Upper Bounds on Algebraic Immunity of Boolean Power Functions

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


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.


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


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Copyright information

© Springer-Verlag Berlin Heidelberg 2006

Authors and Affiliations

  • Yassir Nawaz
    • 1
  • Guang Gong
    • 1
  • Kishan Chand Gupta
    • 2
  1. 1.Department of Electrical and Computer EngineeringUniversity of WaterlooWaterlooCanada
  2. 2.Centre for Applied Cryptographic ResearchUniversity of WaterlooWaterlooCanada

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