This paper presents a symmetric key cryptosystem using the sum of real type functions which allows to increase the cryptographic strength. Both transmitter and receiver choose Key Functions with the same argument, the interval for setting the argument, and the step for changing it. The symbol of the transmitted message is encrypted in an array where each element is the sum of Key Functions with random amplitudes. This sum includes those Key Functions for which the corresponding bit is one. Decryption uses disproportion functions. The system is suitable for encrypting both discrete and continuous messages.
Widely used cryptosystems are based on the set of integers. They implement symmetric and asymmetric encryption algorithms. In symmetric systems, the same key is used for both encryption and decryption. The most famous symmetric systems are AES [1] and GOST 28147-89 [2, 3]. To hack such a system, an enumeration of possible keys is required. The brute-force complexity is O(2k), where k is the key length in bits. For symmetric systems, if the communication channel is open, there is a problem of secure key transmission. This problem does not exist for asymmetric open key systems. In these systems, the most widely used algorithms are RSA and El-Gamal [4, 5]. The RSA algorithm is based on the computational complexity of the integer factorization problem. El-Gamal's algorithm is based on the difficulty of computing the discrete logarithm, especially over a group of points of an elliptic curve [6]. For breaking asymmetric cryptosystems, there are cryptanalysis methods which are faster than full search. This circumstance makes it necessary to use longer keys compared to keys in symmetric systems, but it’s not promising due to the intensive development of the quantum computers [7], which will significantly affect the cryptographic strength of existing cryptosystems [8]. The ordinal brute force has complexity O(2k), meanwhile Grover's quantum algorithm [9] reduces it to O(2k/2) [9].
Implementing quantum algorithms will also
reduce the robustness of asymmetric systems. The
RSA system uses the super polynomial
computational complexity of the factorization of
natural numbers. At the same time, there is a
quantum algorithm whose complexity is
polynomial O(n3) [10]. It means the cryptographic
strength of asymmetric systems can be reduced as
a result of the implementation of Shor's quantum
algorithm for computing the discrete logarithm. In
[11], Shor's algorithm is given for the group of
points of an elliptic curve over the field GF(p)
with complexity O(n3). Implementing quantum
algorithms will also reduce the robustness of
asymmetric systems. A method for increasing the
crypto resistance of the system under these
conditions is proposed in [
The above analysis shows that one should look
for other ways to create cryptosystems. In
particular, to complicate the selection of keys
using the simple enumeration method, one should
switch from using integers to real ones. It is
known [
So, in [
In [
In [
A completely different encryption principle
was proposed in [
The cryptosystems [
The message that is encrypted is a sequence of T numeric character codes from the ASCII table (or numeric values of the pixel brightness components in the case of a graphic image transmission). Each of them is encrypted using one-dimensional arrays of length N values. These arrays are obtained using one and the same step h of changing the argument of m Key Functions of the real variable. In this case, the value y(j, i) of the matrix y(T, N) has the form: , , (1) where: j is the number of the character in the transmitted message; fq(i) = fq(ih), (i = 1, 2, … N > m), (q = 1, 2, … m) - an array of values of the q-th Key Function; kqj - coefficients that are generated during encryption of the j-th element and can be either equal to zero or represent random numbers which are unknown to the recipient.
Key Functions can be either continuous or
discrete. These functions should be the same for
the transmitting and receiving sides and have the
same numbering. Also, the step h of changing the
argument of the Key Functions should be the
same. An encrypted message in the form of a
matrix y(T,N) is transmitted over an open
communication channel. The task is to decrypt the
message using the matrix received at the receiving
end. To solve it, the integral disproportion of the
first order is used [
One of the first publications in which
disproportion functions were proposed was [
!
Here the @ symbol is chosen to denote the operation of calculating disproportion. The symbol "d" stands for "derivative". The order is indicated in parentheses. The left part (2) reads "et d n y with respect to x". The order n ≥ 1 is an integer. If for any value of x, the function y(x) has the form y = kxn, then disproportion (2) is equal to zero regardless of the value of the coefficient k.
For the case when n = 1, ,
For defining the functions parametrically, when, x = φ(t), y = ψ(t), where t is a parameter, disproportion (3) is described by the expression (4) ⋯ , ,
For ψ(t) = kφ(t) disproportion (4) is equal to
zero in the entire area of existence x = φ(t),
regardless of the value of k. In [
where f1(x), f2(x), … fm(x) are known functions; k1, k2, … km are coefficients whose values are unknown.
It is shown that the disproportion functions
allow calculating the values of the unknown
coefficients in (5) from the data obtained for the
current value of the argument. This opportunity is
used to create cryptosystems [
In practice, often the first derivative of the
function does not exist or is equal to zero on some
interval. This excludes the possibility of using
disproportions over the first-order derivative
(24). In this case, it is advisable to use the integral
disproportion of the first order [
In this case, y(x) and f(x) are represented by one-dimensional arrays. If the approximate values of the integrals in (6) are calculated using the trapezoid formula, then for the one and the same step h for y(x) and f(x), disproportion (6) takes the following form (7): ,
(7) and decrypting
The transmitting and receiving sides must have the same system of m Key Functions of the real variable, their numbering, the interval of changing the argument and step h of its change. The number of elements N of the one-dimensional array corresponding to the encrypted character must also be set. These can be both characters from the ASCII table, and components of pixel brightness when transmitting color graphic images. Each of them is represented by an integer. The required number of Key Functions depends on the maximum value of this number. For example, to encrypt characters from the ASCII table, m = 8 Key Functions are required. They can be either continuous or discrete. If the Key Functions are continuous, it is necessary to calculate N elements of one-dimensional arrays of their values, changing the argument from the initial xmin to the final xmax value with a step h. When encrypting characters from the ASCII table or the pixel brightness, their numerical representations differ by one. In these cases, the step h of changing the argument must be equal to one.
An m-bit binary code corresponds to each encrypted character. Each bit in this code is associated with a specific number of the Key Function. If the bit is zero, the value of the corresponding Key Function is also zero. If the bit is equal to one, then a random value of the amplitude of the corresponding Key Function is played. The character to be encrypted is represented by the sum (1). 4.1.
1. The following is a character encryption algorithm:
2. Calculate arrays of N > m values of Key Functions: fq(x), q = 1, 2… m.
3. Enter the encrypted j-th character and calculate its cipher in the form of values of the where h - is the preset time interval. In the discrete representation of signals, this is a time quantization step. , 1 F j, i , 1 , 0 , , k , , ,
Taking into account that the disproportion of F21(j, i) with respect to F21(j, i) is equal to zero, we get:
Third level: The disproportion of F0121(j, i) with respect to F3121(j,i) is calculated in the following way
It is equal to zero because, as can be seen from (14), there is a proportional relationship between F0121(j, i) and F3121(j, i). This fact allows calculating from (14) k3j and k2j, k1j for the j-th message symbol. , , , ,
, , , (14) (15) (16) (17) (18) one-dimensional array y(j, i), i = 1, 2... N according to (1).
4. Repeat this point for all characters of the message of length T.
5. A sequence of T arrays is an encrypted message transmitted over an open communication channel. 4.2.
Pre-compute the arrays fq(i) = fq(ih), (q = 1, 2, ... m), (i = 1, 2, ... N > m), of Key Functions and to receive T one-dimensional arrays y(j , i), j = 1, 2, ... T, i = 1, 2, ... N over the communication channel. Further, in order to simplify the description of the decryption process, an example is given when only three functions are used in the cryptosystem - the keys: f1(x), f2(x), f3(x). In this case m = 3. Accordingly, the j-th character of the message is encrypted as , , (8)
1, 2, … 3 ,
The process consists of m = 3 levels in accordance with the number of Key Functions.
First level: It is necessary to calculate the array of disproportions (7) y(j,i) with respect to any of the Key Functions , for example, f1(i):
, , 1
, , ,
(9) ,
(12) 1 where i = 2, 3…N.
Also calculate the disproportions (7) of the remaining key functions with respect to f1(i):
(10) 1 where r = 2, 3.
Considering that the disproportion of the function relative to itself is zero, we get:
, , , , (11)
Second level: It is necessary to select any disproportion from right-hand of (11), for example F21(j, i). It is used to calculate next disproportions: , 1 , ,
Depending on which of these coefficients are nonzero and which are equal to zero, the j-th message symbol is decrypted. In practice, it must be taken into account that there are calculation errors.
Therefore, it is necessary to compare the disproportion (15) calculated at the last level in modulus not strictly with zero, but with an approximate number ɛ. For example, it could be ɛ=10-4. In this case, if | F01213121(j, i) | <= ɛ, then it should be assumed that it is zero.
The value of ɛ is determined during testing of the cryptosystem. Theoretically, this disproportion is equal to zero for all i = 2, 3, ... N, but, taking into account the calculation errors, it is recommended to do calculations using formulas (16-18) for i, at which the modulus of disproportion (15) is minimal. decrypting characters from an ASCII
An example of encrypting and select the form of eight Key Functions and the are given in the form of arrays arranged vertically. level in absolute value exceeds the permissible values of their parameters. Encrypted and decrypted characters “Hello” y ‘H’ ‘e’ ‘l’ ‘l’
‘o’
1. The Key Functions must be of real type.
They can’t be constant and must not take zero values.
When using the key function, there should be no situation where division by a number close to zero occurs, which leads to an unacceptable calculation error. For this purpose, it is recommended to test the cryptosystem for the entire alphabet of characters that will be used in messages.
4. Check that the sum of two or more key functions does not coincide with any other of the key functions.
5. It is recommended to include all parameters in the expression for each key function. In this case, a change in the value of any parameter leads to a change in all key functions, but not one or several of them only.
6. Before sending an encrypted message, first check what the decrypted message looks like in order to avoid errors that may occur as a result of not taking into account the previous points.
A cryptosystem with symmetric keys is proposed. These keys are real variable functions that satisfy the above constraints. They can be either continuous or discrete. The number of functions is equal to the number of binary digits used to encrypt a character, for example, in an ASCII table. Each of the functions corresponds to a certain binary digit. The symbol of the transmitted message is encrypted with a onedimensional array. The elements of this array represent the sum of Key Functions with random amplitudes. This sum includes those Key Functions, for which the corresponding binary digit is equal to one.
Decryption is performed using disproportion functions. The possibility of encryption and decryption of text information is shown. The given examples show the complexity the guessing Key Functions and the cryptographic strength of the proposed cryptosystem. So, for example, a real-type constant, which equals 400 during encryption, to break the system by brute-force, you need to select with an accuracy of 10-4, but there can be any number of such constants. It is very difficult to find all the constants of the real type at the same time with high precision and thus hack the system, even with well-known formulas of functions - keys.
It should also be noted that the codes of the same adjacent symbols are not repeated, which can be seen from Table 2. This also increases the cryptographic strength of the system.
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