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**2005/449**

**Title:**

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

**Authors:**
Longjiang Qu, Guozhu Feng, 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.

**ePrint:**
https://eprint.iacr.org/2005/449

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