This article was addressed to the problem of improving the quality of low-resource systems for cryptographic information transformation. The existing contradictions between the limits on software and hardware implementation and the strength of the cryptographic algorithms can be partially eliminated by applying a set of groups of symmetric operations of cryptographic encoding. Unfortunately, to date, the results of studying symmetric two-bit two-operand operations of cryptographic encoding have a limited non-systemic nature. The lack of the unified generalized method for synthesizing groups of symmetric multi-bit twooperand operations of cryptographic encoding makes it impossible to use the full potential of the practical application of these operations. Thus, the aim of this article is to create a unified method for synthesizing groups cryptographic encoding. Achieving this aim will make it possible to significantly increase both the variability of lightweight low-resource cryptographic strength directly related to it. A new concept of synthesizing groups of operations is proposed, which will make it possible to address the main shortcoming of the previous concept in which each method is based on its splitting into suboperands. Addressing this shortcoming will also make it possible to synthesize both symmetric two-bit two-operand operations of cryptographic encoding and symmetric two-operand operations of arbitrary bitness. Having been applied a new concept, it was synthesized a new previously unknown group of symmetric two-bit two-operand operations up to permutation. A symmetric group of matrix three-bit two-operand operations of cryptographic encoding was synthesized. The developed method for synthesizing information security, cryptography, information encoding, operations of cryptographic encoding, synthesis of operations of cryptographic encoding CPITS-II-2021: Cybersecurity Providing in Information and Telecommunication Systems, October 26, 2021, Kyiv, Ukraine
operations of cryptographic encoding, belonging to different mathematical groups, provides an increase in the variability and strength of cryptographic algorithms.
Lightweight and low-resource cryptography are designed to implement cryptographic algorithms
on devices with limited technical resources. The relevance of this direction and significant interest in
its development are directly related to the expansion of the scope of using the cryptographic
information protection [
The vast majority of recently developed lightweight ciphers are symmetric block ciphers [
Rudnytskyi);
2022 Copyright for this paper by its authors.
algorithms are used [
One of the ways to partially overcome the conflict between the simplification of the cryptographic algorithm and its complexity can be considered increasing the algorithm's variability through the use of cryptographic information coding operations. 2. Overview of the Work-Related Publications
Symmetric two-operand operations of cryptographic encoding can be used to expand the variability of symmetric block ciphers. These operations can change both within the encryption round and when its shift.
In essence, a two-operand operation of cryptographic encoding is a model of an interconnected
group of lookup table sets for the first operand. Determining the lookup table that implements the
transformation is defined by the value of the second operand [
Despite the great practical possibilities for application in lightweight cryptography, symmetric two-operand operations of cryptographic encoding have almost not been studied. There are only partial studies' results of symmetric two-bit two-operand operations of cryptographic encoding.
In [
Table 1 represents two-bit two-operand operations of cryptographic encoding of the first group, 11 − 214 and their interrelation with one-operand operations (F1, F2, F3 – basic operations, F3, F4, F5 0 0 1 1 – operations of permutations, , , , – operations of inversion). In the mathematical 0 1 0 1 models of these operations, it is designated x1 – x2 values of the first operand's first and second bits, k1 – k2 values of the second operand's first and second bits.
The synthesis of the represented operations is based on using a group of one-operand two-bit operations of cryptographic encoding represented in the Table 2.
The method for synthesizing the first group of two-bit two-operand operations based on the
operation of bitwise modulo two addition for symmetric stream cipher is as follows [
x1 k1 x1 k1 О2 О1*2 О2*2 x2 k2 x2 k2 (1) where О2 is the operation of bitwise modulo two addition; О1*2 and О2*2 are the suboperations of processing the first and the second operands, respectively; О41 x2 k2
x1 k1 О116 x2 k2 1
x1 k1 О51 xx12xk22 k1 k2
О117 x2 k2 1 x1 x2 k1 k2
О51 x1 x2 k1 k2
x1 k1 О118 x1 x2 k1 k2 1 x1 k1
x2 k2 1
x2 k2 1 О81 xx12 xk22 1k1 k2 О210 x1 x2 k1 k2 1
x2 k2 1 О91 xx11 kx12 k1 k2 1
О211 x1 k1 1 x1 x2 k1 k2 1
О110 x2 k2
x1 k1 1 О212 x2 k2 1
x1 k1 1 О111 x2 k2
x1 x2 k1 k2 1 О213 x2 k2 1 x1 x2 k1 k2 1
О112 x1 x2 k1 k2
x1 k1 1
О214 x1 x2 k1 k2 1
x1 k1 1
a basic group of two-operand operations is being synthesized based on the transforming the
suboperations of processing the first and the second operands by basic one-operand operations
F1 x2 , F2 x1 x2 , and F3 x1
x1
x2 x1 x2 [
x1 x2 k1 k2 x1 x2 k1 k2 О21 F21(О1*2) F22(О2*2) x2 k2 x2 k2
Basing on the modulo four addition, the second group of symmetric two-bit two-operand
operations of cryptographic encoding was synthesized [
x2 1 x1 x2 1 F20 x2 1
x1 1 F21 x1 x2 1
x2 1 F22
x1 1 x2 1 F23 x1 x2 1
x1 x2 1 F24 x1 1 Two-bit two-operand operations of cryptographic encoding, synthesized on the basis of left-handed modulo four addition (second group of operations)
x1 x2 k2 k1 k2 1 О223 x2 k2 1 x1 x2 k2 k1 k2 1
О122 x1 x2 k2 k1 k2
x1 x2 k2 k1 1 О224 x1 x2 k2 k1 k2 1 x1 x2 k2 k1 1 x1 x2 F6 x1 О128 x1 x2 k2 k1 k2 1 О62 x1 x2 k2 k1 k2
x1 x2 k2 k1 x1 x2 k2 k1
Synthesizing the second group of two-bit two-operand operations based on the modulo four
addition (left-handed modulo four addition) for symmetric stream cipher is implemented similarly to
synthesizing the first group of operations with the following differences [
x2 k2 x2 k2
where О4 is the operation of left-handed modulo four addition; О1*4 and О2*4 are the
suboperations of processing the first and the second operands, respectively;
the synthesis of the basic two-operand operation through transforming the suboperation of
processing the first and the second operands is implemented as [
k1 x2 k2 k2
The processes of further synthesizing the operations of the first and the second groups of operations coincide.
Synthesizing the third group of symmetric two-bit two-operand operations of cryptographic
encoding based on right-handed modulo four addition is considered in [
Synthesizing the third group of symmetric two-bit two-operand operations based on the operation
of right-handed modulo four addition is implemented similarly to synthesizing the first and the second
groups of operations [
x2 k2 x1 k1 x2 k2 x1 k1 where О4 is the operation of right-handed modulo four addition. (2) (3)
Synthesizing the basic group of symmetric two-bit two-operand operations through the operation
of right-handed modulo four addition is implemented as [
The processes of further synthesizing the operations of all three groups of operations coincide.
In [
x1 k1 (x1 x2 ) (k1 k2 )
x2 k2 (x1 x2 ) (k1 k2 )
The results of a computational experiment [
The operation
(5)
(6)
(4)
It turned out to be difficult to apply the considered synthesis methods [
f1 (x1, k1) f1*(x1) f1*(k1) О О1* О2*
f2 (x2 , k2 ) f2*(x2 ) f2*(, k2 ) where f1 , f 2 are the elementary functions for obtaining the first and the second bits of the result, respectively; f1* , f 2* elementary functions for processing the corresponding bit of the suboperand.
The operation О11 (1) is correctly decomposed into suboperands in accordance with (5). Operations О12 (2) and О13 (3) are conditionally decomposed into suboperands, since a complete decomposition in accordance with (5) has not been obtained. Decomposition of the operation О14 (5) into suboperands, causes even more difficulties. At the same time, it should be taken into account that each method is based on its decomposition into suboperands. To address the noted shortcomings, it is necessary to change the synthesis concept of the operations' groups:
If О then Оi Fi (O) Fi f1 (x, k ) f1 (x, k ) f2 (x, k ) f2 (x, k ) .
To reduce the synthesis complexity, it is advisable to synthesize only the operations of the base group through this concept, and the rest to obtain by permutations and inversions, which is implemented in the prototype methods.
By applying the new concept, a group of operations up to the permutation accuracy is being synthesized, on the basis of the operation О14 . The synthesis results are represented in the Table 6. О244 x1 k1 x2 k2 1 x1 k1 (x1 x2)(k1 k2)1
The obtained synthesis results coincided with the results of the computational experiment, in
which the truth tables of symmetric two-bit two-operand operations of cryptographic encoding were
modeled and sorted [
The proposed concept makes it possible synthesizing symmetric two-operand operations of arbitrary bitness, besides the symmetric two-bit two-operand operations of encoding. To do this, in concept (6), it is necessary to expand the number of bits: . . . . f1 (x1 , x2 ,.., xn , k1 , k2 ,..,kn ) f n (x1 , x2 ,.., xn , k1 , k2 ,..,kn ) If О f 2 (x1 , x2 ,.., xn , k1 , k2 ,..,kn ) then Оi Fi (O) f1 ( x1 , x2 ,.., xn , k1 , k2 ,..,kn ) f 2 ( x1 , x2 ,.., xn , k1 , k2 ,..,kn ) . (7)
Based on this suggestion (7), the algorithm of the method for synthesizing symmetric two-operand operations of cryptographic encoding can be represented as follows:
Based on a complete iterating over n-bit one-operand operations of the base group, applying the concept (7), it is synthesized symmetric two-operand operations of cryptographic encoding of the base group;
Having performed the operations of elementary functions permutation over the two-operand operations of the base group, it will be obtained the extended group of symmetric operations;
Having performed the operations of inverting the preliminary transformation results over the two-operand operations of the extended group, it will be obtained the complete group of symmetric two-operand operations of a given bitness.
On the example of synthesizing symmetric two-bit two-operand operations, the proposed method
provides the synthesis of a complete group of operations (24 operations) based on an arbitrary
operation from this group.
4. Implementation of the Method for Synthesizing Symmetric Two-Operand
Operations of Cryptographic Information Encoding
Consider the synthesis of symmetric three-bit two-operand operations of cryptographic encoding.
The number of one-operand operations of cryptographic encoding is being determined [
1( ) = 2 ! 1( ) = об( ) ∙ оп( ) ∙ ои( ) = об( ) ∙ ! ∙ 2 where n is the operation’s bitness, об( ), оп( ) = !, ои( ) = 2 operations, operations of permutations and operations of inversion, respectively. are the number of basic
Based on the expressions (7) and (8), the number of two-bit one-operand operations of cryptographic encoding is determined as: 1(2) = 4! = 24;
1(2) = об(2) ∙ 2! ∙ 22 = 3 ∙ 6 ∙ 4 = 24.
Consequently:
Since, according to the results of the experiment, there are 96 symmetric two-bit two-operand operations, and they make up 4 groups of 24 operations, it can be assumed that 2(2) = 96 = 4 ∙ 22!. where k is the groups' number of symmetric n-bit two-operand operations of cryptographic encoding.
The number of operations in each group of symmetric three-bit two-operand operations in accordance with (9) is determined: 2(3) = ∙ 23! = ∙ 8! = ∙ 40320 and is 40320 operations.
In practice, at the time, it is not possible to synthesize a group from such a number of operations.
This is due to the lack of the unified mathematical apparatus that makes it possible to simulate the
entire set of three-bit one-operand operations [
In accordance with [
It is synthesized a basic group of symmetric three-bit two-operand matrix operations of cryptographic encoding based on the operation
The developed method for synthesizing symmetric two-operand operations of cryptographic information encoding provides an opportunity of increasing the variability of lightweight cryptographic algorithms. In addition, synthesizing symmetric operations of cryptographic encoding which belong to different mathematical groups increases the cryptographic strength of the algorithm. The application of two-operand operations of cryptographic encoding, to which the synthesized
x3
x1 x3 F4 x2
x3 x3 k3 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
x1 k1 x3 k3 (x1 x2 ) (k1 k2 )
О4 x2 k2 (x1 x2 ) (k1 k2 )
operations are related, leads to a slight increase in the complexity associated with the implementation
of the operations' synthesis both at the hardware and software levels [
1. For improving the quality of low-resource cryptographic systems, it was proposed to apply groups of symmetric operations of cryptographic information encoding.
2. For combination of the existing and new results of studying groups of symmetric operations of cryptographic encoding, a new concept for synthesizing operations was proposed.
3. A new method for synthesizing groups of symmetric multi-bit two-operand operations of cryptographic encoding has been developed to increase the variability of lightweight low-resource cryptographic algorithms.
4. Having been applied this method, a new, previously unknown group of symmetric two-bit two-operand operations of cryptographic encoding has been synthesized. For the first time, a symmetric group of matrix three-bit two-operand operations of cryptographic encoding has been synthesized.
5. The obtained results of operations' synthesis coincided with the results of a computational experiment by simulation of the obtained operations.
6. The application of two-operand operations of cryptographic encoding, synthesized on the basis of this method, leads to a slight increase in the implementation complexity of the cryptographic algorithm both at the hardware and software levels.
7. The proposed concept and the developed method for synthesizing symmetric multi-bit twooperand operations of cryptographic encoding provide the construction of operations belonging to different mathematical groups. Applying operations from different mathematical groups provides an increase in both the variability and the encryption strength.