Algebraic Aspects of the Advanced Encryption Standard

Abstract

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Construction of Substitution Box (S-Box) Based on Irreducible Polynomials on Gf(2^8)

“…The non-linearity of a Boolean function can be defined as the smallest distance between the function and the set of affine functions. It is denoted by Ψ is mathematically represented as: Ψ = min(𝑁 𝜇 ) (6) with 𝑁 𝑗 = min[𝑑(𝑓, 𝑔)], where 𝑔 ∈ 𝐴 𝑛 (7) where 𝐴 𝑛 is the set of all the affine functions. And 𝑑(𝑓, 𝑔) = 2 𝑛−1 − 2 −1 (〈𝜂, 𝛽〉) (8) where 𝜂, 𝛽 represent the binary sequence of 𝑓, 𝑔 respectively and 〈𝜂, 𝛽〉 define the scalar product of sequence, Hence, for a function 𝑓: 𝐵 𝑛…”

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“…This modified S-box formula may then be used to produce an AES encryption expression by using a type of continuing fractions. As a result, each byte of the state space is unchanged after five AES cycles, i j , (5)  , by [5]. The term "modified S-box formula" refers to a modified version of the substitution box (S-box) used in the AES algorithm.…”