## 3 Theoretical Background of Our Novel Approach
Theory of Galois Field (GF (\(\bf{p^{d}}\)))  
[PERSON], who was a great French algebraist who died at the age of 20 years, discovered Galois Field theory [18; 19; 20; 21; 22; 23]. According to Galois Field theory, if p is a prime number, then it is also possible to define a field with p\({}^{4}\) elements for any q. A field is a set function F with two composition laws, plus (+) and multiplication (*), such that  
1. (F, +) is a commutative [18; 19; 20; 21; 22; 23]. This means that any element of a set F (for example, the \(a,\ b,\ and\ c\) elements of F), when added, holds the following property [18]: \[a+b+c=c+a+b=b+c+a\] (6)
2. (F\({}^{X}\), *) is a commutative [19]. This means that any element of a set F (for example, the \(a,\ b,\ and\ c\) elements of F), when multiplied, holds the following property [18; 19; 20; 21; 22; 23]: \[a*b*c=c*a*b=b*c*a\] (7)
3. The distributive law holds, and each element has an inverse [18]. This means that any element of a set F (for example, the \(a,\ b,\ and\ c\) elements F), when computed, holds the following property [18; 19; 20; 21; 22; 23]: \[a*(b+c)=a*b+a*c\] (8)The total number of elements in the field is defined as the order of a field [20]. The order of GF (p\({}^{\rm q}\)) is p\({}^{\rm q}\). For each Galois Field (GF (p\({}^{\rm q}\))), there exists a prime number (irreducible polynomial) of base p with element q. An irreducible polynomial is a polynomial that cannot be factorized or divided into more than one segment except by 1 or itself [9; 10; 11; 12; 13; 14; 15; 16; 17; 18; 19; 20; 21; 22; 23]. For example, GF (2\({}^{2}\)), has an irreducible polynomial of x\({}^{1}\) + 1 = 11 binary = 3\({}_{\rm decimal}\)[21] (p. 385). This means that from 2\({}^{0}\) to 2\({}^{2}\), the only number that cannot be factored or divided except by 1 or itself is 3. For GF (2\({}^{3}\)), there exist two irreducible polynomials: x\({}^{3}\) + x + 1 = 1011 binary = 13\({}_{\rm decimal}\)[9] (p. 385). Therefore, in GF (**2\({}^{32}\)**), there exist at least six long irreducible polynomials as follows [23] (pp. 25-26), [18]:  
\[ \begin{array}{cccccccccccc}x^{32}&+x^{22}&+x^{2}+x+1=80200007_{hexidecimal} &=2149580807_{decimal}&(9)\\ x^{32}&+x^{22}&+x^{21}&+x^{20}&+x^{18}+x^{17}+x^{15}+x^{13}+x^{12}+x^{10}+x^{8 }+x^{6}+x^{4}+x^{1}\\ &&+1=10076B553_{hexidecimal}&\\ &&\\ x^{32}&+x^{23}&+x^{17}&+x^{16}&+x^{14}+x^{10}+x^{8}+x^{7}+x^{6}+x^{5}+x^{3}+1 =f_{1}(x)&(11)\\ x^{32}&+x^{26}&+x^{23}&+x^{22}&+x^{16}+x^{12}+x^{11}+x^{10}+x^{8}+x^{7}+x^{5}+x^{ 4}+x^{2}+x^{1}+1&\\ &&=f_{2}(x)&(12)&\\ x^{32}&+x^{27}&+x^{26}&+x^{25}&+x^{24}+x^{23}+x^{22}+x^{17}+x^{13}+x^{11}+x^{10} +x^{9}+x^{8}+x^{7}+&\\ &&x^{2}+x^{1}+1=10 FC22F87_{hexadecimal}=4559351687_{decimal}&\\ &&\\ x^{32}&+x^{28}&+x^{19}&+x^{18}&+x^{16}+x^{14}+x^{11}+x^{10}+x^{9}+x^{6}+x^{5}+ x^{1}+1=f_{3}(x)&(14)\end{array}  \tag{13}\]  
It is already stated that GF (**2\({}^{32}\)**) is the chosen mathematical procedure to be used to generate S-Boxes with an output of 32 bits. It is noted that a polynomial \(x^{32}\) + \(x^{22}\) + \(x^{21}\) + \(x^{20}\) + \(x^{18}\) + \(x^{17}\) + \(x^{15}\) + \(x^{13}\) + \(x^{12}\) + \(x^{10}\) + \(x^{8}\) + \(x^{6}\) + \(x^{4}\) + \(x^{1}\) + \(1=10076B553_{hexidecimal}\  
\[ \begin{array}{l}S(1)\\ =(4*10076B553_{hexadecimal}\\ +10F{C22F87}_{hexadecimal})\ mod\ (10000000)_{hexadecimal}\\ =119D04D3_{hexadecimal}\end{array}  \tag{16}\]  
Continuing to convert all 4 output bits to 32 output bits of the simplified S-Box, the new S-Box will be represented by Table 6.  
All S-Boxes of ten algorithms commonly used on IoT (AES, BLOWFISH, CAMELLIA, CAST, CLEFIA, DES, MMB, RC5, SERPENT, and SKIPJACK) will be reconstructed using the GF (\(\mathbf{2^{32}}\)) method to strengthen them against linear cryptanalysis attacks. Algorithms will also be reconstructed to suit newly generated S-Boxes.