 Open Access
 Authors : Shubhankar Vashishta , Lakshay Srivastava , Dr. C. Jothi Kumar
 Paper ID : IJERTV10IS040137
 Volume & Issue : Volume 10, Issue 04 (April 2021)
 Published (First Online): 22042021
 ISSN (Online) : 22780181
 Publisher Name : IJERT
 License: This work is licensed under a Creative Commons Attribution 4.0 International License
An Improved SBox Generation Method using Metaheuristic Optimization Technique
Shubhankar Vashishta Computer Science and Engineering SRM Institute of Science and Technology Chennai, India
Lakshay Srivastava
Computer Science and Engineering SRM Institute of Science and Technology Chennai, India
Dr. C. Jothi Kumar
Computer Science and Engineering SRM Institute of Science and Technology Chennai, India
Abstract Substitution boxes (Sboxes) are a crucial nonlinear component in modern block and stream ciphers' cryptanalytic resistance. Due to their relevance, there is a wide range of Sbox construction techniques. The success of AES (Advanced Encryption Standard) posed cryptographers with new challenges in creating powerful substitutionboxes using various underlying approaches. There are various parameters that play a vital role in creating a robust SBox that is secure enough to use which includes Nonlinearity, differential uniformity, absolute indicator value of global avalanche characteristics, Bits Independence Criterion (BIC), confusion characteristics, transparency order etc. We can obtain the desired value for a parameter by using various optimization techniques like PSO, GSA etc. In the proposed scheme, metaheuristic optimization technique will be used for setting the values of the abovementioned parameters.
Keywords Sbox, Particle Swarm Optimization, Gauss Iterated Map, Cryptography

INTRODUCTION
The protection of private data and visual information have been a major concern for years, given the transparent and vulnerable nature of Web and networking technologies. Since many years, the cryptographers have proposed different types of information protection methods. Depending on how information is interpreted encryption techniques can be divided into stream and block ciphers [1]. A block cipher is a method of text cryptography where the encryption key and algorithm are applied to the block of data at one time, called blocks, with an invariable transformation. The block cipher use permutation and replacement layers to design efficiency that shows high uncertainty and scattering properties. For such networks, substitution boxes are crucial components intended to convey the necessary nonlinear data transformation, that in turn contributes to better uncertainty and strength to various cryptographic attacks. The substitution process utilizes block bits for input and non linearly converts them into various block bits for output [2]. It is indeed a linear conversion of the input sequence, as opposed to shuffling, which refers to the permutation
variable size Sboxes. The scale of the massively bulky search space is one of the essential causes of this difficulty. As a result, a chaotic metaheuristic optimization approach is constructed to develop a competent framework of an Sbox with varying size that can produce effective Sboxes.
The initial conditions and control parameters of chaotic systems are highly sensitive to these systems, which is why they are regarded as good origin of entropy. They have strong responsiveness to preliminary constraints and system specifications, as well as quick autocorrelation and arbitrary nature of produced data [2][4]. Even small adjustments in the preliminary constraints and governing conditions have a significant impact on performance, making chaosbased structures ideal for the development of robust encryption algorithms. Chaos based mechanisms on the other hand, can never exhibit chaotic behavior to every value of the preliminary constraints and governing conditions [5]. The chaos based behavior of the selected scheme is the first requirement inside the architecture of chaotic encryption algorithms. Therefore, when choosing initial conditions and control parameters, caution must be taken. The chaos based structures' preliminary constraints and governing conditions are realvalued in the interval in which they are described. And it has a vacuum of infinite value. Optimization algorithms are needed to pick the optimal chaosbased framework through this unlimited search space.

PRELIMINARIES

Sbox parameters

Nonlinearity
The least separation of a Boolean function f to the collection of each affine function is used to calculate its nonlinearity measure [6]. As a result, standing nonlinearities scores should be reflected in the Sbox constituent functions. The nonlinearity NLfn for each Boolean function fn is evaluated with Eq. (1) [13]:
process.
= 1 (2
()) (1)
2
The advancement and improvement of proposals dedicated to the development of substitution boxes has contributed significantly to the success of the AES block cipher as well as its substitution box [3]. They are stable, with structures that emphasize on algebraic strategies, optimization, chaos function as well as structures, and so on. A dynamic and open problem is the design of effective and
where, Wmax(fn) is known as WalshHadamard transform of Boolean function fn [7]. If a Boolean function has weak nonlinearity, it is called fragile. The increase of nonlinearity of stable Boolean functions is regarded as among the most important steps offering control resisting linear attacks [7]. Table I lists out the nonlinearity of the some the Sboxes
used in this paper to compare the nonlinearity of the proposed scheme.
TABLE I. COMPARATIVE ANALYSIS OF NONLINEARITY OF 8 x 8 SBOXES.
Substitution box
Nonlinearity (min)
Ref [17]
84
Ref [18]
98
Ref [19]
98
Ref [20]
100
Ref [21]
102
Ref [22]
102
Ref [23]
106

Differential Uniformity
The differential uniformity compares an SBox's resistivity to differential cryptanalysis. Biham and Shamir defined the cryptanalysis attack technique, which involves creating a disparity in the I/O distribution for pupose of attacking block ciphers and Sboxes [8]. If the XOR value of every output has similar uniformity to the XOR value of every input, then the cryptanalysis can be completed [9]. When the input/output distribution of an Sbox is uniform, it is said to be immune. In the XOR table, the greatest value of differential uniformity (DU) shall be kept low. DU of any Boolean function fn(x) can determined by [13]:
() =
max (# { () ( ) = }) (2)
0,
where set Y contains every possible input value and its components have a figure of 2n. For an Sbox, the largest value of the XOR table should be small enough to prevent cryptanalysis. Table 2 compares the differential uniformity of various chaotic substitution boxes.
TABLE II. COMPARATIVE ANALYSIS OF DIFFERENTIAL UNIFORMITY OF 8 x 8 SBOXES.
Substitution box
Differential uniformity
Ref [17]
16
Ref [18]
12
Ref [19]
12
Ref [20]
14
Ref [21]
12
Ref [22]
10
Ref [23]
10


Gauss Iterated Map
Gauss Iterated map is one of the most popular 1 dimensional chaotic map. It is a simpl function which displays chaotic behaviour with discrete time domain and real space domain [10]. It is dictated by the Eq. (3):
+1 = (2) + (3)
where and are the control parameters which govern the bifurcation and xn is the function variable. The Gauss Iterated map produces best results when the value of is set between 4.5 to 8 and the value of lies between 1 to 1 [11]. Figure 1 shows great chaotic characteristics of Gauss Iterated map for = 6.2.
Figure 1. Bifurcation of Gaussian map at = 6.2
Figure 2. Flowchart of Particle Swarm Optimization.

Particle Swarm Optimization
Particle swarm optimization (PSO) is a metaheuristic optimization algorithm, that was proposed by Eberhart and Kennedy in the year 1995. The model was influenced by studying the behavior of fishes and birds. PSO has been used in a wide range of optimization problems, both alone and in conjunction with different algorithms [12].
Each particle obeys a specific path, being a positional vector that depends on time. Two major key pieces consist of the swarming particles movement: a stochastic bit and a deterministic bit. By differentiating the paths of these discrete particles PSO explores the domain of an objective function. Every particle is drawn closer to the current global best position (g*) and its personal bestknown position () while at the same time showing a tendency to shift impulsively over time. The particle modifies the position as the best new particle present i when it finds a position that is better than any location previously found. At any given moment, at each iteration, there seems to be current best for every particle. The main goal is to discover the best global solution from all of the existing best solutions until they stop improving or after a certain amount of iterations.
Let the position vector and velocity be represented as xi and vi respectively for the ith particle. The Eq. (4) defines the new velocity vector:
+1 = ( ) + 11( ) + 22( )
according to the heuristics mentioned [14]. The initial population, i.e. N, is hence set to 40.

Setting the fitness parameter:
The fitness value for the problem is determined by a specific parameter. The nonlinearity of the Sbox (particle) is observed as fitness value in the population vector in Eq. (4) and Eq. (5).

Initializing the vectors:
The adjustment vector is defined at the start and modified after each iteration. Each place vector initializes the values of the respective Sbox in the sample. As per the Eq. (4), the adjustment vector is modified. For each Sbox, the best personal vectors are determined. The recently created Sbox in the sample which has a higher fitness value than the preceding Sbox is updated as the personal best vector ( ). The Sbox with the highest nonlinearity in the population is used to
describe the global best vector (gb). The AES sbox is set as global best (gb) in Eq. (4). The nonlinearity for AES sbox comes out to be 112. The reason for selecting AES sbox for global best position is due to the fact that it is one of the most widely used sbox having extraordinary resistance to various cryptanalytic attacks.

Setting parameter values:
PSO parameters like r1 and r2 are kept contant at 0.9 in
(4)
Eq. (4). The control constants, k1 and k2, are randomly
where and denotes the position and velocity of ith particle initialised with the chaotic value provided by the 1
at t times. r1 and r2 are two arbitrary vectors with values ranging from 0 to 1. The parameters c1 and c2 are the constants of acceleration, usually equal to, say, c1 c2 2 for balanced approach in its stochastic and deterministic way. The initial positions of all the particles should be distributed sufficiently uniformly so that they can be sampled across most regions. We can then update the new location using the Eq. (5).
+1 = + +1 (5) Here, vi can take any value. However, it is generally bounded within a certain range [0, vmax].


PROPOSED WORK
The Gauss iterated map is used to spawn the first population of the Sboxes. The two parameters, and , are set to 6.2 and 0.38 respectively in Eq. (3). The reason for utilization of these particular values lies in the chaotic nature of the map. The gauss iterated map shows exemplary bifurcation for above specified values. The gauss iterated map is also used to achieve the control parameters, k1 and k2, of the PSO. The research indicates that incorporating chaos through adjustment vector update contributes in a better search area exploration and use and can improve standard of the output influenced the chaosbased initialization and updating of PSO parameters [13]. Steps for the PSO technique are as follows:
A. Creating the vector of population:
Every distinct Sbox is considered a single entity. Using chaotic maps, the initial population of Sboxes is created. The size of the original generated population is kept small
dimensional gauss iterated map for the Eq. (4). The selection of control parameters is left upon the chaotic map for increased unpredictability and dynamic range of output. During the optimization process, these parameters are modified after each iteration. The PSO inertial parameter z is initialised to a 0.7 in Eq. (4) for the best optimization results. This parameter helps in concentration of the Sboxes in accordance to the respective nonlinearities.

Preprocessing and Adjustment:
The Equations (4) and (5) are used to update the adjustment vector and Sboxes for each iteration. This procedure generates certain repeating and negative values. The solution values for the Sbox architecture, on the other hand, are limited to a range [0,255]. As a result, we use a preprocessing and adjustment method to eliminate the possibility of repeating and negative values. Negative values are manifested in the desired range during preprocessing using some mathematics. To preserve the bijectivity of the Sboxes, we look for repeating values during the adjustment process and replace them with missing values [15]. The nonlinearity values are updated for the newly created sample. Optimum nonlinearities are maintained and sent to the recently updated sample, which is double the size (i.e. 80) of that of the
original sample, from the current sample. and gb are revamped, as previously stated. Adjustment vector is denoted by for the ith Sbox and tth iteration.
Initial values for +1 = (2) + :

Gaussian map initial value, xn = 0.231

Gaussian map parameters, = 6.2, = 0.38
Ref [20]
100
14
Ref [21]
102
12
Ref [22]
102
10
(6)
Ref [23]
106
10
Ref [20]
100
14
Ref [21]
102
12
Ref [22]
102
10
(6)
Ref [23]
106
10
+1 = () + 0.91( ) + 0.92( )

Inertial parameter, z = 0.7

In order to simulate, evaluate, and optimize the output of the proposed method of generating an Sbox, we use the input arguments of the method proposed. The suggested approach is evaluated by changing sample range, iterations,and inertial parameter under various scenarios and conditions. The number of iterations for optimization is set as 100.


PERFORMANCE ANALYSIS
A total of 4000 Sboxes were generated and optimized by particle swarm optimization technique. The Sbox in Table IV is the best outcome among the generated Sboxes. For the values of various initial parameters, the minimum value of nonlinearity of Sbox comes out to be 104. The same Sbox displays the differential uniformity of 8. However another optimized Sbox, show in Table V, with nonlinearity 102, displayed better differential uniformity with value 6.
TABLE III. COMPARATIVE ANALYSIS OF PARAMETERS OF 8 x 8 SBOXES.
Various Sboxes from the literature are compared on the basis of nonlineaity and differential uniformity shown in the Table III. Figure 3 shows the competence of the Sbox security scheme proposed in this paper. The fact that an S box is bijective means that it is a onetoone mapping. That is to say, all feasible resultant vectors shall emerge only once. Since each 256 possible resultant value is unique and show up one time, both Sboxes in Table IV and V preserve bijectivity. For Sboxes that are used in block ciphers.
One of the key problems in the world of cryptography over the last two decades has been the construction of extremely nonlinear Sboxes. By using the suggested PSO dependent approach, we can chieve Sboxes that are very similar to the 8×8 Sbox with nonlinearity as high as 112 [16].
Differential cryptanalysis can be restricted for an Sbox demostrating minimal DU [8]. Our Sboxes are measured with differential uniformities of 8 and 6 for S1 and S2,
respectively. Table III. shows that the proposed Sboxes
Substitution box
Nonlinearity (min)
Differential Uniformity
Proposed S1
104
8
Proposed S2
102
6
Ref [17]
84
16
Ref [18]
98
12
Ref [19]
98
12
Substitution box
Nonlinearity (min)
Differential Uniformity
Proposed S1
104
8
Proposed S2
102
6
Ref [17]
84
16
Ref [18]
98
12
Ref [19]
98
12
have greater capacity than other Sboxes to reduce differential cryptanalysis. The proposed Sboxes therefore have DU efficiency and demonstrate considerable resistance to differential cryptanalysis.
Figure 3. Comparison of nonlinearity
Figure 4. Comparison of differential uniformity
TABLE IV. PROPOSED SBOX S1
99
124
119
123
242
107
111
197
48
1
103
43
254
215
171
118
202
130
201
125
250
89
71
240
173
212
162
175
156
164
114
192
183
253
147
38
54
63
247
204
52
165
229
241
113
216
49
21
4
199
35
195
24
150
5
154
7
18
128
226
235
39
178
117
9
131
44
26
27
110
90
160
82
59
15
179
41
227
47
132
83
209
0
237
32
17
30
91
106
203
190
57
74
76
88
207
208
239
170
251
67
77
51
133
69
40
2
127
80
60
159
168
81
163
64
143
53
157
56
245
188
182
218
33
16
66
84
210
205
12
19
236
95
151
68
23
146
167
126
61
100
93
25
115
96
129
79
220
34
42
144
136
70
177
184
20
222
94
11
196
224
50
58
10
73
6
36
92
194
211
172
98
145
149
214
121
231
200
55
109
141
213
78
169
108
86
219
234
101
122
174
8
186
120
37
46
28
166
180
198
232
221
116
31
75
189
139
138
112
62
181
102
72
3
246
14
97
228
87
185
134
193
29
158
225
248
152
238
105
217
142
148
155
243
135
233
206
85
244
223
140
161
137
13
191
230
65
104
249
153
45
252
176
255
187
22
TABLE V. PROPOSED SBOX S2
99
124
119
123
242
107
111
197
48
1
103
43
254
215
171
118
202
130
201
125
250
89
71
240
173
212
162
175
156
164
114
192
183
253
147
38
54
63
247
204
52
165
229
241
113
216
49
21
4
199
35
195
24
150
5
154
7
18
128
226
235
39
178
13
9
131
45
26
27
110
90
160
82
59
214
179
41
227
47
132
83
209
0
237
32
252
177
91
106
203
190
57
74
76
88
207
208
239
170
15
67
77
51
133
69
249
2
127
80
60
159
168
17
22
64
143
146
157
56
245
188
182
218
33
16
30
40
210
205
12
19
236
95
151
68
23
196
81
117
61
100
93
25
115
96
129
79
220
34
42
144
136
70
126
184
20
222
94
11
219
224
50
58
10
73
6
36
92
194
211
172
98
145
149
163
121
231
167
55
109
141
213
78
169
108
86
244
198
101
122
174
8
186
120
37
46
28
166
180
200
232
221
116
31
75
189
139
138
112
62
181
102
72
3
246
14
97
53
87
185
134
193
29
158
225
228
152
234
105
217
142
148
155
238
135
233
206
85
243
223
140
161
137
248
191
230
66
104
65
153
44
251
176
84
187
255

CONCLUSION
This paper implies that an effective optimisationbased Sbox approach is an alternative to spontaneous and algebraic approaches. The Sboxes for high nonlinearity as fitness value are developed with Particle swarm optimization. This approach uses the chaotic Gauss iterated map for original sample generation and other necessary arbitrary values. The procedure was investigated by changing the parameters of the PSO for various scenarios. The proposed method was shown to be capable of producing solid, wellencrypted Sboxes. Compared with many recent SBoxes, the proposed SBoxes have been found to be upstanding enough than many of their contemporaries. Thus it is possible to create strong nonlinear Sboxes using the proposed technique.

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