FSE 2006: Fast Software Encryption pp 375-389

# Upper Bounds on Algebraic Immunity of Boolean Power Functions

• Yassir Nawaz
• Guang Gong
• Kishan Chand Gupta
Conference paper
Part of the Lecture Notes in Computer Science book series (LNCS, volume 4047)

## Abstract

Algebraic attacks have received a lot of attention in studying security of symmetric ciphers. The function used in a symmetric cipher should have high algebraic immunity ($${\cal AI}$$) to resist algebraic attacks. In this paper we are interested in finding $${\cal AI}$$ of Boolean power functions. We give an upper bound on the $${\cal AI}$$ of any Boolean power function and a formula to find its corresponding low degree multiples. We prove that the upper bound on the $${\cal AI}$$ for Boolean power functions with Inverse, Kasami and Niho exponents are $$\lfloor \sqrt{n}\rfloor + \lceil \frac{n}{\lfloor \sqrt{n} \rfloor}\rceil -2$$, $$\lfloor \sqrt{n} \rfloor + \lceil \frac{n}{\lfloor \sqrt{n} \rfloor}\rceil$$ and $$\lfloor \sqrt{n} \rfloor + \lceil \frac{n}{\lfloor \sqrt{n} \rfloor}\rceil$$ respectively. We also generalize this idea to Boolean polynomial functions. All existing algorithms to determine $${\cal AI}$$ and corresponding low degree multiples become too complex if the function has more than 25 variables. In our approach no algorithm is required. The $${\cal AI}$$ and low degree multiples can be obtained directly from the given formula.

### Keywords

Algebraic attacks Algebraic immunity Inverse exponent Kasami exponent Polynomial functions Power functions Niho exponent

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