**On the Boolean functions With Maximum Possible Algebraic Immunity : Construction and A Lower Bound of the Count**

*Longjiang Qu and Guozhu Feng and Chao Li*

**Abstract: **This paper gives a construction method which can get a large class
of Boolean functions with maximum algebraic immunity(AI) from one
such giving function. Our constructions get more functions than any
previous construction. The cryptographic properties, such as
balance, algebraic degree etc, of those functions are studied. It
shows that we can construct Boolean functions with better
cryptographic properties, which gives the guidance for the design of
Boolean functions to resist algebraic attack, and helps to design
good cryptographic primitives of cryptosystems. From these
constructions, we show that the count of the Boolean functions with
maximum AI is bigger than ${2^{2^{n-1}}}$ for $n$ odd, bigger than
${2^{2^{n-1}+\frac{1}{2}\binom{n}{\frac{n}{2}} }}$ for $n$ even,
which confirms the computer simulation result that such boolean
functions are numerous. As far as we know, this is the first bound
about this count.

**Category / Keywords: **foundations / Algebraic Attack, Algebraic Degree, Algebraic Immunity, Annihilator, Balance, Boolean Functions

**Date: **received 25 Nov 2005, last revised 7 Apr 2006

**Contact author: **ljqu_happy at hotmail com

**Available format(s): **PDF | BibTeX Citation

**Version: **20060408:023007 (All versions of this report)

**Short URL: **ia.cr/2005/449

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