=Paper=
{{Paper
|id=Vol-3654/paper12
|storemode=property
|title=Information-Driven Permutation Operations for Cryptographic Transformation
|pdfUrl=https://ceur-ws.org/Vol-3654/paper12.pdf
|volume=Vol-3654
|authors=Vira Babenko,Tetiana Myroniuk,Artem Lavdanskyi,Yaroslav Tarasenko,Oleg Myroniuk
|dblpUrl=https://dblp.org/rec/conf/cpits/BabenkoMLTM24
}}
==Information-Driven Permutation Operations for Cryptographic Transformation==
https://ceur-ws.org/Vol-3654/paper12.pdf
Information-Driven Permutation Operations
for Cryptographic Transformation
Vira Babenko1, Tetiana Myroniuk1, Artem Lavdanskyi1, Yaroslav Tarasenko1,
and Oleg Myroniuk1
1 Cherkasy State Technological University, 460 Shevchenko str. Cherkasy, 18006, Ukraine
Abstract
In the work, the authors proposed one of the techniques of using information-driven
permutation operations for the implementation of cryptographic data transformation. An
algorithm for implementing the proposed method of cryptographic data transformation
based on the use of a basic group of information-driven permutation operations has been
developed. The process of cryptographic transformation of three bytes of data based on the
proposed algorithm is shown by a flowchart containing information-driven permutations, a
Feistel network, shift and XOR operations, and addition modulo 2. The software
implementation of the developed algorithm in the high-level object-oriented programming
language Python is carried out. The obtained results of the work of the created software
made it possible to conduct further research and carry out a qualitative assessment of the
results of cryptographic data transformation according to the proposed method of using
previously synthesized information-driven permutation operations. The effectiveness of
this algorithm was evaluated based on statistical testing by the NIST STS package, as well as
its suitability for implementing data encryption by hardware and software based on a
comparison of test results with the results of using standard encryption algorithms DES,
AES, Blowfish, Kalyna, Strumok, and Linear Feedback Shift Register.
Keywords 1
Technique, information-driven permutation operations, basic operations, algorithm,
cryptographic transformation, key, round, statistical testing.
1. Introduction Software cryptographic protection tools are
flexible, which gives them a special advantage
The development of new and improvement of over hardware ones. Mobility and ease of use
common encryption methods that would be explain their modern popularity and prevalence.
simple in hardware and software Therefore, among the ways to improve the
implementations, and at the same time provide a stability indicators of cryptographic algorithms,
sufficiently high level of cryptographic strength there are several approaches to the construction
by expanding the range of cryptographic of software ciphers. The most promising for
transformation operations used, obtained by software implementation are flexible ciphers
modifying the basic operations, is one of the based on the use of several modifications of the
urgent tasks of information security. The search encryption algorithm, ciphers with pseudo-
and synthesis of modified operations for probable key selection, and ciphers with
cryptographic transformation will make it permutation of fixed procedures and
possible to build algorithms using them with the customization of transformation operations. In
best cryptographic properties, which makes this addition, one of the well-known ways to increase
study relevant. cryptographic strength is the multi-pass mode of
applying the encryption algorithm.
CPITS-2024: Cybersecurity Providing in Information and Telecommunication Systems, February 28, 2024, Kyiv, Ukraine
EMAIL: v.babenko@chdtu.edu.ua (V. Babenko); t.myroniuk@chdtu.edu.ua (T. Myroniuk); a.lavdanskyi@chdtu.edu.ua (A. Lavdanskyi);
o.m.myroniuk.asp23@chdtu.edu.ua (O. Myroniuk)
ORCID: 0000-0003-2039-2841 (V. Babenko); 0000-0002-7588-1055 (T. Myroniuk); 0000-0002-1596-4123 (A. Lavdanskyi); 0000-0002-
5902-8628 (Y. Tarasenko); 0009-0007-7572-6972 (O. Myroniuk)
©️ 2024 Copyright for this paper by its authors.
Use permitted under Creative Commons License Attribution 4.0 International (CC BY 4.0).
CEUR Workshop Proceedings (CEUR-WS.org)
CEUR
ceur-ws.org
Workshop ISSN 1613-0073
Proceedings
137
2. Background Analysis results of experimental studies of the statistical
security and speed characteristics of stream
In our day to be secure, modern information and ciphers show that the Strumok algorithm is the
communication technology needs reliable most balanced solution, it can provide the
encryption. That is why a large number of modern properties of a random sequence generator and
scientific publications are devoted to the give huge indicators in terms of encryption speed.
development of new and improvement of existing It has been practically proven that the encryption
common cryptographic algorithms. Such works speed of the Strumok algorithm on modern
include scientific publications [1–3, 12–14, 22]. computer systems can reach 10–15 Gbit/s.
In this paper [1] to improve the security In works [9, 11] the results of experimental
effectiveness of electronic information studies of statistical properties of common and
resources, two encryption algorithms (Luna and modern cryptographic encryption algorithms
Neptun) have been developed based on fixed are given. In the article [11], experimental
lookup tables with extended bit depth and studies of the cryptographic properties of the
dynamic key-dependent lookup tables. Strumok stream cipher were carried out by the
Properties of random sequences formed using NIST STS statistical testing methodology. In
the proposed algorithms’ encryption [1] were articles [9–11], a comparative analysis of the
explored in the environment of NIST STS statistical security indicators of the world-
statistical tests. famous and most widespread cryptographic
The paper [2] proposes a method that uses algorithms (Enocoro, HC-128, HC-256, Grain,
substitution tables with increased capacity and MICKEY 2, MUGI, Rabbit, Salsa20, SNOW 2.0,
randomized linear and non-linear operations. Sosemanuk, Trivium, AES, Strumok, Kalyna, etc.),
Based on this method, a new advanced block which are standardized at the international or
cipher was proposed and its specifications were national level, is carried out.
given [2]. The article [12] deals with an actual task for
In this work [3] the Linear Feedback Shift increasing the reliability of information
Register (LFSR) is used to produce nonbinary protection systems by the creation and use of
pseudo-random key sequences. The length of the new four-bit cryptographic transformations
sequence has been enhanced by designing a with nonlinear Boolean functions that have the
hybrid model using LFSR and Genetic Algorithm. property of strict avalanche criterion. In the
Achieving a length more than the maximum article [12], for the first time, there was
length of LFSR is the primary intention of this proposed a method for obtaining inverse four-
work. bit cryptographic transformations with the
It should be noted that the main block and strict avalanche criterion property for
stream symmetric encryption algorithms used in balanced Boolean functions containing two
Ukraine are the Kalyna and Strumok logical operations (inversion and addition
cryptographic algorithms. For the cryptographic modulo two). This method [12] is a method of
transformation in the Kalyna standard [4–8], the selecting the already existing basic Boolean
SPN structure was chosen as a high-level design functions from a predetermined set of
of the cipher based on analytical comparison, the balanced basic Boolean functions for direct
layer of nonlinear transformation of the cyclic and inverse cryptographic transformations,
function was implemented based on S-blocks, whereas the existing methods of searching for
and for the implementation of the linear inverse cryptographic transformation are
scattering block, multiplication by MDV was methods for calculating each element of the
chosen matrix. This method of constructing a Boolean functions for the inverse cryptographic
crypto algorithm ensures compliance with transformation.
modern requirements for the level of The article [13] proposes a modification of the
cryptographic stability and speed compared to algorithm for calculating the reciprocal of a
other international standards. number presented in a finite ring, which makes it
In turn, it should be noted the high statistical possible to reduce the number of elementary
security of the stream symmetric encryption operations and thereby increase the
algorithm Strumok [9–11], which revealed certain performance of encryption algorithms.
properties of the random bit generator. The However, the problem task of using a group of
information-driven permutation operations for
138
the implementation of cryptographic hardware. The results obtained are presented in
transformation has not been given attention in Table 1 [16].
the development of security software. Thus, the To obtain the elementary functions of the
development of methods for applying this group cryptographic transformations given in Table 1,
of operations for data encryption and decryption the bit permutation method should be used, the
and the algorithmization of these processes is main task of which is to replace one bit of the
certainly an urgent task of research. elementary function with two others.
Based on the researched results, the obtained
3. Formal Problem Statement elementary functions will be called elementary
functions of information-driven permutations.
In the work [16], a model of elementary
The usage of controlled operations opens up
functions for cryptographic transformation is
great opportunities for achieving the required
built, which has the following form:
level of cryptographic protection. Considering
that the efficiency of using controlled 𝑌 = 𝑥̃𝑖 𝑥̃𝑗 ∨ 𝑥̃̄𝑖 𝑥̃𝑘 , (1)
operations increases with the number of where 𝑌 is the value of the corresponding
potentially implemented modifications, since digit of the output signal of the result of the
in this case the sub-block of the data being elementary functions of the cryptographic
converted expands, the use of the permutation transformation; 𝑥̃𝑖 , 𝑥̃𝑗 , 𝑥̃𝑘 are the values of the
operation becomes relevant, since it has a very corresponding digits of the input signal.
large number of modifications. Thus, the The following are the properties of the
development of cryptographic tools based on defined model [16]:
controlled permutations is a promising 1. 𝑖, 𝑗, 𝑘 take the values 1, 2, 3, and 𝑖 ≠ 𝑗 ≠ 𝑘.
direction in modern cryptography [15]. 2. 𝑥𝑖 can take a direct or inverse value.
Therefore, the main objective of this work is 3. 𝑥𝑗 , 𝑥𝑘 can be both in direct and inverse
to improve the technique of using basic meaning and 𝑗 ≠ 𝑘.
information-driven permutation operations Table 1
through the use of a modified algorithm of Elementary functions of low complexity
cryptographic information transformation that Number of Execution
allows using the full set of permutation Discrete model
function result
modifications. 83 01010011 𝑥1 ⋅ 𝑥2 ∨ 𝑥̄ 1 ⋅ 𝑥3
163 10100011 𝑥1 ⋅ 𝑥2 ∨ 𝑥̄ 1 ⋅ 𝑥̄ 3
Purpose of the work. Develop a way to apply 46 00101110 𝑥1 ⋅ 𝑥̄ 2 ∨ 𝑥2 ⋅ 𝑥̄ 3
a basic group of information-driven 71 01000111 𝑥1 ⋅ 𝑥2 .∨ 𝑥̄ 2 ⋅ 𝑥3
permutation operations and propose an 139 10001011 𝑥1 ⋅ 𝑥2 ∨ 𝑥̄ 2 ⋅ 𝑥̄ 3
53 00110101 𝑥̄ 1 ⋅ 𝑥2 .∨ 𝑥1 ⋅ 𝑥3
algorithm for implementing cryptographic 58 00111010 𝑥̄ 1 ⋅ 𝑥2 .∨ 𝑥1 ⋅ 𝑥̄ 3
data transformation based on the use of these 184 10111000 𝑥̄ 1 ⋅ 𝑥2 ∨ 𝑥̄ 2 ⋅ 𝑥̄ 3
operations. Investigate and evaluate the 116 01110100 𝑥̄ 1 ⋅ 𝑥2 ∨ 𝑥̄ 2 ⋅ 𝑥3
92 01011100 𝑥1 ⋅ 𝑥̄ 2 ∨ 𝑥̄ 1 ⋅ 𝑥3
suitability of using the proposed algorithm for 172 10101100 𝑥1 ⋅ 𝑥̄ 2 ∨ 𝑥̄ 1 ⋅ 𝑥̄ 3
encryption by software and hardware. 29 00011101 𝑥1 ⋅ 𝑥̄ 2 ∨ 𝑥2 ⋅ 𝑥3
197 11000101 𝑥̄ 1 ⋅ 𝑥̄ 2 ∨ 𝑥1 ⋅ 𝑥3
202 11001010 𝑥̄ 1 ⋅ 𝑥̄ 2 ∨ 𝑥1 ⋅ 𝑥̄ 3
4. Materials and Methods 209
226
11010001
11100010
𝑥̄ 1 ⋅ 𝑥̄ 2 ∨ 𝑥2 ⋅ 𝑥3
𝑥̄ 1 ⋅ 𝑥̄ 2 ∨ 𝑥2 ⋅ 𝑥̄ 3
39 00100111 𝑥1 ⋅ 𝑥3 ∨ 𝑥2 ⋅ 𝑥̄ 3
Let us consider in more detail the synthesis of 141 10001101 𝑥1 ⋅ 𝑥3 ∨ 𝑥̄ 2 ⋅ 𝑥̄ 3
114 01110010 𝑥̄ 1 ⋅ 𝑥3 ∨ 𝑥2 ⋅ 𝑥̄ 3
cryptographic transformation operations 27 00011011 𝑥1 ⋅ 𝑥̄ 3 ∨ 𝑥2 ⋅ 𝑥3
based on elementary information-driven 78 01001110 𝑥1 ⋅ 𝑥̄ 3 ∨ 𝑥̄ 2 ⋅ 𝑥3
permutation functions. 177 10110001 𝑥̄ 1 ⋅ 𝑥̄ 3 ∨ 𝑥2 ⋅ 𝑥3
228 11100100 𝑥̄ 1 ⋅ 𝑥̄ 3 ∨ 𝑥̄ 2 ⋅ 𝑥3
In [16] it was determined that the sets of 216 11011000 𝑥̄ 1 ⋅ 𝑥3 ∨ 𝑥̄ 2 ⋅ 𝑥̄ 3
groups of 3-bit elementary cryptographic
transformation functions are functions To construct a method of synthesis of
consisting of three elementary functions, the information-driven elementary functions, we
formal model of which is similar to the will introduce the following definitions [16].
complexity of the addition modulo 2. This means Definition 1. The main element of the
that these sets are simple and do not require elementary function of information-driven
significant resources for implementation by
139
permutations is the repeating element in the cryptographic transformation operations, it is
right and left parts of the elementary function necessary to calculate the number of basic
in direct and inverse meanings. three-bit operations from the total number of
Definition 2. An additional element or an cryptographic transformation operations.
elementary function of information-driven For further research, elementary
permutations is an element that occurs once information-driven functions were divided into
either on the left or on the right side of the direct and reverse elementary information-
elementary function. driven functions [16].
The method for synthesizing elementary When studying certain groups of
information-driven permutation functions is information-driven permutation operations, it
as follows [16]: was found that the base group can be
1. Determine the indices of the main and considered a group where elementary
additional elements of elementary functions are located in the following way [16]:
functions of information-driven the first elementary function of the operation
permutation. is the function f1, the value of the main element
2. Determine direct and inverse values of of which is x1; the second elementary function
elements of elementary functions of of the operation is the function f2, the main
information-driven permutation. element of which is x2; the third elementary
3. Substitute certain values into expression function of the operation is the function f3,
(1) to obtain elementary information- where the value of the main element is x3.
driven permutation functions. Based on this, it can be concluded that the
4. Applying items 1–3 on this set of main elements of the basic operation, which
elementary indices for direct and inverse can form the basic group of cryptographic
values of elementary functions of transformation operations, should be placed
information-driven permutation, we diagonally [16].
obtain the complete set of elementary Thus, eight operations for two groups were
functions of information-driven defined, which are the basic operations of
permutation. forward (Ff) and reverse (Fr) cryptographic
Subsequently, the analysis of cryptographic transformation. They are represented by
transformation operations [16] was carried expressions (2)–(17) [16].
out based on elementary permutation 𝑥1 ⋅ 𝑥̄ 2 ∨ 𝑥̄ 1 ⋅ 𝑥3
functions controlled by information obtained 𝑓
𝐹92,46,27 = [𝑥1 ⋅ 𝑥̄ 2 ∨ 𝑥2 ⋅ 𝑥̄ 3 ] =
based on the experiment. 𝑥1 ⋅ 𝑥̄ 3 ∨ 𝑥2 ⋅ 𝑥3
1 0 0 − 0 1 (2)
As a result of the computational experiment = [0 1 0] ⇔ [ 1 − 0 ].
[16], it was determined that the total number 0 0 1 1 1 −
of three-bit information-driven permutation
operations for cryptographic transformation 𝑥1 ⋅ 𝑥2 ∨ 𝑥̄1 ⋅ 𝑥3
will be equal to the product of the 𝑟
𝐹83,116,78 = [𝑥̄ 1 ⋅ 𝑥2 ∨ 𝑥̄ 2 ⋅ 𝑥3 ] =
multiplication of the number of operations in 𝑥1 ⋅ 𝑥̄ 3 ∨ 𝑥̄ 2 ⋅ 𝑥3
(3)
1 0 0 − 1 1
each group, which is 764 information-driven = [0 1 0] ⇔ [ 0 − 1 ].
operations. 0 0 1 1 0 −
When researching cryptographic
transformation operations, it was found that 𝑥̄ 1 ⋅ 𝑥2 ∨ 𝑥1 ⋅ 𝑥3
𝑓
the total number of these operations is formed 𝐹53,71,27 = [𝑥1 ⋅ 𝑥2 ∨ 𝑥̄ 2 ⋅ 𝑥3 ] =
by a combination of basic operations, 𝑥1 ⋅ 𝑥̄ 3 ∨ 𝑥2 ⋅ 𝑥3
1 0 0 − 1 1 (4)
permutation operations, and inversion = [0 1 0] ⇔ [ 1 − 1 ].
operations [16]: 𝑁 = 𝑁𝑏 ⋅ 𝑁п ⋅ 𝑁і = 𝑁𝑏 ⋅ 3! ⋅ 0 0 1 1 1 −
23 = 384, where N is the total number of
operations; Nb is the number of basic 𝑥1 ⋅ 𝑥2 ∨ 𝑥̄1 ⋅ 𝑥3
𝑟
operations; Nn is the number of operations 𝐹83,29,39 = [𝑥1 ⋅ 𝑥̄ 2 ∨ 𝑥2 ⋅ 𝑥3 ] =
𝑥1 ⋅ 𝑥3 ∨ 𝑥2 ⋅ 𝑥̄ 3
based on the replacement of 1 or 2 elementary 1 0 0 − 1 1 (5)
group functions and Ni is the number of = [0 1 0] ⇔ [ 1 − 1 ].
inversion transactions. Therefore, to 0 0 1 1 1 −
determine the number of three-bit basic
140
𝑥1 ⋅ 𝑥2 ∨ 𝑥̄1 ⋅ 𝑥3 𝑥̄1 ⋅ 𝑥2 ∨ 𝑥1 ⋅ 𝑥̄ 3
𝑓 𝑟
𝐹83,29,39 = [𝑥1 ⋅ 𝑥̄ 2 ∨ 𝑥2 ⋅ 𝑥3 ] = 𝐹58,116,39 = [𝑥̄1 ⋅ 𝑥2 ∨ 𝑥̄ 2 ⋅ 𝑥3 ] =
𝑥1 ⋅ 𝑥3 ∨ 𝑥2 ⋅ 𝑥̄ 3 𝑥1 ⋅ 𝑥3 ∨ 𝑥2 ⋅ 𝑥̄ 3
1 0 0 − 1 1 (6) 1 0 0 − 1 0 (15)
= [0 1 0] ⇔ [ 1 − 1 ]. = [0 1 0] ⇔ [ 0 − 1 ].
0 0 1 1 1 − 0 0 1 1 1 −
𝑥̄1 ⋅ 𝑥2 ∨ 𝑥1 ⋅ 𝑥3 𝑥1 ⋅ 𝑥2 ∨ 𝑥̄1 ⋅ 𝑥3
𝑟 𝑓
𝐹53,71,27 = [𝑥1 ⋅ 𝑥2 ∨ 𝑥̄ 2 ⋅ 𝑥3 ] = 𝐹83,116,78 = [𝑥̄1 ⋅ 𝑥2 ∨ 𝑥̄ 2 ⋅ 𝑥3 ] =
𝑥1 ⋅ 𝑥̄ 3 ∨ 𝑥2 ⋅ 𝑥3 (7) 𝑥1 ⋅ 𝑥̄ 3 ∨ 𝑥̄ 2 ⋅ 𝑥3 (16)
1 0 0 − 1 1 1 0 0 − 1 1
= [0 1 0] ⇔ [ 1 − 1 ]. = [0 1 0] ⇔ [ 0 − 1 ].
0 0 1 1 1 − 0 0 1 1 0 −
𝑥̄1 ⋅ 𝑥2 ∨ 𝑥1 ⋅ 𝑥̄ 3 𝑥1 ⋅ 𝑥̄ 2 ∨ 𝑥̄1 ⋅ 𝑥3
𝑓 𝑟
𝐹58,29,78 = [𝑥1 ⋅ 𝑥̄ 2 ∨ 𝑥2 ⋅ 𝑥3 ] = 𝐹92,46,27 = [𝑥1 ⋅ 𝑥̄ 2 ∨ 𝑥2 ⋅ 𝑥̄ 3 ] =
𝑥1 ⋅ 𝑥̄ 3 ∨ 𝑥̄ 2 ⋅ 𝑥3
(8)
𝑥1 ⋅ 𝑥̄ 3 ∨ 𝑥2 ⋅ 𝑥3 (17)
1 0 0 − 1 0 1 0 0 − 0 1
= [0 1 0] ⇔ [ 1 − 1 ]. = [0 1 0] ⇔ [ 1 − 0 ].
0 0 1 1 0 − 0 0 1 1 1 −
𝑥̄ 1 ⋅ 𝑥2 ∨ 𝑥1 ⋅ 𝑥3 Having studied the basic group of
𝑟
𝐹53,46,114 = [𝑥1 ⋅ 𝑥̄ 2 ∨ 𝑥2 ⋅ 𝑥̄ 3 ] = cryptographic transformation operations,
𝑥̄ 1 ⋅ 𝑥3 ∨ 𝑥2 ⋅ 𝑥̄ 3 presented above in the form of discrete
1 0 0 − 1 1 (9)
= [0 1 0] ⇔ [ 1 − 0 ].
models, we have determined the following
0 0 1 0 1 − dependencies for the main elements of
operations [16]:
𝑓
𝑥̄ 1 ⋅ 𝑥2 ∨ 𝑥1 ⋅ 𝑥̄ 3 𝐹92,46,27 = 𝑥1 = 1, 𝑥2 = 0, 𝑥3 = 0. ⇒
𝑓
𝐹58,116,39 = [𝑥̄ 1 ⋅ 𝑥2 ∨ 𝑥̄ 2 ⋅ 𝑥3 ] = 𝑟
⇒ 𝐹83,116,78 = 𝑥1 = 1, 𝑥2 = 1, 𝑥3 = 0.
𝑥1 ⋅ 𝑥3 ∨ 𝑥2 ⋅ 𝑥̄ 3 𝑓
1 0 0 − 1 0 (10) 𝐹53,71,27 = 𝑥1 = 0, 𝑥2 = 1, 𝑥3 = 0. ⇒
𝑟
= [0 1 0] ⇔ [ 0 − 1 ]. ⇒ 𝐹83,29,39 = 𝑥1 = 1, 𝑥2 = 0, 𝑥3 = 1.
𝑓
0 0 1 1 1 − 𝐹83,29,39 = 𝑥1 = 1, 𝑥2 = 0, 𝑥3 = 1. ⇒
𝑟
⇒ 𝐹53,71,27 = 𝑥1 = 0, 𝑥2 = 1, 𝑥3 = 0.
𝑓
𝑥1 ⋅ 𝑥̄ 2 ∨ 𝑥̄1 ⋅ 𝑥3 𝐹58,29,78 = 𝑥1 = 0, 𝑥2 = 0, 𝑥3 = 0. ⇒
𝑟
𝐹92,71,114 = [𝑥1 ⋅ 𝑥2 ∨ 𝑥̄ 2 ⋅ 𝑥3 ] = 𝑟
⇒ 𝐹53,46,114 = 𝑥1 = 0, 𝑥2 = 0, 𝑥3 = 1.
𝑥̄1 ⋅ 𝑥3 ∨ 𝑥2 ⋅ 𝑥̄ 3 𝑓
1 0 0 − 0 1 (11) 𝐹58,116,39 = 𝑥1 = 0, 𝑥2 = 1, 𝑥3 = 1. ⇒
𝑟
= [0 1 0] ⇔ [ 1 − 1 ]. ⇒ 𝐹92,71,114 = 𝑥1 = 1, 𝑥2 = 1, 𝑥3 = 1.
𝑓
0 0 1 0 1 − 𝐹53,46,114 = 𝑥1 = 0, 𝑥2 = 0, 𝑥3 = 1. ⇒
𝑟
⇒ 𝐹58,29,78 = 𝑥1 = 0, 𝑥2 = 0, 𝑥3 = 0.
𝑓
𝑥̄1 ⋅ 𝑥2 ∨ 𝑥1 ⋅ 𝑥3 𝐹92,71,114 = 𝑥1 = 1, 𝑥2 = 1, 𝑥3 = 1. ⇒
𝑓
𝐹53,46,114 = [𝑥1 ⋅ 𝑥̄ 2 ∨ 𝑥2 ⋅ 𝑥̄ 3 ] = 𝑟
⇒ 𝐹58,116,39 = 𝑥1 = 0, 𝑥2 = 1, 𝑥3 = 1.
𝑥̄1 ⋅ 𝑥3 ∨ 𝑥2 ⋅ 𝑥̄ 3 𝑓
1 0 0 − 1 1 (12) 𝐹83,116,78 = 𝑥1 = 1, 𝑥2 = 1, 𝑥3 = 0. ⇒
𝑟
= [0 1 0] ⇔ [ 1 − 0 ]. ⇒ 𝐹92,46,27 = 𝑥1 = 1, 𝑥2 = 0, 𝑥3 = 0.
0 0 1 0 1 − Having examined the obtained
dependencies, we can conclude that the
𝑥̄1 ⋅ 𝑥2 ∨ 𝑥1 ⋅ 𝑥̄ 3 diagonal values of the main elements of the
𝑟
𝐹58,29,78 = [𝑥1 ⋅ 𝑥̄ 2 ∨ 𝑥2 ⋅ 𝑥3 ] = cryptographic transformation operations
𝑥1 ⋅ 𝑥̄ 3 ∨ 𝑥̄ 2 ⋅ 𝑥3
1 0 0 − 1 0 (13) included in the group under study form eight
= [0 1 0] ⇔ [ 1 − 1 ]. variants of operations, that is, 23 variants.
0 0 1 1 0 − Hence, it can be assumed that the basic group
𝑥1 ⋅ 𝑥̄ 2 ∨ 𝑥̄1 ⋅ 𝑥3 of cryptographic transformation operations
𝑓
𝐹92,71,114 = [𝑥1 ⋅ 𝑥2 ∨ 𝑥̄ 2 ⋅ 𝑥3 ] = can be formed only by those elementary
𝑥̄ 1 ⋅ 𝑥3 ∨ 𝑥2 ⋅ 𝑥̄ 3 operations for cryptographic transformation,
1 0 0 − 0 1 (14) in which the value of the main elements along
= [0 1 0] ⇔ [ 1 − 1 ].
0 0 1 0 1 − the diagonal is equal to 23 options [16].
The modified matrix discrete model of the
combination of the permutation matrix and the
complement matrix is described as [16]:
141
𝑥11
𝐹𝑓 =
𝑥̄ 11 ∨ (𝑥22 ⊕ 𝑥33 ) 𝑥11 ∨ (𝑥22 ⊕ 𝑥33 )
5. Experiments
= [ 𝑥̄ 22 ∨ (𝑥11 ≡ 𝑥33 ) 𝑥22 𝑥22 ∨ (𝑥11 ≡ 𝑥33 ) ] (18)
𝑥̄ 33 ∨ (𝑥11 ⊕ 𝑥22 ) 𝑥33 ∨ (𝑥11 ⊕ 𝑥22 ) 𝑥33
An algorithm for a 5-round data
transformation process, shown in Fig. 1, was
where xij are the elementary functions of the
developed to experiment to implement the
cryptographic transformation of forward (Ff)
proposed method of using the synthesized
cryptographic transformation; 𝑖, 𝑗 ∈ {1,2,3}.
basic group of information-driven permutation
From here, a generalized discrete model of
operations to perform cryptographic data
basic groups of encoding operations for
transformation.
cryptographic transformation is obtained,
This algorithm was implemented by us
which is presented in the following form [16]:
programmatically for further research and
𝐹𝑓 =
(𝑥1 ≡ 11 ⋅ (𝑥2 ≡ (𝑥̄ 11 ∨ (𝑥22 ⊕ 𝑥33 ))) ∨
(𝑥 )) analysis of the results of its work. The
∨ (𝑥1 ≡ (𝑥11 ⊕ 1)) ⋅ (𝑥3 ≡ (𝑥11 ∨ (𝑥22 ⊕ 𝑥33 )))
development of this software tool was carried
(𝑥2 ≡ (𝑥22 )) ⋅ (𝑥1 ≡ (𝑥̄ 22 ∨ (𝑥11 ≡ 𝑥33 ))) ∨
out using the high-level object-oriented
= (19) programming language Python.
∨ (𝑥2 ≡ (𝑥22 ⊕ 1)) ⋅ (𝑥3 ≡ (𝑥22 ∨ (𝑥11 ≡ 𝑥33 )))
(𝑥3 ≡ (𝑥33 )) ⋅ (𝑥1 ≡ (𝑥̄ 33 ∨ (𝑥11 ⊕ 𝑥22 ))) ∨
[∨ (𝑥3 ≡ (𝑥33 ⊕ 1)) ⋅ (𝑥2 ≡ (𝑥33 ∨ (𝑥11 ⊕ 𝑥22 )))]
where xij are the elementary functions of the
cryptographic transformation of reverse (Fr)
cryptographic transformation; 𝑖, 𝑗 ∈ {1,2,3}.
Having studied the obtained models of
encoding and decoding functions, represented
by expressions (18) and (19), we can conclude
that the essence of the method of synthesis of
basic cryptographic transformation operations is
to change the values, which allows obtaining
eight basic cryptographic transformation
operations for encoding and decoding functions.
Having determined the essence of the
method for synthesizing the basic operations
of cryptographic transformation, we can
conclude that the synthesis of cryptographic
transformation operations based on the
obtained discrete models is as follows [16]:
1. Synthesis of all basic operations of
cryptographic transformation.
2. For each received operation, it is
necessary to perform a permutation,
which will increase their number by six
times.
3. To increase the number of operations, it is
necessary to use inversion operations,
which will increase the number of
transformation operations by another
eight times.
The result of the computational experiment is
Figure 1: Algorithm of the developed software
384 cryptographic transformation operations
for three-digit elementary functions. The essence of this algorithm consists of the
following. The input data stream is divided into
3-byte blocks, on which the following
operations are performed: addition modulo 2,
cyclic shift, and permutations.
142
The algorithm of the software, which was positions that affect the next state) are [20–
developed by us for conducting an 24]. The result is recorded byte by byte to a file.
experimental study, consists of several steps. For the example shown in Fig. 2, we
Step 1. For each byte of the output message estimate the probability Pcrack of cracking one
sequence, we perform an XOR operation with round of bit permutations for step 3 of the
each byte of the key. The keys are the basic cryptographic transformation using the
operations determined as a result of the formula:
computational experiment. A new key is used 𝑃crack =
𝑡ℎ𝑒 𝑛𝑢𝑚𝑏𝑒𝑟 𝑜𝑓 𝑐𝑜𝑟𝑟𝑒𝑐𝑡 𝑝𝑒𝑟𝑚𝑢𝑡𝑎𝑡𝑖𝑜𝑛𝑠 𝑜𝑓 𝑏𝑖𝑡 𝑝𝑎𝑖𝑟𝑠
.
for each group of three bytes. 𝑡ℎ𝑒 𝑛𝑢𝑚𝑏𝑒𝑟 𝑜𝑓 𝑎𝑙𝑙 𝑝𝑒𝑟𝑚𝑢𝑡𝑎𝑡𝑖𝑜𝑛𝑠 𝑜𝑓 𝑏𝑖𝑡 𝑝𝑎𝑖𝑟𝑠
Step 2. The next step is to apply the cyclic The number of all different permutations of
shift operation. At the same time, the message n bits is 𝑛!. For three bytes, this number is 𝑛! =
from the previous step is split into bytes, each 24! = 6.204484017 × 10²³.
of which is cyclically shifted by 4 bits. For the given example in Fig. 2 probability
Step 3. After performing the cyclic shift Pcrack of cracking one round of bit permutations
operation on the message obtained in step 2, a for step 3 of the cryptographic transformation is
permutation operation by key 2 is applied to 8
𝑃crack = =
each 24 bits of information to perform dispersal 6,204484017 × 10²³
−23
of the statistical structure of the message. 5 such = 1.289390057 × 10 .
rounds of bit permutations in three bytes are Therefore, the probability of cracking one
performed. The process of implementing the round of bit permutations Pcrack is quite low.
permutation is shown in Fig. 2. Further research consisted of the
verification of the developed cryptographic
transformation algorithm. For this, we chose
the most common method of testing the
Figure 2: Bit permutation process for three statistical properties of NIST STS.
bytes Binary sequence testing according to the
NIST STS method has the following order [18-
Step 4. At this step, the Feistel network was
21, 28, 29]:
applied, which is based on two main features of
1. It is assumed that the binary sequence
cryptography [15, 24–27, 30]: substitution and
studied during testing 𝑆=
transposition. The sequence obtained in step 3
𝑆0 , 𝑆1 , . . . 𝑆𝑛−1 is random—the null
is split into blocks of 6 bytes. For each of the 6
hypothesis is accepted 𝐻0 .
bytes, the Feistel network is used, which
2. Test statistics are calculated 𝑐(𝑆).
involves splitting the message into two equal
3. The value of a certain probability is
parts (three bytes each), where leftBlock is the
determined using the test statistics
left part and rightBlock is the right part.
An OR operation is performed for each byte of function, 𝑃 = 𝑓(𝑐(𝑆)), 𝑃 ∈ [0,1].
the sequence contained in rightBlock with each 4. The probability value P is compared with
byte of the key. An XOR operation is performed the significance level 𝛼 ∈ [0.001,0.01].
on the result obtained after conversion with each The null hypothesis is accepted in the
byte contained in leftBlock according to the case of 𝑃 ≥ 𝛼, and in the opposite case, a
Feistel network algorithm. conclusion is made that the alternative
For the value to be placed in leftBlock, the hypothesis is accepted.
bits are permuted according to step 3. The built-in tests included in the NIST STS
The number of rounds of transformation package are shown in Table 2 [18, 20–21].
using the permutation and key for step 4 is 5 Using 16 built-in tests included in the NIST
rounds. STS package [21], 189 probabilities P are
Step 5. Before data recording in the calculated. Therefore, the result of testing is
resulting file, each 24-bit block (leftBlock, the construction of some vector of values of
rightBlock) is scrambled with a Linear calculated probabilities P = {P1, P2, ..., P189}.
Feedback Shift Register word (a 24th-degree These probabilities can be considered as
polynomial) by performing the addition separate results of test calculations.
modulo 2. The feedback polynomial is 𝑥 24 +
𝑥 23 + 𝑥 21 + 𝑥 20 . In our case, the taps (the bit
143
Table 2
List of tests in the NIST STS package
Number Test Name
1 Frequency
2 Block Frequency
3 Runs
4 Long Runs of Ones
5 Binary Matrix Rank
6 Spectral Discrete Fourier Transform
7 Non-overlapping Templates Matching Figure 3: Statistical portrait of the sequence of
8 Overlapping Templates Matching
9 Universal
the technique of using information-driven
10 Lempel Ziv Compression permutation operations
11 Linear Complexity
12 Serial
13 Approximate Entropy
14 Cumulative Sums
15 Random Excursions
16 Random Excursions Variant
As a result of testing according to the NIST
STS method, a statistical portrait is formed, the
form of which is a matrix with dimension m×q, Figure 4: Statistical portrait of the sequence of
where m is the number of binary sequences the DES cryptographic algorithm
being tested, and q is the number of statistical
tests used to test each sequence [18–21]. The
elements of the matrix Pij ∈ [0, 1], where
̅̅̅̅̅̅̅̅
𝑖 =(1, ̅̅̅̅̅̅̅
𝑚) and 𝑗 =(1, 𝑞) are the values of the
probability obtained as a result of testing the ith
sequence by the jth test.
According to the obtained statistical
portrait, a part of the sequences that passed
each statistical test is determined. For this, the Figure 5: A statistical portrait of the sequence
level of significance is set 𝛼 ∈ [0.001,0.01] and of the AES cryptographic algorithm
the probability values exceeding the
established level of significance 𝛼 are
calculated for each of the 𝑞 tests. That is,
determine the coefficient [18–20]:
#{𝑃𝑖𝑗 ≥ 𝛼|𝑖 = 1, 𝑚}
𝑟𝑗 = .
𝑚
As a result, a vector of coefficients is formed,
the elements of which characterize in Figure 6: A statistical portrait of the sequence
percentage the passage of the sequence of all of the Blowfish cryptographic algorithm
statistical tests.
The properties of pseudo-random
sequences formed using the developed
software tool were studied in the environment
of statistical tests NIST STS. Statistical portraits
of software implementations of the method of
using information-driven permutation
operations and DES [17, 24], AES [26–27],
Blowfish [14, 29–30] algorithms, and linear Figure 7: A statistical portrait of the sequence
feedback shift register are shown in Figs. 3–7 of scrambling with a LFSR word (𝑥24 + 𝑥23 +
respectively. 𝑥21 + 𝑥20 polynomial)
To design the test, the following parameters
were selected [28–29]:
144
1. The sequence length for testing: n = 106
bits.
2. Number of sequences that are tested:
m = 100.
3. number of tests q = 189.
Thus, the tested sample size was
N = 106100 = 108 bits, and the number of tests
(q) for different lengths q = 189, thus, the
statistical portrait of the generator contains Figure 9: A statistical portrait of Kalyna (mini-
18900 probability values Р. version) cipher
In the ideal case, with m = 100 and = 0.01, According to the NIST STS testing
only one sequence out of a hundred can be methodology, the Kalyna cipher and mini-
rejected during testing. The pass rate for each version of Kalyna cipher have the following
test must be 99%. But this is too strict a rule. results: the number of passed statistical tests
Therefore, a rule based on the confidence according to the 𝑃𝑖 ≥ 0.96 criterion is 187 and
interval is applied. The lower limit is 0.96015. 186 respectively, and according to the 𝑃𝑗 ≥
The structure and common basic 0.99 criterion, it is 132 and 135 respectively.
transformations of the Kalyna block cipher are Let’s enter the data obtained in [6–8]
given in the papers [6–8]. A cipher mini-model regarding the statistical testing of the Kalyna
is developed by scaling common cipher cipher into Table 3 for a comparative analysis.
cryptographic transformations with the The work [9] presents the results of
preservation of their algebraic structure in experimental studies of statistical security and
paper [7]. The developed mini-model [7] is speed characteristics of modern stream
intended to study the general characteristics of ciphers, in particular the Strumok cipher. In
the cipher and was used for statistical testing [11], a study of the statistical properties of
by the NIST STS package. standardized stream encryption algorithms
The paper [8] is devoted to the study of was carried out and a detailed description and
statistical properties of crypto-algorithms by analysis of the obtained test results was given.
the Suite NIST STS and contains images of The data obtained in the course of
statistical profiles of algorithms, in particular, experimental studies in works [9, 11] can be
the Kalyna block cipher. We will use the used to conduct a comparative assessment of
experimental data obtained in works [6–8] the statistical properties of the modern
regarding the study of the statistical properties Strumok cipher and the software
of the Kalyna block cipher by using the Suite implementation of the method of applying in
NIST STS for comparison with the results of the formation-driven permutation operations
research of other algorithms obtained by us. developed by us. Statistical portraits of the
Statistical portraits of the results of the Kalyna results of the Strumok cipher with a key length
cipher are shown in Figs. 8–9 (Kalyna mini- of 256 and 512 bits are shown in Figs. 10–11
version) [6–8]. respectively [11]. According to the NIST STS
testing methodology, the Strumok cipher with
256- and 512-bit keys has the following
results: the number of passed statistical tests
according to the 𝑃𝑖 ≥ 0.96 criterion is 186 and
187 respectively, and according to the 𝑃𝑗 ≥
0.99 criterion, it is 130 and 133 respectively.
Let’s enter the data obtained in the paper
[11] regarding the statistical testing of the
Strumok cipher into Table 3 for comparative
Figure 8: A statistical portrait of Kalyna cipher analysis.
145
Thus, we can conclude that the developed
algorithm for implementing the proposed
technique of using information-driven
permutation operations is suitable for
cryptographic data transformation.
The method of researching the efficiency of
implementation of cryptographic
Figure 10: A statistical portrait of Strumok- transformation by a certain algorithm
256 cipher presupposes the definition and analysis of
encryption speed as one of the main indicators
used when comparing cryptographic
algorithms. The main requirement for speed
measurement is the measurement of the
encryption speed indicator of the same volume
of open texts (in different modes) for all
possible combinations of block size and key
length under the same conditions within one
Figure 21: A statistical portrait of Strumok- interactive process of the user of the operating
512 cipher system. To ensure the same conditions for
Table 3 shows the results of testing the measuring encryption speed and for further
sequences formed based on the applied qualitative assessment, it is necessary to
algorithms for comparison. consider that the results of speed testing of
As can be seen from the results, the cryptographic algorithms are directly related to
generator based on the algorithm developed by the technical characteristics of the selected
us passed the comprehensive control over the hardware and software platform. In addition,
NIST STS methodology and has acceptable speed indicators and their ratio for different
results compared to other generators (Table 3). ciphers can change significantly depending on
the compiler version. More often, speed
Table 3 comparisons are performed for software
Sequence test results implementations of cryptographic algorithms.
Number of tests, in which tests Usually, specially developed optimized versions
have passed
Generator
99% 96%
of the software implementation of the
sequences sequences investigated cryptographic algorithms are used
An algorithm that 138 (73.4%) 185 (98.4%) to evaluate the encryption speed to obtain the
implements a technique highest possible indicators.
of using information-
driven permutation It should be noted that the work carried out
operations a software implementation of the developed
DES 145 (77.1%) 186 (98.9%) algorithm of one of the methods of
cryptographic data transformation based on the
AES 138 (73.4%) 177 (94.1%) use of a basic group of permutation operations
controlled by information containing only 25
Blowfish 129 (68.6%) 186 (98.9%) basic synthesized groups to check the
Scrambling with a linear 150 (79.8%) 185 (98.4%)
possibility of using similar operations when
feedback shift register constructing cryptographic primitives. Since at
word (𝑥 24 + 𝑥 23 + 𝑥 21 + this stage of the study, the development of an
𝑥 20 polynomial)
optimized software version of the
Kalyna 132 (70.2) 187 (99.5)
implementation of the proposed method for
applying information-driven permutation
Kalyna (mini-versions) 135 (71.8) 186 (98.9) operations for the implementation of
cryptographic data transformation was not
Strumok-256 130 (69,1%) 186 (98.9%) carried out, then, accordingly, there is no
Strumok-512 133 (70,7) 187 (99,5%)
possibility to evaluate the speed of the
developed algorithm for comparison with other
algorithms. Investigation of the implementation
146
of this method and the effectiveness of its use in Subsequent research should be directed to
terms of encryption speed is planned to be the study of operations of permutations
carried out in further studies. controlled by information of greater capacity,
as well as the use of the full set of basic groups
6. Conclusions of synthesized operations, which will provide
an increase in the number of transformation
operations and the possibility of processing
The paper proposes one of the techniques of
data blocks of greater length. In addition, it is
using information-driven permutation
necessary to study and evaluate in more detail
operations for the implementation of
such parameters of the cryptographic
cryptographic data transformation. The study
transformation algorithm as cryptographic
of the implementation of this technique and
strength, avalanche effect, and speed.
the effectiveness of its use was carried out only
on the example of 25 basic synthesized groups
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