Paper 2013/585
On Algebraic Immunity of Trace Inverse Functions over Finite Fields with Characteristic Two
Xiutao Feng and Guang Gong
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' 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.
Metadata
 Available format(s)
 Publication info
 Preprint. MINOR revision.
 Keywords
 boolean functionsalgebraic immunity
 Contact author(s)
 fengxt @ amss ac cn
 History
 20141010: revised
 20130914: received
 See all versions
 Short URL
 https://ia.cr/2013/585
 License

CC BY
BibTeX
@misc{cryptoeprint:2013/585, author = {Xiutao Feng and Guang Gong}, title = {On Algebraic Immunity of Trace Inverse Functions over Finite Fields with Characteristic Two}, howpublished = {Cryptology ePrint Archive, Paper 2013/585}, year = {2013}, note = {\url{https://eprint.iacr.org/2013/585}}, url = {https://eprint.iacr.org/2013/585} }