Novel Dynamic S-Box Based on Password Key and Circle Map

Many cryptosystems and security techniques use substitution boxes to ensure the data’s secure communication. A new technique is presented for generating a robust S-box to fulfill security requirements. The AES algorithm represents a block cipher cryptographic algorithm. It was selected by the National Institute of Science and Technology as the optimal cryptographic algorithm in 2011. Through the study of the properties of original S-BOX, this algorithm has been subjected to a number of attacks (linear, differential, statistical, and interpolation), and original S-BOX has been static, which makes the attack strong and shows a weakness in the algorithm. It is necessary to make this algorithm more efficient and powerful through the improvement of the dynamic generation of the steps for the protection of textual data security. This paper proposes a dynamic S-Box based on the user's password key (8 chars), shifting, and a 1D circle map. The results in this work indicated that the suggested approach presents a secure S-BOX, which is considered to have 255 differences identified when 1 bit of the key is changed; therefore, about 99% of the S-Box has been changed. Also, an inverse table of S-Box (16*16) is generated via the S-Box output created from the above-mentioned suggestions for returning the values regarding the union of the column and the row for all the S-Box generated values. We examine the quality of our S-Box through various well-known performance parameters. All of the analysis yields very encouraging results, certifying that the generated S-box meets all criteria that are required for reliable and secure encryption. Just a few milliseconds are needed to implement it.


Introduction
Many parties use cryptography systems to communicate securely.A block cipher helps to accomplish information privacy, which is one of the objectives of cryptography.Text information has a massive impact on all aspects, both in public and in personal life.Because of the increases in the utilization of intrusive programs in the past few years, there is a necessity for improving the algorithms, due to the fact that the majority of modern encryption algorithms have been exposed to consecutive attacks [1].It is one of the most widely utilized tools for ensuring the security of information.Many block ciphers rely on permutation and substitution or the Feistel structure [2].The circle map has the ability to show these behaviors as mode and phase locking, sub-harmonics, and period doubling [3].With regard to the field of cryptography and all of the symmetric cryptographic algorithms, the S-Box is considered a nonlinear unit regarding the encryption algorithms, typically because of the associations between the cipher and the key, which is referred to as "confusion."The key purpose of this study is to use chaos theory for the generation of the random S-BOX and password keys for updating all of the values in the state to the generated S-BOX [4].

Literature Survey 1-In 2011, Jolfaei and Mirghadri [5]:
The chaos is utilized for expanding confusion and diffusion in images due to the fact that it is sensitive to initial conditions.Barker's maps have been utilized for designing dynamic maps of permutations and S-boxes.The suggested algorithm has been measured via a series of tests, which include visual tests, randomization testing, graph analyses, coding quality, information interruptions, and correlation analyses.Results have been quite good for ten rounds and a 128-bit data block.[6]: have suggested an approach for overcoming the weaknesses in the S-BOX and improving AES performance through the replacement of the Mix phase with a chaotic system, the process of the XOR, for the reduction of the high computational costs of the Mix column, and the generation of the S-box based upon the chaotic system.Results have shown that the suggested approach was successful in generating an efficiently encrypted image that has very low coefficients of correlation between neighboring pixels and provides high operation speed.

3-In 2015 (Abulgader et al.) [7],
an approach for overcoming S-BOX drawbacks and improving AES performance was proposed by replacing the Mix phase with a chaotic system in which the XOR process resulted in the reduction of high calculations in the Mix column and the generation of the S-box based on the chaotic system.Results have shown that the suggested approach enabled the generation of an efficiently encrypted image with very low coefficients of correlation between adjacent pixels and provided high operation speed.[8] concentrated on the encryption of colored images using 3D Chebyshev maps to generate secret keys for image diffusion and 2D Arnold Cat maps to generate S-box using the XOR function with old S-box.The suggested algorithm has been tested with the use of the UACI, NPCR, and information entropy.Results have shown that the algorithm has the ability to resist different kinds of attacks.

5.
A study by Kamel and Farhan (2017) [9]: created a new block cipher as a moving structure based on two of the proposed algorithms.The first algorithm utilized complementary DNA functions (shift and two S-box layers).In the second algorithm, it included the degree of shift and addition of the DNA.In engine implementation, it resulted in the generation of a secret key using a chaotic generator agreed upon by sender and receiver, and text was encrypted using multiple measurements and the NIST's five statistical analyses, where the benefits of all tests outweighed their low efficiency.

Circle map
Circle maps can be defined as mapping examples showing various significant factors in complex dynamical behavior.They have the ability to show these behaviors as mode and phase locking, sub-harmonics, quasi periodicity, and period doubling, along with routes to chaos through recurrent disruption to quasi periodicity or period doubling.In particular, it is adequate for studying and generating sustained undamped sounds when the possible iterations' space is confined by the map to functions regarding such nature through constructions [10].

Iterated Maps from Circle to Itself
The most generalized circle map type is: The mapping from bounded intervals to identical bounded intervals is represented by Ф. Commonly, the unit interval is taken and indicated as Ф: [0; 1) [0; 1), that otherwise might be considered to be periodically-closed.Also, it is done through taking quotient regarding real numbers via integers, and repeat the real's in interval [0; 1), then indicated Ф: R/Z R/Z (Milnor's 2006, p161).Topologically, one can also say that Ф maps points on circle back onto circle.In the case of wanting to model an excellent sinusoidal oscillator that has been perturbed via a certain coupled nonlinear function, which becomes: In which, Ω represents a constant which is considered as a fixed angular progress with regard to a sinusoidal oscillator, while k represents the strength of the coupling that is related to the nonlinear perturbation f (.).Y0 represents the starting phase.The option of f (.) is extremely flexible, whereas discontinuous functions' examples are identified in smooth cases and literature as the smooth cases.The canonical theoretical example has been represented by the standard circle map: [11] For studying the long term behavior regarding the iterated map Ф(.), one might consider the winding number Which measures average angle that is added in long-term.In the case when such an added angle that has been notated over a [0; 1) interval represents a rational value p/q with p; q ℇ N, after that following q iterations, one will have some recurrence; thus, this map will be periodical.In addition, the irrational winding values have been referred to as "quasi-periodic," which of course is the frequency that is associated with an unperturbed oscillator and assessed as ФΩ = ΩS where S represents the rate of sampling, or time interval between 2 of the time steps for Ωℇ [0; 0.50].In the case when Ω is greater than 0:5, one can get aliasing, and the effective frequency will be reduced once more, with the opposite sign of the phase, as shown in Figure 1.[12]

Relation to the Other Maps
Some of the maps that have been mentioned in the literature include circle maps.For instance, Di Scipio had considered the term that he had referred to as "sine map" (Di Scipio 1999): [13]  +1 = ((2  ) )

Proposed Dynamic S-Box
The phases to generate an INV S-Box and a new S-Box might be indicated as follows:

S-Box Generation
Because of the encryption algorithm's repeated exposure attacks, there is a requirement to reinforce the algorithms to be steadfast against all attacks [15].The algorithm is going to be enhanced via a dynamic S-Box as follows: Step 1: Enter the password key (8 characters) and expand it to 64 characters by converting the ASCII code for each character in the password key to binary and then generating a random number using the circle map.The method of generating random numbers is that the user must select the number of rounds (number of i) through the generation of a circle map.Suppose the user selects the number of rounds as "22."Now cut 8 numbers after the comma.In this case, a new word was generated, which will be the new password key, and the same operation as in step 1 was repeated until 64 characters were generated.The circle map equation is utilized for generating the random numbers: θn+1= θn + Ω -k/2  * sin (2  θ n) (8) [16] where θ value lies between 0 and 1.It has 2 parameters, which are, coupling strength K and driving phase Ω.As model for the phase-locked loops, Ω could be interpreted as driving frequency.For K = 0 and Ω irrational [17].
Step 2: Generating 512 numbers from the password key (64 char), which is converted to binary (512 bit), taking the first 16 bits from 512 bits and converting them into integers, shifting the 16 bits to the left 15 times for another 15 integer numbers, and so on until the last 16 bits.Note: If the number of shifts = 9, then the number of shifts = 0.
Step 3: Remove duplicate numbers and fill it with the remaining numbers between [0, 255].Then generate a 2D matrix containing 256 random numbers.
Step 4: Apply the IP (Initial Permutation) of the DES algorithm [18].

INV S-Box Generation
The inverse table S-box has been invoked by returning row and column values for every number.

Example of S-box Generation
1.The first step is to perform a 1D circle map, where θ = 0.5, Ω = 0.33, k = 0.8 and  = 3.14.The result has been listed in Table 1.The same operation for the second 16 bits and so on until the last 16 bits.The resulted 2D matrix contains 256 random numbers.
4. Remove duplicate numbers and fill it with the remaining numbers between [0,255] as shown in Table 3. 6.The final result of the S-box is shown in Table 5, and the S-Box Inverse has been shown in Table 6 after converting the values in Table 4 to HEX.

Experimental Results
The proposed new S-BOX is going to be put to the test against many standard statistical parameters.

Avalanche Criteria (AC)
Changing the input amount will totally impact the amount of output when the avalanche value ranges from [0, 1], and assessed from equation: Avalanche Effect = .    (/)ℎ  .   (/)ℎ  Note the signification differences between the outputs of Tables 5 and 6 when changing just one character in the password key until the difference reaches 99% using the equation.Avalanche effect= 254/256 = 0.99

Complexity
Through the generated results, one can identify the S-Box complexity, in which the expansion phase regarding the password key is extremely complex since it is going to depend on equation ( 8) of the circle map.For instance, when changing just one character in the password key and using the same value parameters, the S-BOX results in a different percentage of 99%, as shown in Tables 7 and 8.

Time
For implementation, the time is small and requires only a few milliseconds, as listed in Table 9.

Conclusions
In this work, S-Box was developed on the basis of a password key, a circle map, and shifting.The results of this study show that changing 1 byte in the password key without changing the circle map parameters will change the password expansion process, which affects S-Box output.This refers to S-Box generation complexity.The major point is that a dynamic S-Box isn't static, yet it is based on the input, indicating that the security was generated during the generation.Using IP in this proposal contributed to increased randomness, which led to an increase in diffusion and confusion.

Figure 2 :
Figure 2: Sensitivity to the start position of the iteration yo at k =6:4, Ω = 0:11, 2 orbits (top and bottom) are stable, 1 (center) is chaotic.All of the orbits are nonsingular.

Table 1 :
result of 1D Circle Map …… 2. Expansion password key.Suppose the initial password key is "computer".The result of expansion is shown in Table2.

Table 2 :
result of expansion password key

Table 3 :
Generate 16 Numbers from 16 Bits by Shift and Rotates after remove duplicate

Table 4 :
Update Values by IP of DES algorithm

Table 5 :
Dynamic S-Box

Table 6 :
Dynamic S-Box Inverse

Table 7 :
Dynamic S-Box when password key is "Bomputer"

Table 8 :
Dynamic S-Box Inverse when password key is "Bomputer"

Table 9 :
Results of execution time {0,1}x →{0,1}y has been defined as least value of non-linearity of all of the non-zero linear combinations of x Boolean functions fi : {0,1}→{0,1}, i= x-1,…,1,0.The non-linearity of the S-box has to be high in order for it to resist the linear cryptanalysis.