Some lower bounds on the algebraic immunity of functions given by their trace forms

Some lower bounds on the algebraic immunity of functions given by their trace forms The algebraic immunity of a Boolean function is a parameter that characterizes the possibility to bound this function from above or below by a nonconstant Boolean function of a low algebraic degree. We obtain lower bounds on the algebraic immunity for a class of functions expressed through the inversion operation in the field GF(2 n ), as well as for larger classes of functions defined by their trace forms. In particular, for n ≥ 5, the algebraic immunity of the function Tr n (x −1) has a lower bound ⌊2√n + 4⌋ − 4, which is close enough to the previously obtained upper bound ⌊√n⌋ + ⌈n/⌊√n⌋⌉ − 2. We obtain a polynomial algorithm which, give a trace form of a Boolean function f, computes generating sets of functions of degree ≤ d for the following pair of spaces. Each function of the first (linear) space bounds f from below, and each function of the second (affine) space bounds f from above. Moreover, at the output of the algorithm, each function of a generating set is represented both as its trace form and as a polynomial of Boolean variables. http://www.deepdyve.com/assets/images/DeepDyve-Logo-lg.png Problems of Information Transmission Springer Journals

Some lower bounds on the algebraic immunity of functions given by their trace forms

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Publisher
SP MAIK Nauka/Interperiodica
Copyright
Copyright © 2008 by Pleiades Publishing, Ltd.
Subject
Engineering; Communications Engineering, Networks; Electrical Engineering; Information Storage and Retrieval; Systems Theory, Control
ISSN
0032-9460
eISSN
1608-3253
D.O.I.
10.1134/S0032946008030071
Publisher site
See Article on Publisher Site

Abstract

The algebraic immunity of a Boolean function is a parameter that characterizes the possibility to bound this function from above or below by a nonconstant Boolean function of a low algebraic degree. We obtain lower bounds on the algebraic immunity for a class of functions expressed through the inversion operation in the field GF(2 n ), as well as for larger classes of functions defined by their trace forms. In particular, for n ≥ 5, the algebraic immunity of the function Tr n (x −1) has a lower bound ⌊2√n + 4⌋ − 4, which is close enough to the previously obtained upper bound ⌊√n⌋ + ⌈n/⌊√n⌋⌉ − 2. We obtain a polynomial algorithm which, give a trace form of a Boolean function f, computes generating sets of functions of degree ≤ d for the following pair of spaces. Each function of the first (linear) space bounds f from below, and each function of the second (affine) space bounds f from above. Moreover, at the output of the algorithm, each function of a generating set is represented both as its trace form and as a polynomial of Boolean variables.

Journal

Problems of Information TransmissionSpringer Journals

Published: Oct 18, 2008

References

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