Security and Communication Networks

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Machine Learning and Applied Cryptography

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Research Article | Open Access

Volume 2020 |Article ID 8883884 | https://doi.org/10.1155/2020/8883884

Sajjad Shaukat Jamal, Dawood Shah, Abdulaziz Deajim, Tariq Shah, "The Effect of the Primitive Irreducible Polynomial on the Quality of Cryptographic Properties of Block Ciphers", Security and Communication Networks, vol. 2020, Article ID 8883884, 14 pages, 2020. https://doi.org/10.1155/2020/8883884

The Effect of the Primitive Irreducible Polynomial on the Quality of Cryptographic Properties of Block Ciphers

Academic Editor: Tom Chen
Received23 May 2020
Revised03 Aug 2020
Accepted28 Aug 2020
Published24 Sep 2020

Abstract

Substitution boxes are the only nonlinear component of the symmetric key cryptography and play a key role in the cryptosystem. In block ciphers, the S-boxes create confusion and add valuable strength. The majority of the substitution boxes algorithms focus on bijective Boolean functions and primitive irreducible polynomial that generates the Galois field. For binary field F2, there are exactly 16 primitive irreducible polynomials of degree 8 and it prompts us to construct 16 Galois field extensions of order 256. Conventionally, construction of affine power affine S-box is based on Galois field of order 256, depending on a single degree primitive irreducible polynomial over . In this manuscript, we study affine power affine S-boxes for all the distinct degree primitive irreducible polynomials over to propose 16 different substitution boxes. To perform this idea, we introduce 16 affine power affine transformations and, for fixed parameters, we obtained 16 distinct S-boxes. Here, we thoroughly study S-boxes with all possible primitive irreducible polynomials and their algebraic properties. All of these boxes are evaluated with the help of nonlinearity test, strict avalanche criterion, bit independent criterion, and linear and differential approximation probability analyses to measure the algebraic and statistical strength of the proposed substitution boxes. Majority logic criterion results indicate that the proposed substitution boxes are well suited for the techniques of secure communication.

1. Introduction

The exchange of digital data through the Internet has revolutionized the communication parameters over the years. But this rapid communication also provides opportunities to access this digital data illegally. For this reason, the security of this content on the Internet has become a serious challenge for the researchers of different fields. To counter the emerging challenges of security, cryptography and steganography are used to hide the secret information whereas watermarking is used for copyright protection. In this manuscript, we discuss cryptography and relevant aspects of this field. For convenience, cryptography is divided into two types named symmetric key cryptography and asymmetric key cryptography. In symmetric key cryptography, two parties share secret information and keys during encryption and decryption procedures. The private key is shared by both sender and receiver. In addition to this, block ciphers and stream ciphers are two main branches of symmetric key cryptography. In 1949, Shannon gave the idea of block cipher and some examples of block ciphers are Advanced Encryption Standard (AES) [1], Data Encryption Standard (DES), International Data Encryption Algorithm (IDEA), and many more [2, 3]. In AES, there is availability of three different key sizes such as 128, 192, and 256 bits, whereas in DES, the only available key size is 56 bits. The AES has 10, 12, and 14 rounds for key sizes of 128, 192, and 256 bits, respectively. All these rounds have four basic steps, that is, subbyte, shift row, mix column, and add round key. Subbyte is the step which substitutes the plaintext data with substitution box (S-box). This S-box is the only nonlinear part of block cipher used in different well-known cryptosystems. It is used to create confusion to make plaintext data obscure for any attacker and hence S-box is an integral part of any cryptosystem. S-box is a function which has input and output from the Galois field. The Galois field is a finite field having order 256 and denoted by .

1.1. Related Work

S-box is used to create confusion as observed in AES, International Data Encryption Algorithm (IDEA), DES, and many more cryptosystems [4]. It is an established fact that the strength of block cipher depends on the standard and quality of S-box. Due to the necessary immersion of S-box to generate nonlinearity, intricacy persuades different researchers to design strong S-boxes to enhance the security level of cryptosystems. Among different available methods, the algebraic structure-based construction of S-boxes has much more attention. These S-boxes have strong cryptographic features and are robust against linear and differential cryptanalysis.

In the literature, different structural advancements are viewed to improve the quality of S-boxes. The algebraic complexity of AES S-box has been improved with the extension of this S-box, that is, affine power affine (APA) [5]. Furthermore, the symmetric group S8 has also been applied to AES S-box to improve the quality and numbers of S-boxes [6]. Similarly, the application of transformation using binary gray codes on AES S-box gives Gray S-box [7]. In [8], S-boxes are constructed by using the projective general linear group (PGL). Moreover, the construction scheme of chaotic S-boxes using DNA sequence and chaotic Chen system is given in [9, 10]. Different analytical, algebraic, and chaos-based techniques for the construction of S-boxes are given in [1116]. Conventionally, AES uses a polynomial of 8 terms which have all the required properties and improves the security for AES. But the Gray S-box has a -term polynomial. Moreover, residue prime, Xyi, and Skipjack S-boxes are frequently used for the encryption and decryption schemes [17, 18].

It is assumed that the model of Boolean functions and primitive irreducible polynomial has an impact on the strength of S-box. In [19], different primitive irreducible polynomials have been used to identify the effect of primitive irreducible polynomial. To investigate this fact, we want to study all the primitive irreducible polynomials to understand whether there is an impact of irreducible polynomial or not. Archetypally in the synthesis of an S-box, the numbers and in affine transformation belong to Galois field . As the polynomial ring has 16 primitive irreducible polynomials of degree 8, it shows that only 16 opportunities are available for constructing Galois fields . In this paper, we have constructed 16 different robust S-boxes over the elements of these 16 irreducible polynomials. Firstly, we define 16 affine power affine transformations on these different Galois fields which can be given as ; here, for values, we would be able to get 16 distinct S-boxes.

1.2. Motivation

Due to the role of S-boxes in cryptosystems, it is essential to explore all of its aspects. The motivation behind this work is to study all primitive irreducible polynomials and their role in the construction of S-boxes.(1)The Mobius transformation used in a different construction of S-boxes has certain limitations and restrictions in its structure [7]. For example, the condition on the parameters, i.e., squeezes the remaining cases. Hence, there is a need for any other transformation.(2)There are 16 primitive irreducible polynomials in the principal ideal domain whose impact was not studied yet regarding their impression on analyses of S-boxes.(3)By exploring all primitive irreducible polynomials, we have a better opportunity to obtain the cryptographically strong cryptosystems.

1.3. Our Contribution

In this manuscript, we studied all binary degree 8 primitive irreducible polynomials for the construction of S-boxes. The quality of the proposed work can be seen from the different security analyses and resistance against malicious attacks. This whole study can be summarized as follows:(1)We constructed S-boxes associated with the 16 binary degree 8 primitive irreducible polynomials.(2)The APA transformation is used in this work, which is bijective and has no restrictions on the parameters.(3)To evaluate the strength of the proposed S-boxes, we have performed different analyses along with differential cryptanalysis. The outcomes of these analyses are compared with the well-known S-boxes.

The remaining part of the paper is planned as follows: Section 2 presents the preliminaries and construction scheme of the proposed S-boxes. In Section 3, algebraic and statistical analyses are calculated in detail. Section 4 presents definitions of the balanced Boolean function. Section 5 concludes the paper.

2. Primitive Irreducible Polynomials of Degree 8 and GF (28)

2.1. The Galois Fields

We summarize here some well-known facts from the theory of rings and fields. Let be a commutative ring with identity. A nonempty subset of is called an ideal of if is an additive subgroup of and for every , where . If, furthermore, there does not exist a proper ideal of properly containing , then we say that is a maximal ideal of Besides; is said to be a field if each of its nonzero elements has a must inverse in . If is a field of prime characteristic , then is an extension of the prime field . A polynomial is said to be irreducible if it cannot be factored in into two polynomials of strictly smaller degrees. The principal ideal,generated by a monic irreducible polynomial is a maximal ideal in . If is of degree , then the quotient ring,is an extension field of of degree consisting of elements. This field is called a Galois field and is denoted by and is said to be the field extension of defined by the irreducible polynomial . A representative of each element of can be chosen to be of degree strictly less than . If is a root of in an algebraic closure of , then is isomorphic to the field:and so we can identify the two fields. Furthermore, if is a generator of the cyclic finite multiplicative group of nonzero elements of , then we say that is primitive.

The Galois field is particularly of specific interest in cryptographic applications, especially in S-boxes constructions. For our cryptographic purposes, we are interested in such a field whose defining irreducible polynomial is “primitive” (of degree 8, of course). It is well known that there are such polynomials over , for example, , which we list in Table 1. In the following section, we construct 16 S-boxes out of the Galois fields corresponding to the aforementioned sixteen primitive irreducible polynomials.


Primitive polynomials roots Galois field


2.2. The Proposed S-Box Construction Method

For each , consider the affine power affine map (APA):where and are two affine maps with , and

Among other things, the map , which is obviously bijective, was introduced by [5] to produce confusion in the scheme. For our S-boxes, we choose , and and Figure 1 demonstrates the flow chart of the construction of the 16 different S-boxes. Moreover, the construction of S-boxes in correspondence to polynomial 1 (P1) to polynomial 16 (P16) is shown in Figure 1. All the S-boxes are given in Tables 217, corresponding to P1 to P16. These tables are before the conclusion section.


Proposed S-box 1

17614113924975179469486224319749105167250
240601891821882301781012362115311038155127207
91213295588782442158122115821974201210255
11990374243147678396229925314411569
63223205801407112112512054168187822021070
261916118618365232641724193444013734128
16424116325872147619613145819897102246
27723523351571181001172081433013821766157
211245151341801261141428914825421812385154169
10316561958422418410720320014522907231226
58353212191159352425220216247192204136228
791159222724873501121321082251625223130185
33133237611731712786113181175116251177170234
121501814681741493917124199321312816220
160111135131041292399822210914672152206190238
5295166242094647591221561061511949436193


Proposed S-box 2

186170249532081064922014720314320725017713319
24084152381821585967219152097015041121
64255231139182117251004712510977190113110148
1181971981463913810816175131032007391105204
80104130124213221463144481121817219637151
313510129221184261114022211127223312987
861892052325311489218176193229180201158469
221882288524123614023916164577985624102
211160142153546821321023071238235822525298
33765425132162212192175901724543119
17844364616617151234195621454524714168174
21712215901012376027163202243991161201316
2242322012334551372061552268117393169107172
16719919194136961831492332461791569835095
61282548814116512635128667613421697242187
2443074215121151851947815422725122524892157


Proposed S-box 3

12625016210232143129192282004742155131177221
2401422949138120944620645991162750167
67165117133108175960174545825122316418144
1548823869891942011932275611371237286169
12171144181599016189105315126814917638
6511048183210196217591632371189313241140208
173247119332332286617080222166161160146236224
1796311581982311509724322917821517614245
2111992531978541437636212855914012460
731347721110721818020411215230246104151130
168220227145103185191218282522023419025152
13911418718614130122113239211110112239127213
8715618821618415337249701022625595484206
1482141722252544513524351153391959252248
203362051002422442352091257423220721215814719
1095621915717171065723483136791326819878


Proposed S-box 4

49103220208587017323824216015416915811013994
240200781491551172191315016122614481891147
1151202183451710418216715019264178955791
17631001592051346385243226923617714421020
4479206899674661461971501081321525181
2359319422173182465614010624424714823788232
191251234999335925245179222125132109229136
77411628188126202976721716398841712942
211382131194616851162041297531152239166172
681011635832121864310524919925519611132
87184156242231901951539223011220939528012
1131986040138164761017261145122174231986
170183214245135231365422481143124302284114
265312720314113318790118201271807113017185
82175137165282162273755472157248233123254
14212125312810222519316107250241622076515121


Proposed S-box 5

6116310517730219248584179127151118169196
24034202517719112623321539254197204893192
1221289795181217656417323910198439016
17587701621689224910024519324920510755139
1083311114748153156110206137200104247116
19182144660115158167130113172044218141176
147113821621317922615025331856623118724357
7123624422545381654017220118811922424229232
2112145418542227190821594415725023522117128
155355212412084781365912336882518313421
31431532231021292091111311401441849880178220
121481869914921023994234371522306324167189
22194238203562071172686112182376847212229
1661461992281211011801747672321606920810125
2467325210316487910925272410614215418362
96255132156751455013513133222161170519589


Proposed S-box 6

162120205183162013719189951282391591021994
24015518724819692139223121361822280140131158
10053236501927211816713242351901381528
24924617422111161191261500254961349181112
81163108261352162349985147862177104179207
18922255271782252122371471142531657517244184
19916687208712421951751012477924314944217201
38363752516312566692181908610330148123
21119833482321971059933110714519418541227
401462145822877144152573518017224160556
8612422916911681711542045223321213119245
561331131274598115656413417320010220211074
18215714176916834224592032061702151537339
67291301421161868816047106254193121231210
725211725017648443124235209151122230226129
3223188111156220164238829714370834610925


Proposed S-box 7

86101177891991882361651981451122329216476137
24024458715198181129117942301083929184206
103123561803514224616848366422218711119615
5419012010417312081051622242511225317257170
114315315918520419245128509714022712721444
17621811820945324768186249693251502888
02541151478515422199694570136130328134
1711392502432481741912031243332552215227
2117161109954792261356714312416012522818
74926792201792312231482105124114111320016
19219723914624183252821671262252425821952107
233551492357717815720165402167420713313178
831227310266213138801118234461951019317
23775119110238156217220317523441189158116215
2051391202106382293081233763144965942
6210084166146021716116913216390212194155121


Proposed S-box 8

416472810924810013922714111160712517831
2402245312391636723771217112052521336122
9717218223312451196236255170128318617983119
2920623421021811512202456813213542143015
2398958218921248120187219239222202192154246
14324131181102272041841291371147412257188199
94752079761589213039199613410310833229
8921175501661958720316446174121253424116
21123810411223514116251191862532302255215538
581771734045148624220114919025021615759244
1691531501171421011459207208247160140864180
1948837162115351565903684106152232159110
1447380126200165113561614912712168066226
14732132146243443138551631671352238193209
6999107441981052281065772151971931761817
24978242857072171183262212542311181361852


Proposed S-box 9

1641778216114139985131895824322124915918
24064151225120210180135571152047262155227195
10995127141762291782461214747235166217122212
20526271381487017113014021520315614611106233
24216262061421611921825316910421310241245231
19725065869015223724116725441420818271101
23268111101345520910318961741162040118186
42192131226431136315738698601494425553
2118222411032944923680160196222185108188154
176123238117302478815018114413235371989697
136922375121811128312523987214319105201
137214155617217913325236220851845023439194
7399173791244625100168333228191145165161
199771262818359170225420222366163219187174
177244175512867230793452921158131299
04820010734242512481190845219315391207


Proposed S-box 10

1092616321377207155873496136401772512811
240147166252115158185123146682391491601806111
22198208243103972418717922813211018810151130
621191316021914824591018120532229820319
1541786611724610822613520222459236192156141112
217283916717210444230542538216823415058
1076443482011818110217325420076237209143204
63193221214148816126223901578546169196191
2114211991761002161521838937997523372122
1845279206235194225711516250210242127153234
57701652481645675521291241892117124186
251159125134329311612255106175731703513984
411201827781718217453215832479422731137
296111314521223813810530471219274780142
232951972201952314916136869916513244218
51025014422911419045249133140383318620156


Proposed S-box 11

1281911321671111201592182568173217323999239
24094157957511219021315220240101220714064
19238351511541971996061187442017237126118
24801241411641193160710716312924866189221
10109761501102551717017184317415313113206
2442193203121344824516413023041445318278
1552272531582686186627921175162222215247208
981372106357671389613924225438921889752
211106692285212198234225422231191369033
461212491843414225120214216502091358774176
16157736229831497316818119611519411568
822315111486519517223177228013125288170
2202769711232324922623658104143178166117205
541691089120412711645165893055318424615
1561222241921031452431141851022001802508123347
5912513310014714183146235237292810593179241


Proposed S-box 12

14024911032536191601022472352191716618434
240198221159212199188148270527317615493
12832214195238216242299224136209207129
252451399710941225371051067110020616512959
171702081261112421811192152218323314222204150
226144821214087672281162348113180596237
425195169979216931301255589143123178
122158806610133462523311221015713120312743
211541322175815515219253128844488236194168
17023923121823220510315121732614520251136117
146498335202902431647810820116717519168163
65851852551041822717775391611821875798186
6424617919617438118135101193772239424872250
1076218915384875616960231114914129156
1471201542541341302442001722131971621901134586
11423047138611152205013722476631421249174


Proposed S-box 13

24177151257712261531271372097325421396237
2402341021312011421186416249176651822295
5236821602242251447467129113105143186248
1811132311085821643114156371689170603246
4012221724722385163465152311897012173210
20723319710783115164996656231111206125146
205154255167198818710920135596810039220172
14511132185106531621122535418033116175104134
21112320426190908921511011929188230219199
7679174130147166632152414124414112403598
21824524272212394794208238101651719114192
7724314815880203454219414615923521416132169
931011403097508412622919620048551395744
6912849155196223225223613625027157195125206
341797838193227281841338818717875212138117
19120214922228891501212211718286511831032


Proposed S-box 14

9313118253482101401631003418128119053114
24022724210611213814792250206141561525794212
50149235212332541021311561421731943145244195
23743183168121959713712020462119174122186
1232910513619119691402298644112522224162
1553022316139169146352133199362411245965
19012721988214115126963823955719773242
23144474668726185109176198201234222203
211175157113110217853249255116104258217997
15813544781171641602216623614510360230232134
182671771881581077921612246512131805817
18925221524569218192102257454838461166193
161398113219101762082311511814918799202228
205220154198111167202620064165247248209207148
226028184331592715324172171752382437787
35170321371432511291781461508089108130570


Proposed S-box 15

1222018271422399625136246211622231033540
24013232219116160243281281681592061939550154
3823447661261925416965175254894512323339
171941112022422261782101856477218108127147222
1912142371101841553020862121561255119167141
10525217663205139151533318917021318322420475
69100230196188314310483911456011311715294
15712910223513549151811631955824913417110737
211342423867222368825310920722116420313819
482291227176931201438110626137215200187217
118722442451502091211441991418627701158455
4117380179791461817361728561367861
4252851661334122825999015812557292248
1023192520118068177174231612089862232140
511652160241101997131148198247130114250112
44462251315331682190124591971492277487


Proposed S-box 16

613668218166118481142173303920481144134
2404611583912021203106314548911221317
478176149117181381335687162201997860157
2541419229200178140692412116014815825267228
11110310219433122442056110022921580187217189
28124147923176451711372399515442645155
1192367724201275216841512204910115012953
155198174146902338822554179574246138110230
21115320821440184132557116362130175232231128
1561413415215212165411069716772177169234107
17012024710424519571721965853670108209222
19713224457201182965212521018312724112623526
582271431123886135191661682373591233227
105207221222482512435023161869882250180164
1932199379351962491881392531619937223224226
1312110925731164320694113190142185159918

In the proposed work, we present an APA S-box corresponding to each where the APA map gives the lookup tables. We, then, show that these S-boxes have strong cryptographic properties certified with the help of analyses such as nonlinearity, strict avalanche criterion (SAC), bit independent criterion (BIC), linear approximation probability (LP), and differential approximation probability (DP) [20].

3. Security Analysis

In this section, we present some algebraic and statistical analyses of S-box followed [21]. Such analyses indicate the strength of all the proposed S-boxes and give an idea for their application in image encryption and other modes of secure communication.

3.1. Nonlinearity

Nonlinearity analysis of a function is the minimum hamming distance between the Boolean function : and its all n-bit affine functions. In the truth table of Boolean function , the nonlinearity of represents the degree of dissimilarity between and all affine function. If the function has high minimum hamming distance, it indicates it has high nonlinearity. It is an established fact that high nonlinearity provides resistance to any kind of linear approximation attacks [22, 23]. The calculated upper bound of nonlinearity is so that, for, the optimal value of nonlinearity is. Table 18 shows the nonlinearity of 16 S-boxes corresponding to all primitive irreducible polynomials. From this table, it can be seen that the value of nonlinearity has not been affected due to background irreducible polynomial.


S-boxesf1f2f3f4f5f6f7f8Average

S-box 1112112112112112112112112112
S-box 2112112112112112112112112112
S-box 3112112112112112112112112112
S-box 4112112112112112112112112112
S-box 5112112112112112112112112112
S-box 6112112112112112112112112112
S-box 7112112112112112112112112112
S-box 8112112112112112112112112112
S-box 9112112112112112112112112112
S-box 10112112112112112112112112112
S-box 11112112112112112112112112112
S-box 12112112112112112112112112112
S-box 13112112112112112112112112112
S-box 14112112112112112112112112112
S-box 15112112112112112112112112112
S-box 16112112112112112112112112112

3.2. Strict Avalanche Criteria

In [24], Webster and Tavares introduced the strict avalanche criteria (SAC) on the concepts of completeness and avalanche. If a single input bit changes, the output bits change with almost 0.5 probability. It helps to show that the resulting output vector is highly random, and no single pattern can be predictable by minor variation in the input vector [25]. By seeing the performance indexes of S-boxes, the proposed S-boxes successfully satisfy SAC. Table 19 depicts the value of SAC for all the proposed 16 S-boxes. It shows that the maximum value of SAC is 0.562500 for the first 9 S-boxes including 11th, 14th, and 16th S-boxes. Similarly, the minimum value of SAC is 0.453125 for the first 10 S-boxes including 12th and 14th S-boxes. The average value of SAC lies in the interval [0.4856, 0.509766].


S-boxMaxMinAverageSquare deviation

S-box 10.5625000.4375000.4960940.0172495
S-box 20.5468750.4531250.4951170.0128725
S-box 30.5625000.4531250.4941410.0152856
S-box 40.5625000.4531250.507080.0118748
S-box 50.5468750.4531250.5031740.0153901
S-box 60.5625000.4531250.5017090.016637
S-box 70.5625000.4531250.5024410.0170951
S-box 80.5625000.4531250.5039060.0165152
S-box 90.5625000.4531250.4855960.0153978
S-box 100.5468750.4531250.5097660.0123912
S-box 110.5625000.43750.504150.0191487
S-box 120.56250.4531250.5012210.016475
S-box 130.5468750.43750.5009770.0127235
S-box 140.56250.4531250.5083010.0158654
S-box 150.5468750.4375000.4987790.0143727
S-box 160.56250.4375000.4965820.0143171

3.3. Bit Independent Criterion

Another algebraic criterion (BIC) is used to evaluate the strength of S-box, which is presented by Detombe and Tavares in [26]. In Table 14, the outcomes of BIC to SAC and BIC for the proposed S-boxes are given. The minimum BIC to SAC value is 0.47070 for 12th S-box and the highest minimum value is 0.49219 for 2nd S-box. The average BIC to SAC lies between 0.49679 and 0.50739. Similarly, the square deviation values for all the proposed S-boxes are given in Table 20. The maximum and average value of BIC is 112 for all S-boxes. It is depicted that the proposed S-boxes give the nearest best value of BIC analyses.


BIC-SACBIC
S-boxMinAverageSquare deviationMaxAverageSquare deviation

S-box 10.474610.501190.011321121120
S-box 20.492190.506000.008451121120
S-box 30.480470.502020.010151121120
S-box 40.478520.506560.012011121120
S-box 50.484380.501050.009241121120
S-box 60.476560.498190.007841121120
S-box 70.486330.505930.009251121120
S-box 80.488280.502510.008351121120
S-box 90.488280.502580.006441121120
S-box 100.484380.507390.009811121120
S-box 110.476560.497840.009501121120
S-box 120.470700.496790.010261121120
S-box 130.482420.500210.010851121120
S-box 140.490230.507180.009011121120
S-box 150.484380.502650.008831121120
S-box 160.486330.505440.008611121120

3.4. Linear Approximation Probability

Matsui defines the extreme value of the imbalance of an event as the linear approximation probability. It is notable that the parity of the input bits that is, the mask , is equal to the parity of the output bits, i.e., the mask . The linear approximation probability of a given S-box is defined in the following equation:where and are input and output masks, respectively, and the set represents the set of all possible inputs; is the number of elements of. The value of linear approximation indicates the strength of S-box against various linear attacks. In Table 21, the maximum count and the LP value for all proposed S-boxes is 144 and 0.0625. These values of LP of the proposed S-boxes are appropriate against linear attacks.


S-boxLinear approximation probabilityDifferential approximation probability
Max countLPMax valueDP

S-box 11440.062540.015625
S-box 11440.062540.015625
S-box 21440.062540.015625
S-box 31440.062540.015625
S-box 41440.062540.015625
S-box 51440.062540.015625
S-box 61440.062540.015625
S-box 71450.066440.015625
S-box 81440.062540.015625
S-box 91440.062540.015625
S-box 101440.062540.015625
S-box 111440.062540.015625
S-box 121440.062540.015625
S-box 131440.062540.015625
S-box 141440.062540.015625
S-box 151440.062540.015625
S-box 161440.062540.015625

3.5. Differential Approximation Probability

The degree of differential uniformity is known as differential approximation probability (DPs) of S-box. Mathematically, it can be given as

Briefly, it can be explained as follows: an input differential must be mapped to an output differential uniquely for each i. Here, represents all the possible input values and the number of its elements is given by . Table 21 depicts the results of DP, which include the maximum and DP value.

Moreover, Table 22 represents the values of proposed S-boxes along with AES, Skipjack, Xyi, APA, Gray, and residue prime S-boxes.


S-boxesNonlinearitySACBICBIC-SACDPLP

AES S-box1120.50581120.5040.01560.062
APA S-box1120.49871120.4990.01560.062
Gray S-box1120.50581120.5020.01560.062
Skipjack S-box105.70.4980104.10.4990.04680.109
Xyi S-box1050.5048103.70.5030.04680.156
Residue prime99.50.5012101.70.5020.28100.132
Reference [27]1060.4978103.92
Reference [28]0.505
Reference [29]1040.52411030.501810.16250.0486
S-box 11120.4960941120.501190.0156250.0625
S-box 21120.4951171120.506000.0156250.0625
S-box 31120.4941411120.502020.0156250.0625
S-box 41120.507081120.506560.0156250.0625
S-box 51120.5031741120.501050.0156250.0625
S-box 61120.5017091120.498190.0156250.0625
S-box 71120.5024411120.505930.0156250.0625
S-box 81120.5039061120.502510.0156250.0664
S-box 91120.4855961120.502580.0156250.0625
S-box 101120.5097661120.507390.0156250.0625
S-box 111120.504151120.497840.0156250.0625
S-box 121120.5012211120.496790.0156250.0625
S-box 131120.5009771120.500210.0156250.0625
S-box 141120.5083011120.507180.0156250.0625
S-box 151120.4987791120.502650.0156250.0625
S-box 161120.4965821120.505440.0156250.0625

3.6. Statistical Analyses

To evaluate the visual strength of the substitution with the help of the proposed S-boxes, various statistical analyses are made on the host and substituted images. In this proposed work, statistical analyses like homogeneity, entropy, contrast, energy, and correlation are used to evaluate the substitution ability of the 16 proposed S-boxes. These analyses are given aswhere give the row and column locations of an image. The pixel value at kth row and lth column is represented by and is the probability of the image pixel. In equation (8), are mean and standard deviation, respectively.

Correlation analysis helps to find the similarity between the host and substituted image. The correlation analysis provides the range which indicates the perfect, negative, and positive correlation. This is interval for correlation and value of 1 indicates the perfect correlation.

The randomness of the digital image can be calculated with the help of entropy. The higher value of entropy from the interval represents the higher amount of randomness in a digital image. For any viewer, it is only possible with the help of contrast analysis to intensely recognize the objects in the texture of an image. With the help of contrast analyses, one can observe the maximum distinction in image pixels. The range of the contrast can be given by. For constant image, the value of contrast is zero. The goal of finding close distribution between the matrix and its diagonal is obtained in homogeneity analysis. The matrix used in this analysis is named gray level cooccurrence matrix (GLCM) and the range of homogeneity lies between 0 and 1. The range for energy analysis also lies in the interval [0, 1]. The results of Table 23 are obtained by applying these analyses on the original and encrypted images. For all the proposed 16 S-boxes, we calculated the values of the statistical analyses.


S-boxesEntropyCorrelationContrastEnergyHomogeneity

S-box 1
S-box 2
S-box 3
S-box 4
S-box 5
S-box 6
S-box 7
S-box 80.12569.66350.01820.4509
S-box 97.27630.10998.96530.01790.4609
S-box 107.27630.07479.71060.01770.4453
S-box 117.27630.09279.79050.01820.4529
S-box 127.27630.08799.87200.01820.4531
S-box 137.27630.08688.55690.01880.4606
S-box 147.27630.08628.63080.01790.4510
S-box 157.27630.12459.38290.01840.4522
S-box 167.27630.08439.58980.01790.4533

A 256 × 256 JPEG image of Lena is considered for MLC analysis. Figure 2 shows the results of image encryption with 16 proposed S-boxes.

4. Balanced Boolean Function

4.1. Balance Property

The imbalance of a Boolean function weak system against linear cryptanalysis highlights the importance of balance property. The balance property indicates that the higher the magnitude of a function’s imbalance, the more the chances of a high probability linear approximation. A Boolean function is balanced. If the cardinality or Hamming weight of these two functions, that is, is the same, then it is named the balance function.

4.2. Balance Property of the Proposed S-Box

All the Boolean functions involved in proposed S-boxes are balanced just like the Boolean functions of AES, , AES and other well-known S-boxes. The nonlinearity of the proposed S-boxes is equal to 112.

5. Conclusion

In this paper, a scheme for the synthesis of S-boxes over 16 isomorphic Galois fields is presented. Here, we fixed all the parameters of affine power affine transformation, that is, for 16 S-boxes. We have 16 primitive irreducible polynomials of degree 8 and they prompt us to construct 16 Galois field extensions of order 256. By using elements of the Galois field, corresponding to each different pair of the parameters, one can construct different S-boxes. These S-boxes obtained as a result of APA transformation which is bijective, pass nonlinearity test, and out bit independent criterion (BIC) which demonstrates that the existing S-boxes have high confusion producing capability. The evaluation of constructed S-boxes is done with some algebraic and statistical analyses. The results of these analyses highlight the characteristics of all the proposed S-boxes and later these S-boxes are equated with some of the existing S-boxes. In addition to this, we also ensured that all these constructed S-boxes are balanced that guarantee the strength of our S-boxes. Hence, we have concluded that a large class of S-boxes can be obtained by varying parameters of affine power affine transformations. These S-boxes can be used for secure communication.

Data Availability

The data that support the findings of this study are available from the corresponding author upon reasonable request.

Conflicts of Interest

There are no conflicts of interest among the authors.

Acknowledgments

The authors extend their appreciation to the Deanship of Scientific Research at King Khalid University for funding this work through research groups program under Grant no. R.G.P. 1/234/41.

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Copyright © 2020 Sajjad Shaukat Jamal et al. This is an open access article distributed under the Creative Commons Attribution License, which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited.


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