Optimal Covert Communication Technology

Due to advancement in hacking/reverse engineering tools, threat against transfer of sensitive data or highly classified information is always at risk of being intercepted by an attacker. Covert communication outwit this malicious breach of privacy act better than cryptography as it camouflage secret information inside another innocent looking information, while cryptography shows scrambled information that might arouse attention of an attacker. However, the challenges in Steganography are the modification of carrier that causes some abnormalities, which is detectable and often the methods are not optimize. This paper presents an approach in Covert communication Chanel, which utilizes mathematical concept of combination to optimize time of transmission using sets of multiple transmitter’s, and receiver’s addresses where each abstractly represents a set of bits or characters combination without modifying the address. To minimize the number of physical address to be use, a combinatorial and permutation concept of virtual address generation from physical address is introduce. The paper in addition presents some technique like relationship and their application in both re-enforcing resistivity against Steganalysis and generating combinations. Furthermore, a concept of dynamical clockwise and anti-clockwise rotation of combination over addresses after every transmission is introduce to further improve on resistivity against Steganalysis. A simple test was performed for demonstrating relay address, combination and permutation concepts. Based on test results and analysis, the method is effective as expected and it is quite easy to use as it can be implemented in different platform without much difficulties.

Domains by using inter-word and inter-domains correlation using semantics analysis [16]. They took into account word embedding and same part of speech in addition to detecting using frequency distribution of words and part of speech. Furthermore, a paper [17], presented a method based on time interval or delays, which takes the interval of the time such as ∆ = − −1 and ∆ is then compare with chosen sequences of keys to decode or encode hidden binary. The paper further discusses the drawback of most steganography methods, which are due to modification of carrier, and that it gives loophole for some sophisticated algorithm or statistical method to detect steganography flow. Nevertheless, this problem can be solve in this proposed novel methods, which does not modify anything. In addition, the fact that most method hides a single bit one after another, it is extremely difficult to attained optimal transmission based on time for transmitting these floods of bits and this proposed method introduces bit/string and address combination techniques to combat such problem.

METHODOLOGY
This paper presents theoretical mathematical approach in security(Network steganography), which utilizes mathematical concept of combinatorial, and binary string concatenation by combining bits to optimize time of transmission, using multiple transmitters and receivers addresses, for example email addresses, mail addresses, phone numbers, or network ports or addresses and many others where each is assigned a bits combination. In addition, the paper present the concept of dynamical clockwise and anticlockwise rotation of bits combination over the given transmitter-receiver addresses after every transmission to reinforce resistivity against any form of security analysis like (Steganalysis) or cryptanalysis.

Combination
This section presents the concept of combination of bits or characters such that maximum bits or characters can be send at once by making a given address represent such bit combination. Below shows examples of bits combination. In addition, how a formulae for calculating total number of possible bits or character combination "C" given that a number of bits or character to be combine is "n". Let W be set of string of possible combinations resulting from a given combinatorial such that W = { 0 , 1 , 2 , 3 , . . −2 , −1 } and is the total number of elements of excluding empty set w = {∅}. Below indicates bit combination. One by one bit combination; w = {0 ,1} so total combination are = 2 and = 1 . However for Two binary combination, the possibility are W = {00 ,01 ,10,11} and total combination =2 where n = 2 For three binary combination, W = {000 , 001 ,010 ,100 ,011,101,110 ,111} total combination = 2 3 where =3 For four binary combination are; W = {0000, 0001, 0010, 0100,1000,0011,0101, 1111, … . } and total combination = 16 where n=4 Therefore, the pattern keep on increasing such that the power of possible binary combination "C" for a given bit combine "n" can be express as = 2 . From the above, it is clearly evidence that, for binaries combination, the maximum possible binary combination " " is as below where "n" is the total number of binaries combination. given that W = { 0 , 1 , 2 … , (2 −2) , (2 −1) }. Total element of set can be express as follows. We know that total elements (cardinality) of a set can be express as = ( ). = For example = {0,1}, total elements of set is two ( ) = 2 , meaning binary or base two number system, so = 2 Therefore can be express as; = (2.1)

Relationship
Given two sets of addresses , such that {( , ): ∈ , ∈ } and each element of one set is maximally relating to all the elements of the other sets and the inverse relationship holds true( = { 0 , 1 , 2 , … . } ∶ ∈ ℕ = { 0 , 1 , 2 , … . }: ∈ ℕ)The relationship of the two sets " " and " " can be describe as; ℛ ⊆ = {( , ): ∈ , ∈ }. For the inverse case where the receiver wants to reply to the sender, the relationship inverse can be express as ℛ −1 = {( , ): ( , ) ∈ ℛ} and are total elements of sets and respectively. Therefore since the transmission is related, such that,{ , } ∈ ℛ, a ℛ b, Maximum crossing among addresses" ℒ" for sending information can be express as ℒ = ( ) * ( ) ⟹ 0 < ℒ; ℒ ∈ ℕ Or ℒ = ( ) ( ) And to calculate total number of cross transmission is as ℒ. Please see Figure 1 below for the address relationship involving only Transmitters and Receiver addresses.

Transmitter's to Receiver's Address Relationship
Please see figure 1 for relationship without relay address directly from sender to recipient address without intermediate address.

Maximization of Address
Here, an idea of how to maximize total number of address based on concept of combinatorial and permutation to produce more virtual addresses. This approach is idea that given a set of address ( ) with more than one distinct element i.e. ( ) ≥ 2, a given combination of virtual address can be generated. For example. Given address such as = { , , }, virtual address ′ can be generated combinatorial as ′ = { , , , } and for permutation as These are all distinct element although some are virtual and others are real address, so total address available for use has increased to + ′ . Three real addresses has generated four virtual addresses and in total seven new addresses are available for use when using combinatorial concept, However, for permutation twelve virtual addresses are generated from three real physical addresses. 15 addresses available for us.

Combinatorial of Address
For combinatorial nCr of address to generate virtual address, here, order of virtual address combination does not matter much as transmission is simultaneous, so indexing address is difficult if order of address combination is to be taken into account such as in permutation.
For an address with two real elements, virtual can be generated as = { , } where virtual is ′ = { } only one distinct virtual address element can be generated. So formulae for finding total elements of virtual address ′ i.e. ( ′ ) that can be used from a given real address is shown here (2.5) where number of real address total is ( ) and is Combination xCr of address, is the selected address in combination.
For relationship involving relay addresses, please see equation (2.31). In addition, for = 0 represent address for = represents Address , for 0 ≤ ≤ represent relay addresses.

Relay Address (R)
For address can be express as in (2.32) However, for address involving relay address please see (2.33)

Permutation of Address
In permutation, the order combination of physical address forming virtual address does matter very much because transmissions are sequential not simultaneous, and it is time index i.e. is different from . So two or more address combination representing one virtual address can be rearrange in such a way that the order of those address distinctively represents different address for example. A transmission from physical address and can be from the virtual address transmission received at 0 and 1 respectively given that 0 < 1 . For virtual address transmission received at 2 and 3 respectively given that 2 < 3 . It should be noted that to differentiate between virtual address sequential transmissions, time of transmission from the same combination should be within a defined range ∆ or see (2.34). Where values set such that ∆ ≤ ∆ = − −1 (2.34) From equation (2.5), permutation, nPr of such combination is the permutation of entire virtual address plus physical address generated from combination can be written or express in (2.35).
Where is total number of address and number of address chosen. Just like in (2.30~2.33) total relationship and addresses involving relay address can be written as in (2.36) and (2.37).
Please note: handling zero factorial in address here is when its zero. Zero factorial is defined as a mathematical expression for the number of ways to arrange a data set with no value in it, which equals one by definition (0! = 1). So, for this address, since total number of address is greater than zero, so ≥ 0 so ( ) much be greater than zero.

Concatenation
The concept of concatenation is mainly use in formal language theory like in programming languages and pattern. Concatenation of two strings and is often denoted as , || , or, in the Wolfram Language, <> [19]. However, throughout this text, it is denoted as a|| . From the two sets of strings of binary assigned to addresses A and B, the concatenation || consists of all strings of the form || where "a" is a binary string from A and "b" is a string from B, or formally A||B= {a||b: a ∈ A, b∈ B} for concatenation of a string set and a single string, and vice versa. A||b = {a||b: a ∈ A} and a||B = {a||b: b∈ B} However, as given by the work [19], the concatenation of two or more numbers is the number formed by concatenating their numerals. He gave an example, the concatenation of 1, 234, and 5678, whichare12345678. In addition, the value of the result depends on the numeric base. He further presented the formula for the concatenation of numbers and in base as in (2.38). Therefore, throughout this paper, binaries, or stream of bits are treated as string and string concatenation formulae and rules/law are applied as below.

2.5.
Associative Law Rules of Binary operation applicable to string Concatenation here below. For the binary operation, is associative and repeated application of the operation produces the same result regardless how valid pairs of parenthesis are inserted in the expression. A product of two elements (addresses or bits combination) (( , ): ∈ , ∈ )may be written in five possible ways as below.

Order of Concatenation
Below, we discussed some of the order of concatenation that needs to be followed in order not to lose track of the transmitted code of combination. See Table 1 showing sample bit combination  Given time series of transmission, as = { 0 , 1 , 2 , … , } such that ( > −1 > −2 > −3 > ⋯ > 1 > 0 ) By defining sender to receiver order as and receiver to sender order as Therefore from sender to receiver order = ( || ) 0 ||( || ) 1 ||( || ) 2 || … … ||( || ) And from receiver to sender order = ( || ) 0 ||( || ) 1 ||( || ) 2 || … … ||( || ) Above is an example of bit combination in table form shown in Table 1 Base on combination concept, to send letter 'H'=01101001, Can be separated into two 4 by 4 bits combination and sent at once in a single transmission so each transmission will carry one character of 8-bits (1byte)

Application of Relationship for Generating Combination
From the rule of string concatenation given above.
ALGORITHM 1: Homogenous Relationship an example see Table 1 Function: Combination( , ) Please note from the above algorithm 1, function Array_size(x,y) is for pre-allocation of array of size however in some programming language, array size are allocate dynamically and use of this function is not needed. See Figure 3 for example of above algorithm 1 implementation.   Rotation over Addresses Let represent sets of bit combination over addresses A, or B. for subscribe of W " " is the current position of bit combination over a given addresses A or B of index , and "J" represents total transmission time such that( ∈ ℕ, ∈ ℤ ).In addition, Q is the maximum addresses ( ∈ ℕ) or in other word, totals elements of set A or B.for example if 2 = 2 = 2 Given a bit combination, "C" can rotate over elements of either set A or B after every transmission such that the rotation is either clockwise or anti-clockwise. = { −1 , −2 , . . 2  Below are pseudo-code created functions that uses the above two function for rotation, to rotate array of string. Since stream of bits (binary) are treated as string so it can be converted into it subsequence array and the array index manipulated such that it is rotated as prescribe above. The idea creates two functions where each is input initial index , , and the function returns a numeric index after rotation for each element of the array. Furthermore, another function which call the two rotation function is created which determine the value of Where if it is negative, it calls anti-clockwise rotation, else it calls clockwise rotation, and the function returns array of all string after its rotation see Fig 7 for the output of the algorithm being tested.

EXPERIMENTAL RESULT
In this section, three experimental results are presented where the first one is done in a very simple environment to make it easily understandable by non-specialist is covert channel communication and easy to perform the experiment. It is based on the idea of using many phone numbers from two different locations where the confidential information is to be transmitted from and to a given location with those numbers representing the address. The second experiment was done using multiple client machines located in given location where information is to be send from to another location where recipient is located, in the recipient location are locate multiple server machine through this use of email forwarding functionality and use of intermediate proxy server is used.

Experimental Result Based on Rotation
In this, at the sender side, six phone numbers was set up as sender addresses and receiver side six phone numbers also was set up as receiver addresses. At sender side, only two bit combination were used and the same four at the receiver side. See Table 2  To withhold the identity of the user's phone number from exposing, only six digits are shown without their location of calls only just labeled as sender and receiver's location. Furthermore an extra phone number is used in addition to the one assigned bit combination, those phone number are label as null which means it does not carries any bit combination and any phone called from or to such carries no bit combination whether it is directed to the one with assigned bit combination see Table 2 for more details.

Experimental Result 2 Based on Relay Address and Permutation Without rotation Electronic Mail Forwarding functionality:
Here from the email address sender, send an open message to recipient via relay email by forwarding the email. Please see table 5 below contains sample email for transmission through relay address. To withhold the identity of email address involve in this practically, notation is use to represent email address such as A1@email.com. In Table 4 10 11 The following message were recorded from the above emails address in the order (A2-R2~R1-B1)+ (A1-R2-B1~B2) + (A2-R1-B2~B1) + (A1~A2-R1-B2) decoding this into binary (011100000110010011010001) 4. ANALYSIS 4.1. Optimal Performance Since this combine bits such that two or more bit can be send at once, so time of transmission reduces significantly, as many bits can be send at once.
Let " " be time required for sending total of " " bit or character combination for both relay and or without relay address at once. And "k" is a constant unit time per one bit transmission such that: ∝ ⟹ = and = = −1 (4.1) Therefore lim →∞ ( = ) ≈ 0 ; giventhat = 1 From the above, it is evidence that when number of bits combination increases so does the time required for transmission reduces.

4.2.
Secure Analysis Using Probability This section shows how resistive the method is from someone or a program that need to detect it by applying probability theory. Probability without Relay Address Let sample space be for a total address at either sender or receiver's side where a bit combination can occupy at a given time. Therefore, the probability that a chosen bit combination occupy such address is as below: For the probability that it is not is given as ′ = (1 − 1 ) lim →∞ (1 − 1 ) ≈ 1 However, the probability that a chosen transmission line carries the right bits combination is different from above as the sample space is from equation and equal to maximum crossing, a combination of bits between sender and receiver addresses send at once. Let sample space be maximum crossing between sender and receiver's addresses (1 − 1 ℒ ) ≈ 1 This indicates that, increase in number of bit combination → ∞, makes probability ′ ≈ 1 Probability with Relay Address In this part, we shows the probability that a chosen line of transmission from sender to receiver through relay addresses are as below, it's an expansion of the probability for a chosen line transmission above. Here find the probability within each relay from the sender up to receiver. Consider the following: − 1 − 2 − 3 − ⋯ . − − From transmitter address to relay address 1 the probability that the line carries hidden information is else the probability that it does not carries is denoted as ′ for the next node of relay address it is given as 1 and ′ 1 respectively this continues up to the last point of receiver. Overall probability that the entire transmission carries hidden message involving relay addresses can expressed as an independent probability which is the product of all individual probability [20] and express as below (4.2); = ( ) ( 1 )( 2 ). . ( ) (4.2) And the probability that it does not carries is given as (4.3) ′ = ( ′ ) ( ′ 1 ) ( ′ 2 ) … . . ( ′ ) ( ′ ) (4.3) From above it can be noted that as more relay address is added, the probability that a chosen line of transmission caries hidden information tend to almost zero proving theoretically that this approach is good. 5. SAMPLE EXAMPLE Example 1: Given a bits combination n=3 after J=11 times of transmission in clockwise rotation. In addition, an initial position of the bits I= 3. a) What is the new position of the bits combination W 3 ? Solution: Therefore: = 2 3 ∶ = 3 < ( , w, . ) = W ( ⊘2 )+ , ≥ 2 ( , w, , ) = W (11⊘2 3 )+3 ( , w, , ) = W 6 The new position or index is six (6). So if on sender addresses is 3 6 or for receiver address, it is from 3 6 . b) Assuming after example 1, and rotating in anti-clockwise for 19 times transmission. What is the new position of the bits? Solution: Therefore: = −2 3 ; = 3 ≤ − and position I=6 is a new position after clockwise rotation. And since anti-clockwise, J=-19 ( , w, , ) = W ( ⊘−2 )+ , ≤ −2 ( , w, , ) = W (−19 ⊘ − 2 3 )+6 ( , w, , ) = W −3+6=3 So new position= 3 The new position or index is six (6). So if on sender addresses is 6 3 or for receiver address, it is from 6 3 . Example 2: A person want to send a hidden message by post office mail, from country "A" to country "B" using 8-bits binary system such that the mail carries normal message without being modified: a) How many mail addresses are requires from sender and receiver country so that at least each mail sent carries a character of 8-bits? Solution: