Artículos

Algebraic Immunity of Boolean Functions – Analysis and Construction*

Inmunidad Algebraica de Funciones Booleanas –Análisis y Construcción

Deepak Kumar Dalai¹ and Subhamoy Maitra²

¹ Department of Mathematics National Institute of Science Education and Research, Sachivalay Marg, Bhubaneswar 751005 INDIA. E–mail: deepak@iopb.res.in

² Applied Statistics Unit, Indian Statistical Institute 203 B T Road, Calcutta 700108, INDIA. E–mail: subho@isical.ac.in

Article received on March 1, 2008
Accepted on June 14, 2008

Abstract

In this paper, we first analyse the method of finding algebraic immunity of a Boolean function. Given a Boolean function f on n–variables, we identify a reduced set of homogeneous linear equations by solving which one can decide whether there exist annihilators of f at a specific degree. Moreover, we analyse how an affine transformation on the input variables of f can be exploited to achieve further reduction in the set of homogeneous linear equations. Next, from the design point of view, we construct balanced Boolean functions with maximum possible AI with an additional property which is necessary to resist the fast algebraic attack.

Keywords: Algebraic Attacks, Algebraic Normal Form, Annihilators, Boolean Functions, Fast Algebraic Attacks, Homogeneous Linear Equations.

Resumen

En este artículo, analizamos primero el método que permite encontrar la inmunidad algebraica de una función Booleana. Dada una función Booleana f de n variables, identificamos un conjunto reducido de ecuaciones lineales homogéneas resolviendo cuál de ellas puede ser usada para determinar si existen nulificadores de f de un grado específico. Además analizamos cómo una transformación afín de las variables de entrada de f puede ser aplicada para alcanzar una mayor reducción en el conjunto de ecuaciones lineales homogéneas. En seguida, y analizando desde el punto de vista de diseño, construimos funciones Booleanas balanceadas con inmunidad algebraica máxima y una propiedad adicional necesaria para resistir versiones rápidas de ataques algebraicos.

Palabras Claves: Ataques algebraicos, froma normal algebraica, nulificadores, funciones Booleanas, ataques algebrados rápidos, ecuaciones lineales homogéneas.

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Note

* This is a substantially revised and merged version of two conference papers, (i) "Reducing the Number of Homogeneous Linear Equations in Finding Annihilators", in Sequences and Their Applications, SETA '06, pages 376–390, volume 4086, Lecture Notes in Computer Science, Springer Verlag, 2006. Section 3.1 and Appendix A are added over the conference version, (ii) "Balanced Boolean Functions with (more than) Maximum Algebraic Immunity", in International Workshop on Coding and Cryptography, WCC '07, pages 99–108, INRIA, Rocquencourt, France in April 16–20, 2007. The proceedings of WCC '07 is only a workshop record and it is not printed by any publisher.