Reliability and accuracy of information transmission are critically important in today's digital society. At the same time, error-free data transmission should not only guarantee security and efficiency but also optimize the use of resources, such as time, energy, and network bandwidth. To achieve these results, mathematical methods are used, which become the foundation for the development of new algorithms and solutions in the field of information and communication technologies, in particular in the theory of information coding. This article considers the possibility of using two-symbol encoding for efficient data transmission and storage. The paper presents a two-symbol encoding system for real numbers, which is a unique partial case of a system with two bases of different signs. Probabilistic and fractal theory of binary representations is covered. It was determined that this two-symbol information encoding system can be used in cryptography, using numbers recorded in this way as components of a series of pseudo-random numbers, and in steganography for hiding information in video images or other open data streams of both static and dynamic types based on fractal theory. The approaches considered in this study can be used in the training of specialists in the field of information and cyber security.
1. Introduction Encoding is the process of converting information from one form to another that conforms to a specific algorithm, standard, or protocol. Encoding information is an important component of information and cyber security. Encoding information ensures compatibility, efficiency, and security during data transmission, processing, and storage [1]. In other words, as a result of encoding, if there is no interference, the message is presented in a more compact (compressed) form, otherwise, the encoding process detects and corrects errors that arise due to interference. The latter ensures the reliability of the information. A set of message characters receives a unique numerical code that corresponds to a certain encoding standard.
“A code is a system of symbols or signals for transmitting, processing, and storing information” [2, p. 551]. In information transmission systems, codes can be classified according to various characteristics:
In information theory and programming, there are a variety of encoding methods, many of which are named after prominent scientists who made significant contributions to their development. These codes are used for efficient and reliable data representation, including compression, error protection, and information transmission:
encoding by using variable-length codewords and is used in many file formats, such as
compression, is a predecessor of the Huffman code and also uses variable-length codewords.
This list is not exhaustive and there are many other codes named after their authors. Some of them may be less well-known but are also important for the development of information technology. The main goal in the process of encoding information, scientists identify:
The first direction is studied by the theory of economic coding [3–7], and the second—by the theory of noise-resistant coding [8–10]. Obviously, the solution to these problems is best solved by encoded information, so this problem will be relevant in the future. The current state of coding is characterized by the implementation of artificial intelligence, in particular machine learning. Mathematical methods will play a key role in creating efficient and reliable coding systems that ensure the transmission and storage of information [11].
Today, exact sciences boldly operate with both finite and infinite sets and data arrays. At the same time, the ideas of coordination and coding are effectively used. As a result, powerful families of mathematical objects are described by a small number of basic (reference) objects and relations. Meaningful information about dependencies, and relations (correspondences) takes on a digital form, encrypted by codes: sets, matrices, and sequences of elements of the alphabet, which can be finite and infinite, constant and variable [12]. 2. Representation of real numbers through a two-symbol alphabet In mathematics and its applications today, various representations and images (encodings) of real numbers are used, which use finite and infinite, constant and variable alphabets, i.e. a real number has different forms of existence. One of the simplest is the representation and image in the s-base (binary, ternary, decimal, etc.) number system. Two-symbol information encoding systems traditionally use the alphabet A ={0 ,1 }. Two-symbol systems deserve special attention, in particular, due to the minimal alphabet. Next, we will focus on encodings of real numbers.
The coding of real numbers of the interval [ 0 ; 1] using the alphabet A is called the correspondence between the sets [ 0 ; 1] and L= A × A × A × …, in which each number x ∈[ 0 ; 1] corresponds to at least one element of the set L and each element of the set L is the image of at least one number of the interval [ 0 ; 1]. The sequence itself (an)=( a1 , a2 , ... , an )∈ L, which corresponds to the number x, is called its representation (or code), and а an is the nth digit (or symbol) of its representation.
It seems that the coding system has zero redundancy since each number can have no more than two images, and what is important is the majority of numbers in one representation.
The binary representation of numbers is the simplest example of two-symbol continuous
encodings with zero redundancy [13].
2.1. Existence of a G-representation of a natural number
Definition 1. If for a natural number a there exists a set ( a1 , a2 , … , an ) of zeros and ones such that
(
Note. As we can see, the G-representation of the same natural number can have a different number of digits. For example, 8=(1000)G=(11000)G, and in general
n a=2n+∑ [(−1)1+σk ak 2n−k ] ≡ (1 a1 … an)G ,
k=1 where σ k=a1+ …+ ak−1, then we will say that the number a has a G-representation, which is symbolically written a=(1 a1 … an)G and is ( n+1 )-digit.
For example, each of the numbers within ten has a G-representation, and there are exactly two of them, which is easy to verify.
1=20=(
( ) ( )
n G n G
a=2n+1=2n+1−2n+1=(11 0⏟… 0 1) =(11 0⏟… 0 11) . (
Theorem 1. Every natural number has exactly two G-representation, i.e. for any natural number
a there exists a set of zeros and ones ( a1 , a2 , ... , an ) such that (
Let us assume the truth of the statement for n=k , i.e. a∈( ak−1 ; 2k ] has the decomposition k a=2k +∑ [(−1)1+σi ai 2k−i ] ≡ (1 a1 … ak )G ,
i=1 where σ i=a1+ …+ ai−1, and there are exactly two of them.
Consider n=k +1, i.e. the number a∈ ( 2k ; 2k+1 ]. If a=2k+1, then a= 1 0⏟… 0 (
) =(11 0⏟… 0) . k+1 G k+1 G
(
Then a=2k +u=2k+1−2k +u= 11 0⏟… 0 c1 … cm . Since the number u ≡ a−2k satisfies the ( )
k−m G inequality 0<u<2k, then by assumption, it has exactly two G-representation. Therefore, exactly two G-representation has the number a. According to the principle of mathematical induction, the statement is proved for any natural number a.
Definition 2. The canonical G-representation of a natural number is the G-representation, which has the minimum number of digits and the maximum number of zeros at the same time.
16=(10000)G=(110000)G , a=2n−1=(1 0⏟… 0 1) =(1 0⏟… 0 11) n−1 G n−2 G is the first of the G-representations. 2.2. G-representation of the fractional part of a real number Theorem 2 [14]. For any number x ∈[ 0 ; 1 ] there exists a sequence of zeros and ones ( α n ) such that 2 a= α21 +∑k=∞2 α k (−2k1)σk
= α21 +∑k=∞2 2k−σkα( −k2)σk ≡ ΔGα1α2….αn … , where σ k=a1+ …+ ak−1.
Corollary. For any number x∈ [ 0 ; 1] there exists a sequence of zeros and ones ( α n ) such that x= 12 α 0+ α21 +∑k=∞2 α k (−2k1)σk ≡ Δα0 α1…αn … .
(
Theorem 3 [15, 16]. For any number x ∈[ 0 ; g0] there is a sequence of zeros and ones ( α n ) such that
∞ k−1 G x=δ α1+∑k=2 ( δ αk ∏j=1 gα j )≡ Δα12α2…αk … , δ αk=α k g1−αk .
Corollary 1. If g0= 12 , then δ αk= α2kk ,
∏kj=−11 gα j= (−12)αk1+−…1+αk−1 (
The symbolic notation ΔG2
α1α2…αk … of a number x and its decomposition into an alternating series
(
Lemma 1. Lebesgue measure of a set
G
C ≡ C [ G2 ; s1 … sm ]={ x : x= Δα12α2…αn… , α k … α k+m−1 ≠ s1 … sm ∀ k ∈ N }
(
E={x : x= Δα12α2…αn… , α km+1 α km+2 … α km+m ≠ s1 … sm ∀ k ∈ N }.
G
It is obvious that C ⊂ E. Let us prove that the Lebesgue measure λ ( E )=0. From which it follows that λ (C )=0. To do this, let us denote by F0 ≡ [ 0 ; g0], Fk the union of all cylinders of rank km, among the interior points of which there are points of the set E, and Fk+1 ≡ Fk ∖ Fk+1. Then Fk= Fk+1∪ Fk+1, λ ( Fk+1)= λ ( Fk )− λ ( Fk+1 ), E⊃ Fk⊃ Fk+1 ∀ k ∈ N and λ ( E )= lim λ ( Fk ) .
k →∞ Let us express λ ( Fk )= gλ0( λF(kF−1k)) ⋅ λλ (( FFkk−−12)) ⋅ …⋅ λλ (( FF10)) = ∏i=k1 λλ( F(Fi−i)1) = ∏i=k1 λ ( Fλi()F−i−λ1()Fi) =
k =∏ (1− i=1 λ ( Fi) λ ( Fi−1)
).
Whence
∞ λ ( E )= lim λ ( Fk )=g0 ∏ k →∞ k=1 λ ( Fk−1) λ ( Fk ) ∞ =g0 ∏ (1− k=1
).
(
G c1 ≤ |ΔGa12…ak i| ≤ c2 ,
|ΔGa12…ak| |ΔGα12…αkm| 0<c1m ≤ |ΔGα12…αkm s1… sm|=∏|gsi|≤ c2m<1.
m i=1 where c1=min {g0 ;−g1 }, c2=max {g0 ;−g1 }, and therefore
Then the ratio is separate from 0, and therefore the difference 1− is separate λ ( Fk−1) λ ( Fk−1) from 1, i.e. the necessary condition for the convergence of the infinite product is not met, and therefore it converges to zero.
Corollary 2. The set C is a null set of Cantor type with a self-similar structure.
Theorem 4. Almost every number from the interval [ 0 ; g0] in its G2-representations use each of the sets of digits of the alphabet as consecutive digits an infinite number of times.
Proof. Let ( s1 , … , sm) be an arbitrary ordered set of zeros and ones, H be the set of all numbers [ 0 ; g0 ], in the G2-representation of which the set of digits s1 s2 … sm appears an infinite number of times as consecutive digits, D be the set of all numbers that use this set as consecutive digits of the G2-representation only a finite number of times.
Let us prove that H is a set of full Lebesgue measure, that is, that λ ( H )=g0. To do this, it is
λ ( Fk )
enough to prove that λ ( H )=0, where H =[ 0 ; g0 ]∖ H . It is obvious that
(
H =Un=1 Dn , where Dn={x : x= ΔG2
α1…αn … , α k+1 … α k+m ≠ s1 … sm ∀ k ≥ n}.
Let us calculate λ ( Dn ). Since
Dn= U α1∈A … U [ ΔGα12…αn ∩ D ] , ΔGα12…αn ∩ D= ΔG2
α1…αn ∩ H , αn∈A
G then according to the previous lemma 1 λ ( Δα12…αn ∩ D )=0, and therefore, λ ( Dn)=0. Therefore λ ( H )=0, since
∞ λ ( H ) ≤ ∑ λ ( Dn)=0.
n=1 2.4. Probabilistic theory of binary representation The probability theory of real numbers in their binary representation deals with solving problems related to probability distributions on sets of numbers defined by the properties of the binary representation of their elements [17].
G Theorem 5. If the random variable ξ = Δξ12ξ2… has a uniform distribution on the interval [ 0 ; g0 ], then the digit (ξn) of its G2-representation are independent and have distributions P (ξn=0)=g0, P (ξn=1)= – g1.
Theorem 6. If the distribution function Fη ( x ) of a random variable η= ΔG2 η1…ηk … with independent binary digits ηk has a positive derivative at the points 2−1 , 2−2 , …, then η has an exponential distribution on а [ 0 ; 1] with density where the distributions of digits ηk are given by the formulas β ( eβx ) eβ−1 f ( x )=
.−∝ < β <∝ , P (ηk=0)= p0 k= P (ηk=1)= p1k= 2.5. Fractal theory of binary representation The essence of the fractal theory of numbers in a given system of their coding consists in studying fractal properties of a set of numbers, determined by conditions (restrictions) on their representation (use of digits). These conditions can be formulated in terms of prohibitions on the use of digits and their combinations, in terms of frequencies of digits or combinations of digits in the representation of a number, etc.
The methods of fractal theory are based on the ideas of self-similarity, in particular selfsimilarity and self-affinity, tools of the theory of metric dimensions, and measures of fractional orders. The Q2-representation of numbers is effectively used in fractal theory [17]. cylinders, [ 0 ; g0]⊃ E
Let the G2-representation of numbers in the interval [ 0 ; g0 ] be given, W G is the set of all G 2 is a fixed set, 1 ≥ α is a positive number, ε >0. Let lε ≡ inf {∑|ωi|α : E⊂U ωi , ωi∈W G} be given, where |ωi| is the length of the cylinder ωi, and α
|ωi|≤ ε i i the lower bound is taken as all possible coverings of the set E by G2-cylinders whose lengths do not exceed ε.
It is obvious that the numbers lαε ( E ) and mαε ( H ), generally speaking, are different ( lαε ( E ) ≥ mαε ( H ), since the class of all possible covers of the set E by intervals includes the class of covers by G2-cylinders). Let us prove that despite this, the functions H α ( E ) and Lα ( E )=εl→im∝ lαε ( E ) of the variable α take the values 0 and ∞ simultaneously. And this is a reason to believe that when determining the Hausdorff-Bezikovitch dimension of an arbitrary set E⊂ [ 0 ; g0 ] one can limit oneself to covers of the set by G2-cylinders, which follows from the statement.
Theorem 7. For arbitrary E⊂[ 0 ; g0 ], 0< α ≤ 1, ε >0, the double inequality holds mαε ( E ) ≤ lαε ( E ) ≤ 2 (2m+1)⋅ mαε ( E ) , (20) m m is the smallest natural number satisfying the inequality g0 ≤ 1 – g0. 3. Two-symbol system in some variants of information encoding The proposed two-symbol system has great prospects in mathematics, but its possibilities in the field of coding were not considered. Let us try to highlight those areas of its application in information coding for further research:
The two-symbol system involving Q2-representation has a chance to be used in the interests of steganographic closure (hiding) of information in video images or other open data streams of both static and dynamic types based on fractal theory. In this case, the pixels carrying information (part of the message code) are placed in the plane of the picture randomly.
It should be noted that this representation can be used as a primary code. But for such codes, one of the important properties is conciseness—the minimum number of bits of representation of a particular state of an object or process values. In this sense, the two-symbol system is inferior to the classical positional binary code, because it uses additional digits with a negative sign in the representation of a number, which introduces redundancy. This increases presentation time, and memory load, and increases channel transmission time. However, when using a two-symbol system, the Hamming distance (a measure of dissimilarity of code combinations—the number of bits in which the symbols of one of the compared combinations differ from the other) between the combinations may increase slightly (on average). It is a consequence of the introduction or presence of redundancy. Combinations appear that are inherent in this particular representation and those that cannot exist in this representation. For example, the number 5 can be written: 5=23−22+1=(1101)G=23−22+2−1=(1111)G , while the classic 1001 cannot be used and is a prohibited combination within the proposed twosymbol system. From this position, it can be suggested that there is a chance of using a two-symbol system as a simpler, not very efficient, but conditionally noise-resistant code. Additional research is needed to assess this possibility. First of all, this concerns the quantitative ratio between the allowed and forbidden combinations that arise in a two-symbol system. This is also related to the value of the possible code distance (the minimum Hamming distance between any two allowed code combinations). The greater the number of forbidden combinations, the potentially greater the code distance between the permitted combinations (dk), and the greater the error rate (number of distorted symbols in the accepted code combination) that the code allows to detect ( ydetect) or correct ( ycorrect): for detection for error correction
dk detect ≥ ydetect +1 ; dk correct ≥ 2 ycorrect +1 . (21) (22) (23)
The number of allowed values with the required code distance is important, and it increases with increasing code redundancy.
The given example of the representation of the number 5 also indicates the possibility of using the specified representation as a variant in the binary-decimal code. The information society requires effective information protection. Coding plays a key role in ensuring the confidentiality, integrity, and availability of information, and the basis of coding rules are mathematical methods and models. The authors of the article proposed directions for applying mathematical theory related to two-symbol number coding systems in the field of information coding.
We see directions for further exploration in a more detailed study of the possibilities of implementing this system, using the G-representation not only of a natural number, but also of the fractional part of a real number.
Declaration on Generative AI
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