=Paper=
{{Paper
|id=Vol-3646/Paper_19.pdf
|storemode=property
|title=Modeling of a Cryptographic Protocol for Matching a Shared Secret Key-Permutation of Significant Dimension with its Isomorphic Representations
|pdfUrl=https://ceur-ws.org/Vol-3646/Paper_19.pdf
|volume=Vol-3646
|authors=Volodymyr Saiko,Vladimir Krasilenko,Svitlana Kiporenko,Illia Chikov,Diana Nikitovych
|dblpUrl=https://dblp.org/rec/conf/iti2/SaikoKKCN23
}}
==Modeling of a Cryptographic Protocol for Matching a Shared Secret Key-Permutation of Significant Dimension with its Isomorphic Representations==
https://ceur-ws.org/Vol-3646/Paper_19.pdf
Modeling of a Cryptographic Protocol for Matching a Shared
Secret Key-Permutation of Significant Dimension with its
Isomorphic Representations
Volodymyr Saiko 1, Vladimir Krasilenko 2, Svitlana Kiporenko 2, Illia Chikov 2 and Diana
Nikitovych 2
1
Taras Shevchenko National University of Kyiv, 60 Volodymyrska Street, Kyiv, 01033, Ukraine
2
Vinnytsia National Agrarian University, st. Sonyachna, 3, Vinnytsia, 21008 Vinnytsia Oblast, Ukraine
Abstract
A protocol for agreement by user parties of secret keys-permutations of significant dimension
and their new isomorphic matrix representations is proposed. Features and advantages of such
representations are considered. The need to create such secret permutation keys to improve the
cryptographic stability of matrix affine-permutation ciphers and other cryptosystems of the
new matrix type is well-founded. The results of modeling the basic procedures of the proposed
key agreement protocol in the form of an isomorphic permutation of a significant dimension,
namely the processes of generating permutation matrices and their degrees, are given. Model
experiments of the protocol as a whole, including accelerated methods of raising permutations
to significant degrees, were performed. Such methods use sets of fixed permutation matrices,
which are degrees of the underlying permutation matrix, and all these matrices are given in
their isomorphic representations. The values of the fixed exponents correspond to the
corresponding weights of the digits of the binary or other code representations of the selected
random numbers. The results of simulation modeling demonstrated the adequacy and
advantages of isomorphic representations of the processes of functioning of matrix-algebraic
models of cryptographic transformations and the proposed secret key-permutation agreement
protocol.
Keywords 1
Matrix-algebraic model, matrix representations, isomorphic permutation key, cryptogram,
cryptographic transformations, affine-permutation cipher, protocol, matrix-type cryptosystem.
1. Introduction, overview and analysis of publications
Introduction. Generalization of known cryptosystems [1-14] with scalar-type data formats to the
cases of matrix-tensor formats, emergence and research of a new class of matrix-type cryptosystems
(MTC) [15-18] based on their matrix-algebraic models (MAM) of cryptographic transformations (CT)
2D (3D) - arrays, images (Is), which have a number of significant advantages, contributed to the
intensification of MTC, MAM research and the demonstration of a number of new improvements and
applications [11-14, 16, 18, 19-21]. MAMs in their hardware implementations are more easily displayed
on matrix processors, have extended functionality, improved crypto-resistance, allow checking the
integrity of cryptograms of black and white, color images [16, 18-20], and the presence of distortions
in them [16], create block ones [17], parametric [18], multi-page [18] models with their significant
stability [16, 18]. Secret key generation protocols for known non-matrix type ciphers were considered
in [2, 6, 12, 22-29], and for matrix type ciphers were partially considered in our previous works,
including in works [30, 31], where some improved matrix modifications of known key matching
protocols were proposed. Generalized MAM, matrix affine and affine-permutation ciphers (MAPCs),
Information Technology and Implementation (IT&I-2023), November 20-21, 2023, Kyiv, Ukraine
EMAIL: vgsaiko@gmail.com (A. 1); krasvg@i.ua (A. 2); kiporis11@ukr.net (A. 3); ilya95chikov@gmail.com (A. 4);
diananikitovych@gmail.com (A. 5)
ORCID: 0000-0002-3059-6787 (A. 1); 0000-0001-6528-3150 (A. 2); 0000-0001-5045-5052 (A. 3); 0000-0002-2128-5506 (A. 4); 0000-0002-
8907-1221 (A. 5)
©️ 2023 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
196
their modifications were studied and used in the creation of blind and other improved digital signatures
in [18]. For CT in matrix models of permutations (MM_P), with their basic procedures of matrix
multiplication and some other element-by-element modulo operations on matrices, byte matrices
formed from rows, columns, vectors, which in unitary or other codes display symbols, codes, bytes,
must be multiplied by the permutation matrix (PM). Procedures for rearranging bits, bytes or their
groups are the most common and mandatory for almost all known and newly created algorithms and
ciphers. To increase the entropy of cryptograms images with their CT based on MM_P and change their
histograms, the decomposition of R, G, B components and their bit slices and several matrix keys (MKs)
of the PM type are necessary [16, 18, 21, 30]. A number of such pseudo-random (current, step-by-step,
frame-by-frame) MKs, which would meet the requirements and be quickly generated, is also needed
for masking, CT of video files or stream of blocks from files, images with their significant sizes.
Formulation of the problem. Thus, there is a need for the MAM to form a number of MKs of the
PM-type that would satisfy a number of requirements from the main MK. Since the issue of matching
the main MK (MMK) of a general type, but not the sequence of PMs, was considered in [30, 31], and
the methods of generating a stream of MKs-permutations from the main MK were partially considered
in [31], but only for bit MPs of small sizes (256*256), then the purpose of the work is to propose and
investigate a protocol for the coordination of a secret (main) MK in the form of an PM of significant
dimensions, i.e., an main PM (MPM), to improve and adapt the type and structure of a MPM of such
or even greater dimensions to the images format and to fast hardware solutions, to model this protocol
and the process of formation flow of PMs from such a MPM for MAM CT in MT systems. In addition,
the above review and analysis of publications allows to determine another important task, namely the
need to develop and model such MAM CTs, which would be best suited for implementation based on
vector-matrix multipliers (VMMs), as well as to determine the characteristics and indicators of such
models and implementations.
2. Presentation of the main material and research results.
An overview of MT ciphers, especially multifunctional parametric block ciphers [17], their analysis
shows that it is advisable to use isomorphism of various representations of permutations (matrices or
vectors) that act as a master key (MK) and block or step-by-step, round MKs to achieve the goal of PM-
type, i.e. sub-keys (SKs), which are matrices of permutations of P (its powers!) or vectors isomorphic
to them. It is known from the works [15, 16, 17, 18] that with CT based on the basis of matrix affine-
permutation ciphers (MAPCs) and vector affine-permutation ciphers (VAPCs), cryptograms for some
types of text-graphic documents (TGD) and images (I), especially for block-based MAMs, when using
one personal computer (PC) for all blocks are insufficient in terms of stability, however, a number of
PCs created from MPM solve this problem. And that is why the aspect of coordinating the secret MPM
of the PM-type with a significant dimension is important. Let's consider the situation when for M blocks
with a length of 256*256 bytes, presented in the form of a matrix of a black and white image, it is
necessary to rearrange all bytes in accordance with PM. In this case, PM in the generally accepted form
should be square with N*N elements ("0" or "1"), where N=216=65536. The power of the set of possible
such PMs, i.e. their number, is estimated as N!=65536!, which gives colossal values for this N.
But each byte address of the block can be represented by two bytes indicating two coordinates (row
and column) of the block. This gives us the opportunity to represent any permutation with two blocks
(256*256 elements) of bytes, setting in each identical address of these blocks the corresponding senior
byte (in the first block) and junior byte (in the second block) coordinates of the new address of the byte
selected for permutation. The view of the software module in Mathcad for generating the basic (main)
MK (PM) and the view of its components KeyA and KeyB in the format of two black and white images
is shown in Fig. 1. Therefore, any PM can be uniquely represented by two matrices of size 256*256,
the elements of which take values from the range 0-255, with the peculiarity that each of their 256
gradations of intensity in each of these two matrices (images) is repeated exactly 256 times. The
histograms of KeyA and KeyB PM components are shown in Fig. 2 and have the form of horizontal
lines, as expected. We note that such an isomorphic representation of the PM in the form of two images
gives us the opportunity to use these components KeyA and KeyB as two secret MCs of a general type,
for example, as additive and multiplicative keys in the MAPCs or other MAMs. This is evidenced by
the results of the simulation of the CT image (Im) of the MAPC using the proposed PM and its
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components, as keys, shown in Fig. 3 with the matrices of explicit image (Im), intermediates, its
cryptograms (Cmap) and verifiable images [31]. And the histograms of explicit image, its cryptograms
after each CT with affine components of this PM are shown in Fig. 2.
Figure 1: Software module for generating the basic (main) MK (PM) and the view of KeyA and KeyB
components in the format of two black and white images (Mathcad window).
500
H_CD mg 0.05 400
H_KeyA mg 0.95
300
H_KeyB mg
H_CDa mg 0.5 200
H_CDm mg 0.75
100
0
0 50 100 150 200 250 300
intmg
Figure 2: Histograms H_KeyA and H_KeyB, respectively, of the components KeyA and KeyB of PM, the
histogram H_CD of explicit image, the corresponding histograms H_CDa and H_CDm of the
cryptogram of the image after additive and multiplicative affine transformations of this image using
the same KeyA and KeyB (Mathcad window).
Figure 3: The results of the simulation of MAPC based on PM and its components, as additive and
multiplicative MKs. Top row, from left to right: explicit image, after transformations, cryptogram of
image after MAPC; bottom row: reconstructed, intermediate and difference (right) images of TGD.
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Figure 4: View (2D) of known generated PMs: top (forward), bottom (reverse) permutations.
Figure 5: Program modules (copies from Mathcad) displaying the procedure of iterative permutations
of the initial permutation matrix PM, isomorphic to the elevation of the permutation matrix PM to
the required power (11 !) by side x (Alisa).
These model experiments confirmed that the CT MAPC with the existing 2 components of the PM
give high-quality cryptograms CD_ImAa and CD_ImAm, whose histograms H_CDa and H_CDm are
so close to the uniform distribution law that even for image (Im) with an entropy of 0.738, the entropy
of cryptograms differs from the theoretical maximum (8 bits) by just a fraction of a percent, going all
the way up to 7.99. The results of the simulation of the MAPC and multi-step MAPC for different cases,
when the components of affine transformations are first performed in a different sequence and with
different or one MK from the PM, and then permutation using the PM, or vice versa, also proved similar
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qualitative CTs, when applying the proposed representations of the PM. But for all modifications of the
MAM with such PMs, the power of the set of which is estimated by a significant value N! = (256*256)!,
the issue of agreeing the session secret MPM is paramount.
Here is an analogy with the Diffie-Hellman protocol. In Fig. 5-8 show the results of modeling these
two steps of the protocol for the agreement of the secret MC in Mathcad, and Fig. 9-10 shows the
obtained intermediate and resulting secret MPM in the isomorphic representation of images. The parties
do not know the degrees of the other party, but the MPs obtained by them are identical, which can be
seen from Fig. 10. In this way, raising MPMs (N*N binaries, where N=216 !) to a power is equivalently
replaced by fast permutations, which, moreover, can be even more accelerated for significant powers
due to the use of some basic set of fixed (fixed powers of MPM) and their specific sequence, which
provides significant advantages due to the acceleration of the calculation of degrees of MPM, the
simplicity of possible implementations and the reduction of costs.
Figure 6: Program modules (copies from Mathcad) displaying the procedure of iterative
permutations of the initial permutation matrix PM, isomorphic to the elevation of the permutation
matrix PM to the required power (17 !) by side y (Bob).
In accordance with the MP protocol, values of significant dimensions must be multiplied many
times, that is, raised to a power. And the degrees to which the parties raise these isomorphically
presented MPs must be significant enough to ensure the necessary crypto-resistance against random
attacks. Therefore, taking into account the necessity and expediency of using the above-mentioned
accelerated methods of raising matrices to a power, we show an adequate isomorphic transformation of
this procedure into some sequence of fixed permutations.
Depending on the code in which the value of the degree is given, appropriate permutations are
selected from the formed set of fixed MPs, the degrees of which correspond to the corresponding
weights of the digits of the binary or other code representations of the selected random numbers: xc
(Alisa) and yс (Bob). The results of these simulations, the corresponding formulas, procedures, key
fragments are shown in Fig. 11-12. A comparison of matrix elements in Fig. 12 highlights their equality.
Using the developed functional parametric models of the CT with the help of a secret MK (PM),
agreed with the proposed protocol, shown above, a check of the correctness of their synthesis and
adequacy of the models was performed by means of direct and reverse CT image, which was shown in
Fig. 1-3. The results obtained by modeling in Mathcad confirm the correctness of the protocol, and the
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stability analysis, which will be presented in more detail in the report, shows the impossibility of attacks
due to the huge number of possible PMs.
Figure 7: Program modules (copies from Mathcad) reflecting the procedure of iterative permutations
in the new PM obtained from y, isomorphic to the elevation to the required power (11 !) by side x
(Alisa).
Figure 8: Program modules (copies from Mathcad) reflecting the procedure of iterative
permutations in the new PM obtained from x, isomorphic to the elevation to the required power
(17 !) by side y (Bob).
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Figure 9: New PMs received by the parties (each in the form of their two components) after the
first step of the protocol, those that are forwarded to the other party.
Figure 10: The participants of the session received identical new PMs (each in the form of their
two components) after the second step of the protocol, i.e. essentially one secret PM.
Although the initial MPM is known to both parties, the protocol allows without knowledge of the
secret degrees being chosen sides, form a secret key, PM in a similar isomorphic form in a time
proportional to the number fixed permutations. In addition, stability analysis taking into account the
power of the set formed by this the protocol of the relevant PM of significant dimensions showed the
impossibility of carrying out attacks as a result of a huge set of possible MPs, which is estimated by the
value (216)!
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Figure 11: Formulas and procedures (copies from Mathcad windows) used for modeling isomorphic
formation accelerated processes of degrees of matrix permutations by sides.
Figure 12: Fragments of the keys formed after the second step, which testify to the adequacy of
the accelerated algorithms of the isomorphic formation of degrees of matrix permutations by
sides.
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3. Conclusions
The relevance and necessity of creating secret permutation keys to increase the cryptographic
stability of matrix affine permutation ciphers and other cryptosystems of the new matrix type are
substantiated. A protocol for agreeing a secret key in the form of isomorphic representations of
permutation matrixs of significant dimensions was proposed, model experiments were performed that
confirmed the adequacy of the functioning of the models and the proposed protocol and methods of
permutation matrixs generation, their advantages. The models are simple, convenient, adaptable for
various format and color images, implemented by matrix processors, have high efficiency, stability, and
speed.
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