On Algebraic Immunity of Trace Inverse Functions over Finite Fields with Characteristic Two

Abstract

The trace inverse function \Tr(\lambda x^{-1}) over the finite field $\mathbb{F}_{2^n}$ 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(\lambda x^{-1}). However, currently only some bounds on algebraic immunity of \Tr(\lambda x^{-1}) are given in public literature. In this work we give the exact value of \Tr(\lambda x^{-1}) over finite fields $\mathbb{F}_{2^n}$, that is, \AI(\Tr(\lambda x^{-1}))=\floor{\sqrt{n}}+\ceil{\frac{n}{\floor{\sqrt{n}}}}-2=\ceil{2\sqrt{n}}-2, where $n\ge2$, $\lambda\in \mathbb{F}_{2^n}$ and $\lambda\ne0$, which is just the upper bound given by Y. Nawaz et al. And at the same time our result shows that D.K. Dalai\u27 conjecture on the algebraic immunity of \Tr(\lambda x^{-1}) is correct. What is more, we further demonstrate some weak properties of \Tr(\lambda x^{-1}) in resistance to fast algebraic attacks