## Cryptology ePrint Archive: Report 2012/396

On second-order nonlinearity and maximum algebraic immunity of some bent functions in $\cP S^+$

Brajesh Kumar Singh

Abstract: In this paper, by modifying a subclass of bent functions in $\mathcal P S_{ap}$, we construct another subclass of bent functions in $\mathcal P S^+$ with maximum algebraic degree. We demonstrate that the algebraic immunity of the constructed functions is maximum. The result is proved by using the well known conjecture proposed by Tu and Deng (Des. Codes Cryptogr. 60(1), pp. 1-14, 2011) which has been proved recently by Cohen and Flori (http://eprint.iacr.org/ 2011/400.pdf). Finally, we identify a class of $\cD_0$ type bent functions constructed by modifying Dillon functions whose lower bound on second-order nonlinearity is very high.

Category / Keywords: secret-key cryptography / Boolean function, bent function, algebraic immunity, Dillon functions, $\cD_0$ type bents, second-order nonlinearities.