Algebraic Immunity of Boolean Functions-Analysis and Construction (original) (raw)
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Inmunidad Algebraica de Funciones Booleanas – Análisis y Construcción
2010
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.
Balanced Boolean functions with (more than) maximum algebraic immunity
2007
In this correspondence, construction of balanced Boolean functions with maximum possible algebraic (annihilator) immunity (AI) is studied with an additional property which is necessary to resist fast algebraic attack. The additional property considered here is, given an n-variable (n even) balanced function f with maximum possible AI n 2 , and given two n-variable Boolean functions g, h such that f g = h, if deg(h) = n 2 , then deg(g) must be greater than or equal to n 2. Our results can also be used to present theoretical construction of resilient Boolean functions having maximum possible AI.
1 Modifying Boolean Functions to Ensure Maximum Algebraic Immunity
2014
Abstract—The algebraic immunity of cryptographic Boolean functions is studied in this paper. Proper modifications of functions achieving maximum algebraic immunity are proved, in order to yield new functions of also maximum algebraic immunity. It is shown that the derived results apply to known classes of functions. Moreover, two new efficient algorithms to produce functions of guaranteed maximum algebraic immunity are developed, which further extend and generalize known constructions of functions with maximum algebraic immunity. Index Terms—algebraic attack, algebraic immunity, annihilators, Boolean functions, cryptography I.
Algebraic Immunity for Cryptographically Significant Boolean Functions: Analysis and Construction
IEEE Transactions on Information Theory, 2006
Recently, algebraic attacks have received a lot of attention in the cryptographic literature. It has been observed that a Boolean function used as a cryptographic primitive, and interpreted as a multivariate polynomial over 2 , should not have low degree multiples obtained by multiplication with low degree nonzero functions. In this paper, we show that a Boolean function having low nonlinearity is (also) weak against algebraic attacks, and we extend this result to higher order nonlinearities. Next, we present enumeration results on linearly independent annihilators. We also study certain classes of highly nonlinear resilient Boolean functions for their algebraic immunity. We identify that functions having low-degree subfunctions are weak in terms of algebraic immunity, and we analyze some existing constructions from this viewpoint. Further, we present a construction method to generate Boolean functions on variables with highest possible algebraic immunity 2 (this construction, first presented at the 2005 Workshop on Fast Software Encryption (FSE 2005), has been the first one producing such functions). These functions are obtained through a doubly indexed recursive relation. We calculate their Hamming weights and deduce their nonlinearities; we show that they have very high algebraic degrees. We express them as the sums of two functions which can be obtained from simple symmetric functions by a transformation which can be implemented with an algorithm whose complexity is linear in the number of variables. We deduce a very fast way of computing the output to these functions, given their input.
On Some Necessary Conditions of Boolean Functions to Resist Algebraic Attacks
2006
In this thesis we discuss certain properties of Boolean functions that are necessary for resistance against algebraic and fast algebraic attacks. A Boolean function f(x1, . . . , xn) on n variables may be described as a multivariate polynomial over GF (2) and it is well known that its algebraic degree d should not be low if it has to be used as a primitive in a well designed cryptosystem. Recently, it has been noted that a necessary condition in resisting algebraic attack is as follows: the function f should not have a relation fg = h, where g, h are nonzero n-variable Boolean functions of low degrees. This condition boils down to the situation that the function f should not have relations like fh1 = 0 or (1 + f)h2 = 0, where h1, h2 are nonzero n-variable Boolean functions of low degrees. The function h1 (respectively h2) is called the annihilator of f (respectively 1 + f). The notation AIn(f) is used to denote the minimum degree of the annihilators of f or 1 + f . This is well know...
Modifying Boolean Functions to Ensure Maximum Algebraic Immunity
The algebraic immunity of cryptographic Boolean functions is studied in this paper. Proper modifications of functions achieving maximum algebraic immunity are proved, in order to yield new functions of also maximum algebraic immunity. It is shown that the derived results apply to known classes of functions. Moreover, two new efficient algorithms to produce functions of guaranteed maximum algebraic immunity are developed, which further extend and generalize known constructions of functions with maximum algebraic immunity.
Secondary constructions of Boolean functions with maximum algebraic immunity
Cryptography and Communications, 2013
The algebraic immunity of cryptographic Boolean functions with odd number of variables is studied in this paper. Proper modifications of functions with maximum algebraic immunity are proved that yield new functions whose algebraic immunity is also maximum. Several results are provided for both the multivariate and univariate representation, and their applicability is shown on known classes of Boolean functions. Moreover, new efficient algorithms to produce functions of guaranteed maximum algebraic immunity are developed, which further extend and generalize well-known constructions in this area. It is shown that high nonlinearity as well as good behavior against fast algebraic attacks are also achievable in several cases.
2005
Algebraic attack has recently become an important tool in cryptanalysing different stream and block cipher systems. A Boolean function, when used in some cryptosystem, should be designed properly to resist this kind of attack. The cryptographic property of a Boolean function, that resists algebraic attack, is known as Algebraic Immunity (AI). So far, the attempt in designing Boolean functions with required algebraic immunity was only ad-hoc, i.e., the functions were designed keeping in mind the other cryptographic criteria, and then it has been checked whether it can provide good algebraic immunity too. For the first time, in this paper, we present a construction method to generate Boolean functions on n variables with highest possible algebraic immunity n 2 . Such a function can be used in conjunction with (using direct sum) functions having other cryptographic properties. In a different direction we identify that functions, having low degree subfunctions, are weak in terms of algebraic immunity and analyse some existing constructions from this viewpoint.
On the Algebraic Immunity of Symmetric Boolean Functions
Lecture Notes in Computer Science, 2005
In this paper, we analyse the algebraic immunity of symmetric Boolean functions. We identify a set of lowest degree annihilators for symmetric functions and propose an efficient algorithm for computing the algebraic immunity of a symmetric function. The existence of several symmetric functions with maximum algebraic immunity is proven. In this way, a new class of function which have good implementation properties and maximum algebraic immunity is found. We also investigate the existence of symmetric functions with high nonlinearity and reasonable order of algebraic immunity. Finally, we give suggestions how to use symmetric functions in a stream cipher.