# American Institute of Mathematical Sciences

May  2017, 11(2): 373-377. doi: 10.3934/amc.2017031

## Fast algebraic immunity of Boolean functions

 1 Department of Mathematics, University of Paris Ⅷ and Paris ⅩⅢ and Télécom ParisTech, LAGA, UMR 7539, CNRS, Sorbonne Paris Cité 2 Télécom ParisTech, department INFRES/MIC2, CNRS, UMR 5441

Received  February 2016 Revised  March 2016 Published  May 2017

Since 1970, Boolean functions have been the focus of a lot of attention in cryptography. An important topic in symmetric ciphers concerns the cryptographic properties of Boolean functions and constructions of Boolean functions with good cryptographic properties, that is, good resistance to known attacks. An important progress in cryptanalysis areas made in 2003 was the introduction by Courtois and Meier of algebraic attacks and fast algebraic attacks which are very powerful analysis concepts and can be applied to almost all cryptographic algorithms. To study the resistance against algebraic attacks, the notion of algebraic immunity has been introduced. In this paper, we use a parameter introduced by Liu and al., called fast algebraic immunity, as a tool to measure the resistance of a cryptosystem (involving Boolean functions) to fast algebraic attacks. We prove an upper bound on the fast algebraic immunity. Using our upper bound, we establish the weakness of trace inverse functions against fast algebraic attacks confirming a recent result of Feng and Gong.

Citation: Sihem Mesnager, Gérard Cohen. Fast algebraic immunity of Boolean functions. Advances in Mathematics of Communications, 2017, 11 (2) : 373-377. doi: 10.3934/amc.2017031
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