=Paper=
{{Paper
|id=Vol-2608/paper51
|storemode=property
|title=Cryptosystem based on a key function of a real variable
|pdfUrl=https://ceur-ws.org/Vol-2608/paper51.pdf
|volume=Vol-2608
|authors=Viktor Avramenko,Volodymyr Demianenko
|dblpUrl=https://dblp.org/rec/conf/cmis/AvramenkoD20
}}
==Cryptosystem based on a key function of a real variable==
https://ceur-ws.org/Vol-2608/paper51.pdf
Cryptosystem Based on a Key Function of a Real
Variable
Viktor Avramenko1[0000-0002-6317-6711] and Volodymyr Demianenko2[0000-0002-1512-970X]
1,2
Sumy State University, Rimsky-Korsakov st., 2, Sumy, 40007, Ukraine
1
avramenko1938@gmail.com, 2vldemyan@gmail.com
Abstract. Using the function of a real variable in cryptosystems as a key allows
you to increase its cryptographic strength, because it is more difficult to pick up
such key. Therefore, the development of such systems is relevant. A cryptosys-
tem with a symmetric key is offered. This key is some function of a real vari-
able that satisfies some restrictions. It can be either continuous or discrete.
The transmitting and receiving parties select the key-function, the first
transmitted character or the first transmitted value for the analog message, the
function area of the key function, and the step of changing the function argu-
ment. A Disproportion over first-order derivative is used to encrypt an analog
message.
The Cauchy problem is solving for decrypting this message. Discrete mes-
sages are encrypted using the first-order disproportionality integral function.
Decryption is performed by the inverse transformation of the formula for inte-
gral disproportion.
Algorithms for encrypting and decrypting messages are presented. The abil-
ity to encrypt and decrypt text information, 2D graphic images, as well as ana-
log messages are shown. The examples show the complexity to pick up the key
function and the cryptographic strength of the proposed cryptosystem.
A cryptosystem, in which the function of a real variable is used as a key and
as well as disproportion functions are used, is suitable for encryption of both
discrete and continuous messages. To “crack” such a system, it is required to
pick up the form of the key function and to find the values of its parameters
with very high accuracy. That is, the system has high cryptographic strength.
Keywords: cryptosystems, disproportion functions, real variable func-
tions, key-function, encryption, decryption, text messages, 2D images,
analog messages.
1 Introduction
It is difficult to imagine the modern world without systems of secrecy of the transmit-
ted information in both the military and business fields. Therefore, it is not surprising
that they are constantly being improved. In addition, new ways of hiding information
are emerging. The most widely used encryption algorithms are divided into symmet-
Copyright © 2020 for this paper by its authors. Use permitted under Creative
Commons License Attribution 4.0 International (CC BY 4.0).
ric and asymmetric. In symmetric systems, encryption and decryption are performed
using the same key. Such systems include AES [1], GOST 28147-89 [2, 3]. Hacking
them requires enumeration of possible keys. The complexity of exhaustive search is
estimated as O (2k), where k is the key length in bits.
In asymmetric systems (public key systems), RSA and El-Gamal algorithms are
most widely used [4, 5]. The RSA algorithm is based on the computational complex-
ity of the integer factorization problem. The reliability of the El-Gamal algorithm is
based on the difficulty of computing the discrete logarithm. Of particular interest is
the El-Gamal algorithm over the group of points of an elliptic curve [6].
For asymmetric algorithms, there are cryptanalysis methods that work faster than
full search. Therefore, asymmetric algorithms have to use longer keys than keys in
symmetric systems.
The disadvantages of both symmetric and asymmetric systems include the fact that
they are based on a set of integers. This allows you to develop various methods of
hacking such cryptosystems up to the implementation of a simple selection of keys.
To increase cryptographic strength, you have to gradually increase the length of the
keys.
However, due to the increasing capabilities of computer technology, the way to in-
crease the length of keys is unpromising. Currently, great efforts are being made to
create quantum computers [7]. It is expected that their use will significantly affect the
cryptographic strength of existing cryptosystems [8]. The well-known Grover quan-
tum algorithm [9] for restoring the key of a symmetric encryption algorithm from the
message text and cipher text. Full search on a regular computer has complexity O
(2k), where k is the key length. For a quantum computer, this complexity is 2 times
less and amounts to O (2k / 2) [9]. That is, the efficiency of the key length is reduced
by 2 times.
Quantum algorithms also pose a threat to systems with asymmetric encryption. The
cryptographic strength of the RSA encryption system is based on the super-
polynomial computational complexity of factoring natural numbers. However, there is
a quantum algorithm, whose complexity is polynomial O (n3) [10].
There is also a Shore quantum algorithm for computing the discrete logarithm. In
[11], the Shor algorithm for a group of points of an elliptic curve over a field GF (p)
with complexity O (n3) is presented.
At the same time, it is known [12] that the set of real numbers has a greater power
in comparison with the set of natural numbers. Therefore, the development of crypto-
systems using real numbers is of practical interest. It can be expected that they may be
more persistent in the event of an attempt to select a key by enumeration.
This paper presents one of the many possible options for such the cryptosystem.
2 Problem statement
It is necessary to develop algorithms for encrypting and decrypting analog and dis-
crete messages using a function of a real variable as a key.
3 Literature review
Mostly modern cryptosystems use a block cipher. It operates with groups of bits of a
fixed length - blocks. Their size can be from 64 to 256 bits.
The widely used block symmetric ciphers (BSS) are based on the use of several
symmetric cryptographic transformations (elementary ciphers).
When constructing them, three main approaches are used:
- based on Feistel chains;
- IDEA-like ciphers;
- SPN (Substitution Permutation Network) structure. The AES system belongs to
SPN based BSN [1]. In it, cryptographic conversions are performed in the simple
substitution mode over 128-bit blocks (16 bytes), which has a block length of 128
bits. The key length has several options: 128, 192, 256 bits.
The number of cycles of conversion depends on the length of the key. It is, re-
spectively, 10, 12 or 14 cycles.
The encryption algorithm GOST 28147-89 [2, 3] also belongs to the category of
block ciphers, where two parts of the selected block of information are of equal size.
It is a classical symmetric encryption algorithm based on the Feistel network and is
characterized by high cryptographic strength. However, this algorithm also has disad-
vantages:
1) In comparison with the byte-oriented algorithm, AES on 8-bit platforms GOST
loses in speed by 4 times.
2) In the text of the standard GOST 28147-89, there are no clear criteria for the se-
lection of replacement nodes. Quite often, fears are expressed that there are weak
replacement nodes.
In 1978, three authors: Ronald Rivest, Adi Shamir, Leonard Adleman proposed the
RSA algorithm [4, 5]. This algorithm was the first full-fledged public key algorithm.
Encryption is carried out without the transfer of secret keys.
In addition to classical cryptosystems using integers as keys, systems based on
functions of real variables were proposed [13]. Symbols from the ASCII code table
are encrypted with a sum of 10 key functions with coefficients before them zero or
one. The amplitudes of these functions during encryption of each new character are
taken randomly. At the receiving end, using the disproportion functions [14-16],
fragments of key functions that are present in the received encrypted signal are recog-
nized. It allows to decrypt the character transmitted at the current time.
In [17, 18], a variant was proposed when, three key functions of a real variable are
used to encrypt a binary codes. Next symbols are coded: "1", "0", "space", "the transi-
tion to a new row". Any other character is recognized as the transition to a new row.
For unauthorized access to the intercepted message, you need to select the type and
parameters of the key functions.
Both cryptosystems in the process of computer simulation showed high crypto-
graphic strength when trying to select the parameters of key functions even if their
type is known.
However, using the sum of key functions and the need to recognize them at the re-
ceiving end complicates the algorithm. In addition, the encrypted message is much
longer than the original. Therefore, the task is to develop a cryptosystem using only
one key function of a real variable.
4 Mathematical statement of the problem
A message that is encrypted is a sequence of numerical character codes (or numerical
values of the components of the brightness of the pixels in the case of a graphic im-
age), which is described by a discrete function yi, i = 0, 1, ..., N-1, where N is the
number of characters in the message. It is replaced by a sequence of ciphers that are
calculated using the key function, which can be either discrete fi, i = 0, 1, ..., M
(M > N), or continuous f(x). Here x = iꞏh, where i is the sequence number of the char-
acter in the message, and h is the step with which the argument x changes. This step
should be the same for both sides of the message transfer. Disproportion functions are
used for encryption.
5 Disproportion functions
Several types of the disproportion function are known: the disproportion over n-th
order derivative, the disproportion over n-th order value, relative and sequential dis-
proportions. Virtually all of them are characteristics of numerical functions. Below is
a summary of those that are used in this work.
The disproportion over n-th order derivative of the function y(x) with respect to x is
described by the expression:
y 1 dny
@ d x( n ) y (1)
x n n! dx n
Here, the @ symbol is chosen to indicate the operation of calculating dispropor-
tion. The symbol “d” means “derivative”. The order is indicated in parentheses. The
left side of (1) reads “at d n y with respect to x”.
If for any value of x, the function y(x) has the form y = kxn, then disproportion (1)
is equal to zero regardless of the value of the coefficient k.
The disproportion over 1-st order derivative (n = 1) has the form:
y dy
@ d x(1) y (2)
x dx
In case of a parametric specification of functions, when x = φ(t), y = ψ(t), where t is
a parameter, disproportion (2) takes the form:
d
(t ) dt
@d (1)
(t ) (3)
(t )
(t ) d
dt
For ψ(t) = kφ(t) disproportion (3) is equal to zero in the entire area of existence
x = φ(t), regardless of the value of k.
If y(x) is a sum of known functions taken with unknown coefficients, then the dis-
proportion functions allow us to calculate the values of these coefficients from the
data obtained for the current value of the argument. This opportunity was used both
for creating cryptosystems [13, 17, 18], and in solving a number of more general
problems [15, 16].
For the case when the first derivative does not exist or is equal to zero on any in-
terval, it is proposed to use the first-order integral disproportion [19, 20]. This dispro-
portion of the function y(x) with respect to f(x) has the form:
x
@I (1)
y ( x)
xh
y ( x)dx
y ( x)
, (4)
f ( x) x
f ( x)
xh
f ( x)dx
where h is the preset time interval. In the discrete representation of signals, this is a
time quantization step.
In this case, y(x) and f(x) are represented by one-dimensional arrays. If the ap-
proximate values of the integrals in (4) are calculated using the trapezoidal formula,
then for the same step h for y(x) and f(x), disproportion (4) takes the form:
yi 1 yi yi
@ I (f1i ) yi (5)
f i 1 f i f i
6 Encryption and decryption of text and image messages
Text messages are a sequence of character codes, for example, from an ASCII ta-
ble. That is, it is a sequence of integers. When transmitting color graphic images, the
brightness components of pixel-integer numbers from 0 to 255 are transmitted. As a
key function, you can take any function of a real variable. It can be either a continu-
ous or a discrete function. However, due to the fact that the message is represented by
a discrete function that does not have a first derivative, in any case, integral dispro-
portion should be used for encryption (5).
If the key function is continuous, you need to calculate an array of its values,
changing the argument from the initial xmin to the final xmax values in increments of h.
The parameters xmin, xmax, h must be the same for the transmitting and receiving sides.
When encrypting the characters from the ASCII table or pixel luminance components,
their numerical representations differ by one. In these cases, the step h of changing
the argument must be equal to one.
In addition, the transmission and, accordingly, the reception must begin with a cer-
tain symbol known to the transmitting and receiving sides.
Therefore, the encryption algorithm is as follows:
1. The start character that is known to both sides must be entered.
2. Enter an array of values of key function.
3. Read from the file or enter from the keyboard a sequence of message characters
and convert them to the numbers. In this case, for text symbols the ASCII table can
be used. As a result, the array yi, i = 0, 1, ... N-1 will be get.
4. Using the code of the given initial character y0, calculate the disproportion
I i @ I (f1) yi , i = 1, 2, …, N-1 for each next character in accordance with (5) and
i
transmit them over the communication channel.
7 Decryption algorithm for text and image messages
1. Enter the start character y0 that is known for the transmitting side.
2. If the key function is discrete, enter the array of its values. If the key function is
continuous, calculate the array of its values in accordance with the instructions
above.
3. Read from the file the accepted disproportion values for each of the symbols
Ii, i = 1, 2, ... N-1.
4. Use the known initial symbol y0, and the accepted disproportions Ii, i = 1, 2, ... N-1,
for calculating the estimated codes of the recovered message according to the for-
mula (6).
( yi I i ( f i 1 f i )) f i
yi (6)
f i 1
Round off estimated codes to the nearest integers to get recovered codes.
5. Using the recovered codes found, reproduce the message in symbolic form (when
transmitting a graphic image, reproduce the corresponding pixel attributes).
8 Examples of encryption and decryption of discrete messages
Example 1
An example of encryption and decryption of characters from the ASCII code table
is provided.
The key function has the form:
f ( x) aex sin( x) ln(ax ) (7)
x = iꞏh is an argument;
i is the sequence number of the character in the encrypted message;
h = b + c is the step of changing the argument.
Here a = 0.1, β = 0.01, b = 0.65, c = 0.35 are constants.
Since in the example the characters are encrypted from the ASCII table, in accor-
dance with the above algorithm, the sum of the constants b and c is equal to one.
The sequence of numerical codes yi, i = 1, 2, ... N-1, corresponding to the transmit-
ted characters, is encrypted. When transmitting messages, let's the first character is
“G”. In the ASCII table, its code is 71 (that is, y0 = 71).
Given the discrete nature of the message, integral disproportion (5) is used to en-
crypt it.
Decryption is carried out using (6) and according to the above algorithm.
Table 1 shows the transmitted characters, their ciphers, and the corresponding de-
crypted characters.
Table 1. Codes of transmitted characters and the results of their decryption.
Charac- Source Character De- Charac- Source Character De-
ter charac- code crypted ter charac- code crypted
number ter charac- number ter charac-
ter ter
1 -0,33745 19 @ -14,9626 @
2 C 18,56566 C 20 17,18209
3 o 35,05402 o 21 # -0,50861 #
4 n 23,38864 n 22 2,084013
5 t 45,28726 t 23 $ -1,0013 $
6 r 75,66021 r 24 2,205811
7 o 215,5424 o 25 % -1,38787 %
8 l 3638,486 l 26 2,342513
9 1392,007 27 ^ -21,4272 ^
10 e -49,5932 e 28 22,22681
11 x 36,34984 x 29 & -1,64792 &
12 a 60,50844 a 30 2,295671
13 m 7,687781 m 31 * -2,84342 *
14 p 11,55305 p 32 3,393497
15 l 13,07326 l 33 ( -2,13477 (
16 e 11,94166 e 34 2,607738
17 ! 44,92703 ! 35 ) -2,34347 )
18 2,001215
It can be seen from it that the cipher of each character is a real number with both an
integer and a fractional part. Encrypted and decrypted characters match.
In order to “crack” a message, you need to select the type of the key function and
the values of its parameters.
The following is an example that illustrates the cryptographic strength of the sys-
tem to obtain a key, even if somehow it was possible to find out the type of key-
function. Suppose that the above sequence of characters is encrypted using function
(7), and decrypted using the same type of function, but the constant a instead of the
value 0.1 during decryption is chosen incorrectly and is 0.1005.
Table 2 shows the results, from which it is seen that even such a slight deviation of
the parameter of the key function does not allow to decrypt transmitted characters
correctly.
Table 2. Codes of transmitted characters and the results of their decryption with an incorrectly
selected key function.
Char- Source Character De- Charac- Source Character De-
acter charac- code crypt- ter charac- code crypted
num- ter ed number ter charac-
ber char- ter
acter
1 -0,33745 19 @ -14,9626 D
2 C 18,56566 B 20 17,18209
3 o 35,05402 n 21 # -0,50861 М
4 n 23,38864 l 22 2,084013 Ѓ
5 t 45,28726 q 23 $ -1,0013 -
6 r 75,66021 n 24 2,205811 Г
7 o 215,5424 g 25 % -1,38787 T
8 l 3638,486 5 26 2,342513 П
9 1392,007 ќ 27 ^ -21,4272 ‚
10 e -49,5932 X 28 22,22681 ®
11 x 36,34984 ” 29 & -1,64792
12 a 60,50844 f 30 2,295671 e
13 m 7,687781 ( 31 * -2,84342 ѕ
14 p 11,55305 ґ 32 3,393497 ъ
15 l 13,07326 33 ( -2,13477 A
16 e 11,94166 L 34 2,607738 q
17 ! 44,92703 + 35 ) -2,34347 «
18 2,001215 2
A positive feature of the proposed encryption system is that the cipher of one and
the same symbol is not repeated and depends on its number in the message.
Table 3 shows the repeated characters, their ciphers, and decryption results.
Table 3. Results of decryption of the sequence of identical characters.
Char Sou Character De- Charac- Source Character De-
acter rce code crypt ter charac- code crypt-
num- char ed number ter ed
ber ac- char- char-
ter acter acter
1 a 42,89534 a 23 % -1,07004 %
2 a 55,98126 a 24 % 0,230793 %
3 a 223,1301 a 25 % 0,211779 %
4 a -2722 a 26 % 0,195132 %
5 a 193,6388 a 27 % 0,180462 %
6 a 47,19846 a 28 % 0,167453 %
7 a 21,12258 a 29 % 0,155856 %
8 a 12,02764 a 30 % 0,145461 %
9 a 7,807816 a 31 % 0,136099 %
10 a 5,504747 a 32 % 0,127629 %
11 31,8935 33 1,189454
12 A -12,0762 A 34 9 -5,17547 9
13 A 1,71819 A 35 9 0,164057 9
14 A 1,413497 A 36 9 0,154966 9
15 A 1,186178 A 37 9 0,146586 9
16 A 1,011796 A 38 9 0,138837 9
17 A 0,874885 A 39 9 0,131652 9
18 A 0,765272 A 40 9 0,124967 9
19 A 0,67603 A 41 9 0,118735 9
20 A 0,602313 A 42 9 0,112905 9
21 A 0,540641 A 43 9 0,10744 9
22 9,210041 44 4,816542
From Table 3 it can be seen that the ciphers of the same characters differ from each
other, but at the same time, their decryption is error-free.
This once again shows that for hacking it is necessary to find the type of the key
function, and then select the values of its parameters with high accuracy.
Example 2
The encryption of a graphic message is considered - a 2D image using the key
function of a real variable
f (q) aesin(( b c ) q a ) , (8)
where q is the sequence number of the pixel;
a = 100, b = 0.65, с = 0,35.
The image of a seagull on the seashore is encrypted. According to the algorithm,
transmission begins with a pixel known to the receiving side. Pixel brightness is also
represented by integers with a step equal to one. Therefore, according to the algorithm
in this case, it is required that the sum of a and b is equal to one.
Table 4 shows the disproportions of the brightness components for the first 23 pix-
els.
Table 4. Components brightness pixels’ codes.
pixel disproportion of component red component green component blue
number component A disproportion disproportion disproportion
1 -0,17260194 -0,052654658 -0,079113617 -0,109100438
2 0,249815554 0,122038436 0,147606293 0,183288106
3 -0,25138517 -0,066806224 -0,080056209 -0,073489858
4 -1,54785813 -0,671606862 -0,746100819 -0,895078346
5 -2,11017572 -1,141977451 -1,198335871 -1,339798239
6 0,638962347 0,336698458 0,322878349 0,344784243
7 0,904695026 0,493147485 0,536886675 0,603130017
8 0,35818895 0,185034305 0,203422857 0,224748673
9 -0,0619523 -0,079999226 -0,074320205 -0,069612985
10 -1,08142099 -0,6158245 -0,699207099 -0,762820098
11 -2,35243212 -1,569025352 -1,733111253 -1,953901199
12 -0,07899416 -0,025833335 -0,031099612 -0,000303777
13 1,059101779 0,630917816 0,701524602 0,778998655
14 0,482163337 0,302525074 0,337339747 0,360361396
15 0,079311391 0,042700382 0,052726876 0,071145629
16 -0,68481115 -0,484134182 -0,526938708 -0,575032293
17 -2,19903004 -1,601314543 -1,722045604 -1,879955093
18 -0,9877602 -0,663217085 -0,68687686 -0,706872244
19 1,094772111 0,607081863 0,704520639 0,846261488
20 0,632160636 0,56848388 0,574362813 0,580312445
21 0,193576492 0,084571962 0,09345652 0,077852313
22 -0,3793663 -0,320611934 -0,339514003 -0,33505087
23 -1,80013937 -1,01073232 -1,126174205 -1,185141129
It can be seen from it that the ciphers are real numbers with both an integer and a
fractional part, the nature of the changes of which is difficult to predict.
Figure 1 shows the original (left) and decrypted (in the middle) images according
to the disproportions received at the receiving end, known to the first pixel and the
key function.
Fig. 1 Original image (left), decrypted (in the middle), decrypted with an incorrectly selected
key-function parameter (right).
Figure 1 on the right shows the result when instead of a = 100, when decrypting
the image, the value a = 99.9999 was mistakenly selected. Despite a slight difference
from the value of this parameter during encryption, it was not possible to “crack” the
image. The same result if b + c = 0.999999 is selected instead of unity.
This indicates the cryptographic strength of the encryption system and how diffi-
cult it is to get a key function by enumeration.
9 Examples of encryption and decryption of the analog message
The case is considered when the message y(x) and the key function f(x) are con-
tinuous, smooth, having a first derivative.
The transmitting and receiving parties agree on the key function, the initial value of
the argument x0, the step of its change h and the initial value of the transmitted mes-
sage y(x0).
In practice, most often, there is a need to encrypt signals from moving objects.
In this case, the signal is represented by a function of time y(t). To encrypt it using
the key function f(t), disproportion (3) should be calculated, which in this case has the
form:
dy
y (t )
z (t ) @ d (1)
f (t ) y (t ) dt (8)
f (t ) df
dt
In this case the time change step h is selected from the representation condition y(t)
with the necessary accuracy.
The decrypted values are calculated from equation (9) obtained from (8):
y (t )
y(t ) f (t ) z (t ) (9)
f (t )
The decrypted message is found from (9) by solving the Cauchy problem using the
known y(t0), h and z(t) values obtained via the communication channel (8).
Example 3. Analog message encryption
For simulation, suppose that an analog message is described by the following ex-
pression:
y (t ) k exp((a t ) cos(bt )) sin 2 (bx) (10)
The key function has the form:
f (t ) a exp(bt ) sin(bt ) ln(at b) , (11)
where a = 1, b = 10, k = 100 are constants.
The time t varies from 0.1 to 4 in increments of h = 0.0025.
The derivatives were calculated numerically using the values of three points.
The results of the cryptosystem simulation are shown in table 5.
Table 5. Transmitted and decrypted values of a numerical function.
account transmitted decrypted value
number value
1 0,16938 0,173057
2 0,674778 0,68059
3 1,51054 1,51856
4 2,66904 2,67926
5 4,14073 4,15308
6 5,91425 5,93181
7 7,97656 7,99589
8 10,313 10,3338
9 12,9076 12,9296
10 15,743 15,7656
11 18,8006 18,8234
12 22,061 22,0834
13 25,5038 25,5252
14 29,1082 29,1278
15 32,8525 32,8735
16 36,715 36,7328
17 40,6737 40,6874
18 44,7066 44,7188
19 48,7916 48,7979
20 52,9072 52,9066
21 57,0322 57,0268
22 61,1458 61,1315
23 65,228 65,2037
24 69,2595 69,2267
25 73,2218 73,1769
26 77,0975 77,0394
27 80,8701 80,7977
28 84,5241 84,4363
29 88,0453 87,9424
30 91,4207 91,3003
Obtained results indicate that even with a rough calculation of the derivatives, the
decryption error in this example is a fraction of a percent. This error can be reduced
by decreasing the step h, and by applying more accurate methods of numerical differ-
entiation, for example, the Newton-Stirling method [21].
10 Constraints on the key-function of a real variable
When implementing the proposed cryptosystem, it is necessary to take into account
the restrictions imposed on the key function.
1. The function must be defined on the set of real numbers.
2. The function should not be constant and not take zero values.
3. When using the key function, a situation should not arise when the number is di-
vided by a number close to zero, which leads to the appearance of an unacceptable
calculation error. To this end, it is recommended to test the cryptosystem for the
entire alphabet of characters that will be used in messages.
4. Before sending an encrypted message, first check what it looks like after decryp-
tion. This will avoid mistakes that may occur because of not taking into account
the previous paragraphs.
It should also be noted that in the system there is an effect similar to the “avalanche
effect” in the AES cryptosystem. It consists in the fact that behind an incorrectly de-
crypted character the remaining message is decrypted incorrectly.
11 Conclusions
A cryptosystem with a symmetric key is offered. For the first time, a function of a real
variable is used as a key, which satisfies some restrictions. It can be either continuous
or discrete. Encryption is performed using disproportion functions. The specific ex-
amples show the possibility of encryption and decryption of text information, 2D-
graphic images, as well as analog messages. Examples are also given that demonstrate
the cryptographic strength of the proposed cryptosystem.
In the practical application of the proposed cryptosystem, the above limitations on the
key function should be taken into account.
References
1. Federal Information Processing Standards Publication 197 November 26, 2001 Specifica-
tion for the ADVANCED ENCRYPTION STANDARD (AES).
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