On Algebraic Immunity of Trace Inverse Functions over Finite Fields with Characteristic Two
نویسندگان
چکیده
The trace inverse function Tr(λx−1) over the finite field F2n is a class of very important Boolean functions and has be used in many stream ciphers, for example, SFINKS, RAKAPOSHI, the simple counter stream cipher presented by W. Si and C.S. Ding, etc. In order to evaluate the security of those algorithms in assistance to (fast) algebraic attacks, it is essential to algebraic properties of Tr(λx−1). However, currently only some bounds on algebraic immunity of Tr(λx−1) are given in public literature. In this work we give the exact value of Tr(λx−1) over finite fields F2n , that is, AI(Tr(λx−1)) = b √ nc+ d n b√nce − 2 = d2 √ ne − 2, where n ≥ 2, λ ∈ F2n and λ 6= 0, which is just the upper bound given by Y. Nawaz et al. And at the same time our result shows that D.K. Dalai’ conjecture on the algebraic immunity of Tr(λx−1) is correct. What is more, we further demonstrate some weak properties of Tr(λx−1) in resistance to fast algebraic attacks.
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