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44
Nonlinearity bounds and constructions of resilient Boolean functions
 LNCS 1880, M. Bellare, Ed
, 2000
"... Abstract. In this paper we investigate the relationship between the nonlinearity and the order of resiliency of a Boolean function. We first prove a sharper version of McEliece theorem for ReedMuller codes as applied to resilient functions, which also generalizes the well known XiaoMassey characte ..."
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Cited by 34 (8 self)
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Abstract. In this paper we investigate the relationship between the nonlinearity and the order of resiliency of a Boolean function. We first prove a sharper version of McEliece theorem for ReedMuller codes as applied to resilient functions, which also generalizes the well known XiaoMassey characterization. As a consequence, a nontrivial upper bound on the nonlinearity of resilient functions is obtained. This result coupled with Siegenthaler’s inequality leads to the notion of best possible tradeoff among the parameters: number of variables, order of resiliency, nonlinearity and algebraic degree. We further show that functions achieving the best possible tradeoff can be constructed by the MaioranaMcFarland like technique. Also we provide constructions of some previously unknown functions.
On the Algebraic Immunity of Symmetric Boolean Functions
 In Indocrypt 2005, number 3797 in LNCS
, 2005
"... In this paper, we analyse the algebraic immunity of symmetric Boolean functions. ..."
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Cited by 34 (1 self)
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In this paper, we analyse the algebraic immunity of symmetric Boolean functions.
A larger class of cryptographic Boolean functions via a study of the MaioranaMcFarland construction
 In Advances in Cryptology  CRYPTO 2002, number 2442 in Lecture Notes in Computer Science
, 2002
"... Abstract. Thanks to a new upper bound, we study more precisely the nonlinearities of MaioranaMcFarland’s resilient functions. We characterize those functions with optimum nonlinearities and we give examples of functions with high nonlinearities. But these functions have a peculiarity which makes th ..."
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Cited by 25 (3 self)
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Abstract. Thanks to a new upper bound, we study more precisely the nonlinearities of MaioranaMcFarland’s resilient functions. We characterize those functions with optimum nonlinearities and we give examples of functions with high nonlinearities. But these functions have a peculiarity which makes them potentially cryptographically weak. We study a natural superclass of MaioranaMcFarland’s class whose elements do not have the same drawback and we give examples of such functions achieving high nonlinearities.
Propagation Characteristics and CorrelationImmunity of Highly Nonlinear Boolean Functions
 EUROCRYPT 2000, Lecture Notes in Comp. Sci
, 2000
"... Abstract. We investigate the link between the nonlinearity of a Boolean function and its propagation characteristics. We prove that highly nonlinear functions usually have good propagation properties regarding different criteria. Conversely, any Boolean function satisfying the propagation criterion ..."
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Cited by 24 (7 self)
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Abstract. We investigate the link between the nonlinearity of a Boolean function and its propagation characteristics. We prove that highly nonlinear functions usually have good propagation properties regarding different criteria. Conversely, any Boolean function satisfying the propagation criterion with respect to a linear subspace of codimension 1 or 2 has a high nonlinearity. We also point out that most highly nonlinear functions with a threevalued Walsh spectrum can be transformed into 1resilient functions. 1
New Constructions of Resilient and Correlation Immune Boolean Functions Achieving Upper Bound on Nonlinearity
, 2001
"... Recently, weight divisibility results on resilient and correlation immune Boolean functions have received a lot of attention. These results have direct consequences towards the upper bound on nonlinearity of resilient and correlation immune Boolean functions of certain order. Now the clear requireme ..."
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Cited by 19 (4 self)
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Recently, weight divisibility results on resilient and correlation immune Boolean functions have received a lot of attention. These results have direct consequences towards the upper bound on nonlinearity of resilient and correlation immune Boolean functions of certain order. Now the clear requirement in the design of resilient Boolean functions (which optimizes Siegenthaler's inequality) is to provide results which attain the upper bound on nonlinearity. Here we construct a 7variable, 2resilient Boolean function with nonlinearity 56. This solves the maximum nonlinearity issue for 7variable functions with any order of resiliency. Using this 7variable function, we also construct a 10variable, 4resilient Boolean function with nonlinearity 480. Construction of these two functions was posed as important open questions in Crypto 2000. Also, we provide methods to generate an infinite sequence of Boolean functions on n = 7+3i variables (i 0) with order of resiliency m = 2+2i, algebraic degree 4 + i and nonlinearity 2 n 1 2 m+1 , which were not known earlier. We conclude with constructions of some unbalanced correlation immune functions of 5 and 6 variables which attain the upper bound on nonlinearity.
A New Statistical Testing for Symmetric Ciphers and Hash Functions
 Proc. Information and Communications Security 2002, volume 2513 of LNCS
, 2002
"... This paper presents a new, powerful statistical testing of symmetric ciphers and hash functions which allowed us to detect biases in both of these systems where previously known tests failed. We first give a complete characterization of the Algebraic Normal Form (ANF) of random Boolean functions by ..."
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Cited by 17 (1 self)
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This paper presents a new, powerful statistical testing of symmetric ciphers and hash functions which allowed us to detect biases in both of these systems where previously known tests failed. We first give a complete characterization of the Algebraic Normal Form (ANF) of random Boolean functions by means of the M obius transform. Then we built a new testing based on the comparison between the structure of the different Boolean functions Algebraic Normal Forms characterizing symmetric ciphers and hash functions and those of purely random Boolean functions. Detailed testing results on several cryptosystems are presented. As a main result we show that AES, DES Snow and Lili128 fail all or part of the tests and thus present strong biases.
Results on Rotation Symmetric Bent and Correlation Immune Boolean Functions
 Fast Software Encryption Workshop (FSE 2004
, 2004
"... Abstract. Recent research shows that the class of Rotation Symmetric Boolean Functions (RSBFs), i.e., the class of Boolean functions that are invariant under circular translation of indices, is potentially rich in functions of cryptographic significance. Here we present new results regarding the Rot ..."
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Abstract. Recent research shows that the class of Rotation Symmetric Boolean Functions (RSBFs), i.e., the class of Boolean functions that are invariant under circular translation of indices, is potentially rich in functions of cryptographic significance. Here we present new results regarding the Rotation Symmetric (rots) correlation immune (CI) and bent functions. We present important data structures for efficient search strategy of rots bent and CI functions. Further, we prove the nonexistence of homogeneous rots bent functions of degree ≥ 3 on a single cycle.
Additive autocorrelation of resilient Boolean functions
 In: Selected Areas in Cryptography 2003, LNCS
, 2004
"... Abstract. In this paper, we introduce a new notion called the dual function for studying Boolean functions. First, we discuss general properties of the dual function that are related to resiliency and additive autocorrelation. Second, we look at preferred functions which are Boolean functions wit ..."
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Cited by 11 (4 self)
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Abstract. In this paper, we introduce a new notion called the dual function for studying Boolean functions. First, we discuss general properties of the dual function that are related to resiliency and additive autocorrelation. Second, we look at preferred functions which are Boolean functions with the lowest 3valued spectrum. We prove that if a balanced preferred function has a dual function which is also preferred, then it is resilient, has high nonlinearity and optimal additive autocorrelation. We demonstrate four such constructions of optimal Boolean functions using the Kasami, DillonDobbertin, Segre hyperoval and WelchGong Transformation functions. Third, we compute the additive autocorrelation of some known resilient preferred functions in the literature by using the dual function. We conclude that our construction yields highly nonlinear resilient functions with better additive autocorrelation than the MaioranaMcFarland functions. We also analysed the saturated functions, which are resilient functions with optimized algebraic degree and nonlinearity. We show that their additive autocorrelation have high peak values, and they become linear when we fix very few bits. These potential weaknesses have to be considered before we deploy them in applications. 1
New constructions of resilient Boolean functions with maximal nonlinearity
 Proceedings of FSE 2000, to appear in the Lecture Notes in Computer Science Series
, 2000
"... . In this paper we develop a technique that allows to obtain new eective constructions of highly resilient Boolean functions with high nonlinearity. In particular, we prove that the upper bound 2 ..."
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. In this paper we develop a technique that allows to obtain new eective constructions of highly resilient Boolean functions with high nonlinearity. In particular, we prove that the upper bound 2
On Plateaued Functions and Their Constructions
 in Fast Software Encryption FSE ′ 2003, 10th International Workshop (Lund, Sweden. February 24–26, 2003). Proceedings (Springer
, 2003
"... Abstract. We use the notion of covering sequence, introduced by C. Carlet and Y. Tarannikov, to give a simple characterization of bent functions. We extend it into a characterization of plateaued functions (that is bent and threevalued functions). After recalling why the class of plateaued function ..."
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Abstract. We use the notion of covering sequence, introduced by C. Carlet and Y. Tarannikov, to give a simple characterization of bent functions. We extend it into a characterization of plateaued functions (that is bent and threevalued functions). After recalling why the class of plateaued functions provides good candidates to be used in cryptosystems, we study the known families of plateaued functions and their drawbacks. We show in particular that the class given as new by Zhang and Zheng is in fact a subclass of MaioranaMcFarland’s class. We introduce a new class of plateaued functions and prove its good cryptographic properties.