 where i is the entry index and j is the retrieval index. In (4) is
a schema collection dataset L = {l1, l2, l3, · · · , l8}, where ∃ x ∈ Z+
64.
l1 ={a∗
x (01), a∗
x (02), a∗
x (03), · · · , a∗
x (08)},
l2 ={a∗
x (09), a∗
x (10), a∗
x (11), · · · , a∗
x (16)},
.
.
.
.
.
.
l8 ={a∗
x (57), a∗
x (58), a∗
x (59), · · · , a∗
x (64)}.
(4)
2.2. Square transposition schematic testing
Every combination of input and output schemes in the square transposition will result in a transposition
method, and each combination must produce a random order of index values. All users can design their input
and output schemes. Therefore, random testing needs to be done to ensure that every designed scheme will
produce a good transposition method.
Figure 4 shows a testing scheme, in which if each pair of schemes has not yet reached randomness, a
scheme can be replaced by another scheme. This research uses three tests of randomness (Frequency Monobit
Test, Frequency Test within a Block, and Runs Test) so that if two or three methods are random, the combination
of those schemes can be used as a method of transposition.
Figure 4. Testing of input and output schemes
3. RESULT AND DISCUSSION
3.1. Square transposition entry scheme
Based on (1), 64-bit is used as input and square size 8 × 8. Two input schemes are selected with
randomly selected index values, the two schemes are given in succession in Figures 5 and 6, respectively.
Figure 5. Input scheme 1 Figure 6. Input scheme 2
3.2. Retrieval scheme design
The retrieval scheme is a rule that takes every bit from a square that previously had a bit from the bit
entry process. Here are several retrieval schemes used as pairs of input schemes.
3.2.1. Horizontal retrieval scheme
This design uses the Entry-1 Scheme to insert bits into a square, as shown in Figure 7. The horizontal
retrieval process is carried out from the top left corner to the right corner of the square. The order of each bit
a8i+1 for i = 0, 1, · · · , 7 is always to the left of the first entry of every line to square (i + 1).
Square transposition: an approach to the transposition process in block ... (Magdalena A. Ineke Pekereng)

3388 r ISSN: 2302-9285
The horizontal retrieval scheme results starts from a37 based on the index j = 1 to j = 64 for a55.
Thus, square transposition output is obtained which is based on byte, as shown previously in (4), is a schema
collection dataset L = {l1, l2, l3, · · · , l8} where l1 = {a37, a29, · · · , a48}, l2 = {a30, a39, a41, · · · , a43},
· · · , l8 = {a61, a13, a02, · · · , a55}. Transposition results from the retrieval-1 scheme and the horizontal entry
schema can be visualized in Cartesian coordinates, where each takes index (i) is abscissa and index enter (j)
as ordinate. The results of complete bit retrieval are shown in Figure 8.
Figure 7. Horizontal retrieval scheme
Figure 8. Graphic of the index values
3.2.2. Vertical retrieval scheme
Square transposition also uses an input-1 scheme to input each entry from the square. Retrieval is
done vertically from top to bottom, starting at the top right corner entry to the bottom right of the square. In
general, every bit of ai (j) and the retrieval index j = (8z + 1); z ∈ {0, 1, · · · , 7}. If z is even, the retrieval
is done vertically top-down, and if z is odd, the retrieval will be done from the bottom up. The retrieval results
are based on bits shown in Figure 9.
The vertical retrieval scheme starts from a48 based on the index j = 1 to j = 64 for a37 bits. So
the square transposition output is based on bytes L = {l1, l2, l3, · · · , l8}, where l1 = {a48, a43, a06, · · · , a55},
l2 = {a52, a33, a15, · · · , a27}, · · · , l8 = {a61, a17, a54, · · · , a37}. The visualization of transposition index of
the input-1 scheme and the vertical input scheme is shown in Figure 10.
Figure 9. Vertical Retrieval Scheme
3.2.3. Zigzag retrieval scheme
The input-2 scheme is used in Figure 6 which the retrieval scheme is done in zigzag form from the
lower left to the upper right. The retrieval plots are based on index values j = 1 to j = 64, which the

complete plots are shown in Figure 11. Retrieval starts from a46 to a25, so that the square transposition
output can be obtained based on byte L = {l1, l2, l3, · · · , l8} where l1 = {a46, a42, a16, · · · , a63}, l2 =
{a40, a17, a64, · · · , a29}, · · · l8 = {a5, a11, a09, · · · , a25}. The geometric interpretation of the value of the
zigzag input-2 and zigzag retrieval scheme is shown in Figure 12.
Figure 11. Zigzag retrieval scheme
3.2.4. Rice plow retrieval scheme
The transposition technique by adopting the rice plow process can be done with the assumption of a
square as a rice field plot. Each bit plot is adjusted to the rice plow process starting from the outside point
towards the midpoint, which the complete plots are shown in Figure 13. The input-2 scheme is used to fill in
each input from the square so that the retrieval using rice plow plot can be carried out.
The retrieval process starts from the lower right corner (a08) with a rotating plot around the square
towards the center (a04). The value of the transposition output index of the input-2 scheme and rice scheme
retrieval is L = {l1, l2, · · · , l8}; where l1 = {a08, a21, a56, · · · , a46}, l2 = {a42, a28, a47, · · · , a24}, · · ·
l8 = {a43, a43, a27, · · · , a04}. The visualization of the transposition index value is shown in Figure 14.
Figure 13. Rice plow retrieval scheme
3.3. Testing of randomness on index values
The method used in randomness testing is Mono Bit frequency Test, Bit Block frequency Test, and
Run Test, with α = 0.01. Each transposition index value is declared as random if two or three test results

3390 r ISSN: 2302-9285
have p-value > α. The complete test results are shown in Table 1. The combination of each input scheme
and retrieval scheme is carried out to see how well the pair of schemes are designed or selected so that square
transposition can produce a random index value.
The schemes of input 1 & vertical output obtain the highest p-value with the average value of 0.273,
and the ones with the lowest score are the schemes of input 1 & zigzag scheme with a smaller average of
p-value, which is 0.101. Overall, all pairs of input and output schemes can maintain the p-value that results
in a random index value and the pair of schemes can produce a better index value when compared to the
transposition method in AES and DES algorithms.
Table 1. Randomness test result for each scheme
2*No 2*Input & Retrieval Scheme p-value 2*Result
Mono Bit Block Bit Run-Test
1 Input-1 & Horizontal Scheme 0.1341 0.1230 0.1853 random
2 Input-1 & Vertical Scheme 0.2727 0.2194 0.1432 random
3 Input-2 & Zigzag Scheme 0.1204 0.2282 0.2031 random
4 Input-2 & Rice Plow Scheme 0.1950 0.1917 0.1981 random
5 Input-1 & Zigzag Scheme 0.1012 0.1118 0.2116 random
6 Input-1 & Rice Plow Scheme 0.1018 0.2014 0.1102 random
7 Input-2 & Horizontal Scheme 0.1210 0.1052 0.2056 random
8 Input-2 & Vertical Scheme 0.2044 0.2015 0.1186 random
9 DES 2.569 ×10−8 0.0539 9.172 ×10−6 non random
10 AES 4.251 ×10−8 0.0042 1.456 ×10−4 non random
The use of output-1 and output-2 schemes plays an important role in yielding the output of random
index values. Selection of a pair of schemes using a combination of horizontal, vertical, zigzag, and plow or
others that have a patterned index will generate poor transposition index value. It happens because the input
and output scheme has the same or similar line direction.
3.4. Correlation testing
Correlation value (r) can be used to see the magnitude of the relationship between input (x) and
output (y) of statistically related algorithms. The correlation interval is −1 ≤ r ≤ 1, and if r approaches 0,
then the algorithm is able to make the input and output not statistically related. In this condition, if r < 0, the
absolute value |r| can be used to find out the distance r from 0.
Correlation testing uses three plaintext inputs which it is expected to represent text variations that
might be used by users. Input “fti uksw” is to represent traditional text input because usually, users use it. The
second more extreme test is the use of the same input, which is “xyyyyyyyy” (not “yyyyyyyyy” because this
correlation formula is undefined). The third test is “$aL4t1G4” which also represents a variety of symbols,
numbers, and letters that are used as input.
The results obtained in Table 2 show that the output of each scheme of the square transposition has
an average correlation value close to 0. Thus, it indicates that the relationship between input and output is not
related statistically. Consequently, the square transposition succeeds in disguising the information, so that the
distribution of redundancies occurs well and will certainly increase the diffusion effect on the cryptography
algorithm.
Table 2. Testing result of input-output correlation
2*No 2*Transposition Method Correlation Value |r| 2*Average
fti uksw xyyyyyyy $aL4t1G4
1 Input-1 & Horizontal Retrieval 0.249 0.331 0.217 0.266
2 Input-1 & Vertical Retrieval 0.162 0.127 0.142 0.162
3 Input-2 & Zigzag Retrieval 0.254 0.267 0.324 0.254
4 Input-2 & Rice Plow Retrieval 0.313 0.375 0.252 0.313
5 Input-1 & Zigzag Retrieval 0.112 0.009 0.018 0.112
6 Input-1 & Rice Plow Retrieval 0.016 0.090 0.040 0.016
7 Input-2 & Horizontal Retrieval 0.138 0.268 0.265 0.138
8 Input-2 & Vertical Retrieval 0.076 0.098 0.184 0.076
9 DES 0.342 0.126 0.374 0.342
10 AES 0.376 0.429 0.277 0.376

The transposition of DES and AES algorithms has resulted in higher average correlation values than
the value from the schematic combination of square transposition so that it can be said that each pair of schemes
can generate a better transposition algorithm. Of course, the use of square transposition in cryptography will
increase the strength of overall cryptographic algorithms. Optimization of the transposition process using
square transposition is a part that needs to be done by cryptographers to improve or modify the weak parts of
the algorithm.
4. CONCLUSION
The determination of a pair of input and output schemes in square transposition should be based on
schemes that have different lines to obtain a good transposition process. Combination of schemes that were
carried out produced less patterned geometric visualization that oscillates irregularly, so that the transposition
method could generate random index values. This result is also seen in randomness testing in which the overall
obtained p-value is greater which α = 1% so that the square transposition can produce better a transposition
method when it is compared to AES and DES values which the index is not random. Square transposition
produces an average correlation value closer to 0 for testing the text input when compared to AES and DES
transpositions. Thus, the square transposition manages to disguise the information on the input so that it is not
visible in the output. Besides, the square transposition can spread the distributed redundancies well, so that
it will increase the diffusion effect on the cryptographic algorithm. The result shows that the algorithm in the
square transposition optimizes the transposition process that previously has non-random index values. This
design optimizes algorithm processes by concentrating on the diffusion effect and by not giving a burden on
the complexity of time and space. Algorithm modification is a process that every cryptographer needs to do to
produce a more efficient algorithm in cryptography to secure information.
ACKNOWLEDGEMENT
The researcher would like to thank the Bureau of Research, Publication and Community Service
(BP3M) of Universitas Kristen Satya Wacana Salatiga for providing the funding assistance through the Funda-
mental Internal Research Scheme in 2018/2019.
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