A Novel Construction of Perfect Strict Avalanche Criterion S-box using Simple Irreducible Polynomials

An irreducible polynomial is one of the main components in building an S-box with an algebraic technique approach. The selection of the precise irreducible polynomial will determine the quality of the S-box produced. One method for determining good S-box quality is strict avalanche criterion will be perfect if it has a value of 0.5. Unfortunately, in previous studies, the strict avalanche criterion value of the S-box produced still did not reach perfect value. In this paper, we will discuss S-box construction using selected irreducible polynomials. This selection is based on the number of elements of the least amount of irreducible polynomials that make it easier to construct S-box construction. There are 17 irreducible polynomials that meet these criteria. The strict avalanche criterion test results show that the irreducible polynomial p17(x) =x + x + x + x + 1 is the best with a perfect SAC value of 0.5. One indicator that a robust S-box is an ideal strict avalanche criterion value of 0.5


INTRODUCTION
S-box which is also known as a substitution box has a function in the process of randomizing data bits [1]. The strength of the S-box generated will determine the durability of a message that has been encrypted from linear and differential attacks [2]. The method used in constructing S-box construction in this paper is algebraic techniques. In algebraic techniques, irreducible polynomials play an important role in making S-box construction [3]. An irreducible polynomial is a polynomial that has two multiplication factors, namely itself and 1 [4]. In GF ((2 8 ), the irreducible polynomial that is built has the highest degree of 8 which is used to make a multiplicative inverse. The resulting multiplicative inverse will be used to build an S-box. Hence, the role of the irreducible polynomial is crucial in building a strong S-box construction [5].
One of the criteria for determining the strength of an S-box is the perfect value of the strict avalanche criterion (SAC) [1]. The SAC is used to see changes that occur in input bits and output bits. If there is a change in 1-bit of input, ideally there should be half of the output bit changed. This means that the perfect SAC value is 0.5 [6]. The SAC value produced by the S-boxes construction that has been carried out by previous researchers is various. Girija and Singh [7] developed S-boxes with a double random phase encoding (DRPE) system. Farwa et al. [8] built an S-box with a specific nonlinear and iterative map approach. Çavuşoğlu et al. [9] developed an S-box design by placing an 8-bit value taken from a random number generator (RNG).
In this paper, we will present S-boxes construction using simple irreducible polynomials. S-boxes are built based on 17 simple irreducible polynomials, i.e., p1(x), p2(x), …, p17(x). The resulting S-boxes will be tested using SAC. The proposed S-box is the selected S-box that has the best SAC value compared to previous studies. In the next section, we will discuss the irreducible polynomial.

Irreducible Polynomial
An irreducible polynomial is a polynomial that has two multiplication factors, i.e., itself and 1. Table 1 shows the irreducible polynomials classified according to the order of the number of the smallest polynomial elements [24] as listed in [25], [26], [27], [28], and [29]. Based on Table 1, 17 irreducible polynomials have the least number of polynomial elements, namely five elements. Through 17 selected irreducible polynomials, the multiplicative inverse will be built.
Further discussion of multiplicative inverse construction is presented in the novel Sboxes construction section.

Novel S-Boxes Construction
In this section, we will introduce S-box construction based on simple irreducible polynomial specifically irreducible polynomial which has the least polynomial elements. The S-box construction proposed based on the scheme in Figure 1. According to Figure 1, the S-box construction is generated from a multiplicative inverse applied to affine mapping. The affine mapping consists of an affine matrix and the addition of a constant 8-bit vector as shown in Eq. (1).

Performance Analysis of The Novel S-Boxes
In [8] 0.5066 In [9] 0.5064

CONCLUSION
In this research, a novel method was introduced in building S-box construction to get the perfect SAC value. The S-box construction starts with classifying the irreducible polynomial based on the number of the least polynomial elements (simple irreducible polynomial). The selection of simply reduced polynomials aims to simplify calculations in building multiplicative inverse. The results of each element of the multiplication inverse are applied to the affine matrix and the addition of a constant 8-bit vector to produce an S-box. This result shows that the proposed S-box17 has a perfect SAC value of 0.5. So it can be concluded that the proposed S-box has the best SAC value compared to the S-boxes from the previous research. For the next research, it is expected that there will be further research in S-box construction that has perfect value other than the SAC criteria in a good S-box test.