Construction of self-dual normal bases and their complexity

Abstract

Recent work of Pickett has given a construction of self-dual normal bases for extensions of finite fields, whenever they exist. In this article we present these results in an explicit and constructive manner and apply them, through computer search, to identify the lowest complexity of self-dual normal bases for extensions of low degree. Comparisons to similar searches amongst normal bases show that the lowest complexity is often achieved from a self-dual normal basis

Correlation Distribution Analysis of a Two-Round Key-Alternating Block Cipher

“…The proof of Lemma 5.2 shows how to find x 1 , x 2 , x 3 . The cubes x 3 1 , x 3 2 , x 3 3 are the three roots of p 1 (y) = y 3 + k 2 y + 1, if Tr(k 3 ) = 1 and the roots of p 2 (y) = y 3 + (k 2 + 1)y + 1 if Tr(k 3 ) = 0, where p 1 (y)p 2 (y) divides (y).…”

“…Note: If x 3 1 , x 3 2 , x 3 3 are the three roots of p 1 (y) for a given k, they are the roots of p 2 (y) but for k := k + 1. Also, Tr (k + 1) 3 = Tr(k 3 + 1) = 1 + Tr(k 3 ).…”

“…In this paper we consider the same cipher but with a round function of more general form and we determine the correlation distribution in dependence of the key for arbitrary input and output masks (σ, ω), where σ, ω ∈ F n 2 , n odd. Our methods are based on finite-field theory, see, e.g., [3,10,11,13] and some general results on correlation analysis and linear cryptanalysis [5,8,12].…”

Construction of Weakly Self-Dual Normal Bases and Its Aplication in Orthogonal Transform Encoding Cyclic Codes

“…ii.ϕ α : v → v −1 • α is one-to-one correspondence beetwen the solution for equation (2) and the set of elements in F q n which generate weakly self-dual normal bases.…”

“…. , γβ π(m−1) }, where γ = 1 tr(β 0 ) 2 and π(i) = i or π(i) = i + m 2 mod m. Hence, bases B enable us to calculate its dual bases easily. Hence, we get the following complete approach.…”