Hybrid classes of balanced Boolean functions with good cryptographic properties (original) (raw)
Related papers
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.
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.
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.
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.
Walailak Journal of Science and Technology (WJST)
This paper consists of proposal of two new constructions of balanced Boolean function achieving a new lower bound of nonlinearity along with high algebraic degree and optimal or highest algebraic immunity. This construction has been made by using representation of Boolean function with primitive elements. Galois Field, used in this representation has been constructed by using powers of primitive element such that greatest common divisor of power and is 1. The constructed balanced variable Boolean functions achieve higher nonlinearity, algebraic degree of , and algebraic immunity of for odd , for even . The nonlinearity of Boolean function obtained in the proposed constructions is better as compared to existing Boolean functions available in the literature without adversely affecting other properties such as balancedness, algebraic degree and algebraic immunity.
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.
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.
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.
A family of boolean functions with good cryptographic properties
Axioms, 2019
In 2005, Philippe Guillot presented a new construction of Boolean functions using linear codes as an extension of the Maiorana–McFarland’s (MM) construction of bent functions. In this paper, we study a new family of Boolean functions with cryptographically strong properties, such as non-linearity, propagation criterion, resiliency, and balance. The construction of cryptographically strong Boolean functions is a daunting task, and there is currently a wide range of algebraic techniques and heuristics for constructing such functions; however, these methods can be complex, computationally difficult to implement, and not always produce a sufficient variety of functions. We present in this paper a construction of Boolean functions using algebraic codes following Guillot’s work.