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On a generalized combinatorial conjecture involving addition mod 2 k −1, available at http://eprint.iacr.org/2011/400
"... In this note, we give a simple proof of the combinatorial conjecture proposed by Tang, Carlet and Tang, based on which they constructed two classes of Boolean functions with many good cryptographic properties. We also give more general properties about the generalization of the conjecture they propo ..."
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In this note, we give a simple proof of the combinatorial conjecture proposed by Tang, Carlet and Tang, based on which they constructed two classes of Boolean functions with many good cryptographic properties. We also give more general properties about the generalization of the conjecture they propose.
Almost Perfect Algebraic Immune Functions with Good Nonlinearity
"... Abstract. In the last decade, algebraic and fast algebraic attacks are regarded as the most successful attacks on LFSRbased stream ciphers. Since the notion of algebraic immunity was introduced, the properties and constructions of Boolean functions with maximum algebraic immunity have been research ..."
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Abstract. In the last decade, algebraic and fast algebraic attacks are regarded as the most successful attacks on LFSRbased stream ciphers. Since the notion of algebraic immunity was introduced, the properties and constructions of Boolean functions with maximum algebraic immunity have been researched in a large number of papers. However, there are few results with respect to Boolean functions with provable good immunity against fast algebraic attacks. In previous literature, only CarletFeng function, which is affine equivalent to discrete logarithm function, was proven to be optimal against fast algebraic attacks as well as algebraic attacks. In this paper, it is proven that a family of 2kvariable Boolean functions, including the function recently constructed by Tang et al. [IEEE TIT 59(1): 653–664, 2013], are almost perfect algebraic immune for any integer k ≥ 3. More exactly, they achieve optimal algebraic immunity and almost perfect immunity to fast algebraic attacks. The functions of such family are balanced and have optimal algebraic degree. A lower bound on their nonlinearity is obtained based on the work of Tang et al., which is better than that of CarletFeng function. It is also checked for 3 ≤ k ≤ 9 that the exact nonlinearity of such functions is very good, which is slightly smaller than that of CarletFeng function, and some functions of this family even have a slightly larger nonlinearity than Tang et al.’s function. To sum up, among the known functions with provable good immunity against fast algebraic attacks, the functions of this family make a tradeoff between the exact value and the lower bound of nonlinearity.
Constructing Vectorial Boolean Functions with High Algebraic Immunity Based on Group Decomposition
"... Abstract. In this paper, we construct a class of vectorial Boolean functions over F2n with high algebraic immunity based on the decomposition of the multiplicative group of F2n. By viewing F2n as G1G2 {0} (where G1 and G2 are subgroups of F ∗ 2n, (#G1, #G2) = 1 and #G1 × #G2 = 2 2k − 1), we give a ..."
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Abstract. In this paper, we construct a class of vectorial Boolean functions over F2n with high algebraic immunity based on the decomposition of the multiplicative group of F2n. By viewing F2n as G1G2 {0} (where G1 and G2 are subgroups of F ∗ 2n, (#G1, #G2) = 1 and #G1 × #G2 = 2 2k − 1), we give a generalized description for constructing vectorial Boolean functions with high algebraic immunity. Moreover, when n is even, we provide two special classes of vectorial Boolean functions with high(sometimes optimal) algebraic immunity, one is hyperbent, and the other is of balancedness and optimal algebraic degree.
On the Immunity of Boolean Functions Against Fast Algebraic Attacks Using Bivariate Polynomial Representation
"... Abstract. In the last decade, algebraic and fast algebraic attacks are regarded as the most successful attacks on LFSRbased stream ciphers. Since the notion of algebraic immunity was introduced, the properties and constructions of Boolean functions with maximum algebraic immunity have been research ..."
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Abstract. In the last decade, algebraic and fast algebraic attacks are regarded as the most successful attacks on LFSRbased stream ciphers. Since the notion of algebraic immunity was introduced, the properties and constructions of Boolean functions with maximum algebraic immunity have been researched in a large number of papers. However, it is unclear whether these functions behave well against fast algebraic attacks. In this paper, we study the immunity of Boolean functions against fast algebraic attacks using bivariate polynomial representation. Based on bivariate polynomial representation, we present a sufficient and necessary condition for a Boolean function to achieve good immunity against fast algebraic attacks, propose an efficient method for estimating the immunity of a large class of Boolean functions, including the functions of Q. Jin et al., and prove that the functions of D. Tang et al. achieve (almost) optimal immunity against fast algebraic attacks.
CONCATENATIONS OF THE HIDDEN WEIGHTED BIT FUNCTION AND THEIR CRYPTOGRAPHIC PROPERTIES
"... (Communicated by JoanJosep Climent) Abstract. To resist Binary Decision Diagrams (BDD) based attacks, a Boolean function should have a high BDD size. The hidden weighted bit function (HWBF), introduced by Bryant in 1991, seems to be the simplest function with exponential BDD size. In [28], Wang et ..."
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(Communicated by JoanJosep Climent) Abstract. To resist Binary Decision Diagrams (BDD) based attacks, a Boolean function should have a high BDD size. The hidden weighted bit function (HWBF), introduced by Bryant in 1991, seems to be the simplest function with exponential BDD size. In [28], Wang et al. investigated the cryptographic properties of the HWBF and found that it is a very good candidate for being used in real ciphers. In this paper, we modify the HWBF and construct two classes of functions with very good cryptographic properties (better than the HWBF). The new functions are balanced, with almost optimum algebraic degree and satisfy the strict avalanche criterion. Their nonlinearity is higher than that of the HWBF. We investigate their algebraic immunity, BDD size and their resistance against fast algebraic attacks, which seem to be better than those of the HWBF too. The new functions are simple, can be implemented efficiently, have high BDD sizes and rather good cryptographic properties. Therefore, they might be excellent candidates for constructions of reallife ciphers. 1.
A construction of Boolean functions with good cryptographic properties
"... The two important qualities of a cipher is security and speed. Frequently, to satisfy the security of a Boolean function primitive, speed may be tradedoff. In this paper we present a general construction that addresses both qualities. The idea of our construction is to manipulate a cryptographical ..."
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The two important qualities of a cipher is security and speed. Frequently, to satisfy the security of a Boolean function primitive, speed may be tradedoff. In this paper we present a general construction that addresses both qualities. The idea of our construction is to manipulate a cryptographically strong base function and one of its affine equivalent functions, using concatenation and negation. We achieve security from the inherent qualities of the base function, which are preserved (or increased) and obtain speed by the simple Boolean operations. We present two applications of the construction to demonstrate the flexibility and efficiency of the construction.
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"... Evolving balanced Boolean functions with optimal resistance to algebraic and fast algebraic attacks, maximal algebraic degree, and very high nonlinearity. ..."
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Evolving balanced Boolean functions with optimal resistance to algebraic and fast algebraic attacks, maximal algebraic degree, and very high nonlinearity.
On the Resistance of Primevariable Rotation Symmetric Boolean Functions against Fast Algebraic Attacks
"... Abstract Boolean functions used in stream ciphers should have many cryptographic properties in order to help resist different kinds of cryptanalytic attacks. The resistance of Boolean functions against fast algebraic attacks is an important cryptographic property. Deciding the resistance of an nvar ..."
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Abstract Boolean functions used in stream ciphers should have many cryptographic properties in order to help resist different kinds of cryptanalytic attacks. The resistance of Boolean functions against fast algebraic attacks is an important cryptographic property. Deciding the resistance of an nvariable Boolean function against fast algebraic attacks needs to determine the rank of a square matrix of order ∑e i=0
On the (Fast) Algebraic Immunity of Boolean Power Functions
"... Abstract The (fast) algebraic immunity, including (standard) algebraic immunity and the resistance against fast algebraic attacks, has been considered as an important cryptographic property for Boolean functions used in stream ciphers. This paper is on the determination of the (fast) algebraic immun ..."
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Abstract The (fast) algebraic immunity, including (standard) algebraic immunity and the resistance against fast algebraic attacks, has been considered as an important cryptographic property for Boolean functions used in stream ciphers. This paper is on the determination of the (fast) algebraic immunity of a special class of Boolean functions, called Boolean power functions. An nvariable Boolean power function f can be represented as a monomial trace function over finite field F2n, f(x) = Trn1 (λxk), where λ ∈ F2n and k is the coset leader of cyclotomic coset Ck modulo 2 n − 1. To determine the (fast) algebraic immunity of Boolean power functions one may need the arithmetic in F2n, which may be not computationally efficient compared with the operations over F2. We prove that if λ = αk and α is a primitive element of F2n, or k is coprime to 2n − 1, then the (fast) algebraic immunity of Boolean power function Trn1 (λx k) is the same as that of Trn1 (x k). This may help us determine the immunity of some Boolean power functions more efficiently. We show that Niho functions satisfy the coprime condition, and verify that a number of odd variables Kasami functions also satisfy the coprime condition.