Cryptology ePrint Archive: Report 2007/444

Tight bounds between algebraic immunity and nonlinearities of high orders

Lobanov Mikhail

Abstract: Among cryptographically significant characteristics of Boolean functions used in symmetric ciphers the algebraic immunity and the nonlinearities of high orders play the important role. Some bounds on the nonlinearities of high orders of Boolean functions via its algebraic immunity were obtained in recent papers. In this paper we improve these results and obtain new tight bounds. We prove new universal tight lower bound that reduces the problem of an estimation of high order nonlinearities to the problem of the finding of dimensions of some linear spaces of Boolean functions. As simple consequences we obtain all previously known bounds in this field. For polynomials with disjoint terms we reduce the finding of dimensions of linear spaces of Boolean functions mentioned above to a simple combinatorial analysis. Finally, we prove the tight lower bound on the nonlinearity of the second order via its algebraic immunity.

Category / Keywords: boolean functions

Date: received 27 Nov 2007, last revised 15 Dec 2007

Contact author: misha_msu at mail ru

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Version: 20071215:091046 (All versions of this report)

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