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Paper 2013/585

On Algebraic Immunity of $\Tr(x^{-1})$ over $\mathbb{F}_{2^n}

Xiutao Feng

Abstract

The trace inverse function $\Tr(x^{-1})$ over the finite field $\mathbb{F}_{2^n}$ is a class of very important Boolean functions in stream ciphers, which possesses many good properties, including high algebraic degree, high nonlinearity, ideal autocorrelation, etc. In this work we discuss properties of $\Tr(x^{-1})$ in resistance to (fast) algebraic attacks. As a result, we prove that the algebraic immunity of $\Tr(x^{-1})$ arrives the upper bound given by Y. Nawaz et al when $n\ge4$, that is, $\AI(\Tr(x^{-1}))=\ceil{2\sqrt{n}}-2$, which shows that D.K. Dalai' conjecture on the algebraic immunity of $\Tr(x^{-1})$ is correct for almost all positive integers $n$. What is more, we further demonstrate some weak properties of $\Tr(x^{-1})$ in resistance to fast algebraic attacks.

Metadata
Available format(s)
PDF
Publication info
Preprint. MINOR revision.
Keywords
trace inverse functionalgebraic immunityfast algebraic attacks
Contact author(s)
fengxt @ amss ac cn
History
2014-10-10: revised
2013-09-14: received
See all versions
Short URL
https://ia.cr/2013/585
License
Creative Commons Attribution
CC BY
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