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24
On almost perfect nonlinear functions
 in Proc. IEEE Int. Symp. Information Theory
, 2005
"... Abstract—We investigate some open problems on almost perfect nonlinear (APN) functions over a finite field of characteristic 2.We provide new characterizations of APN functions and of APN permutations by means of their component functions. We generalize some results of Nyberg (1994) and strengthen a ..."
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Abstract—We investigate some open problems on almost perfect nonlinear (APN) functions over a finite field of characteristic 2.We provide new characterizations of APN functions and of APN permutations by means of their component functions. We generalize some results of Nyberg (1994) and strengthen a conjecture on the upper bound of nonlinearity of APN functions. We also focus on the case of quadratic functions. We contribute to the current works on APN quadratic functions by proving that a large class of quadratic functions cannot be APN. Index Terms—Almost bent function, almost perfect nonlinear (APN) function, power function, permutation polynomial. I.
Generalised RudinShapiro Constructions
 WCC2001, WORKSHOP ON CODING AND CRYPTOGRAPHY, PARIS(FRANCE
, 2001
"... A Golay Complementary Sequence (CS) has PeaktoAveragePowerRatio (PAPR) ≤ 2.0 for its onedimensional continuous Discrete Fourier Transform (DFT) spectrum. Davis and Jedwab showed that all known length 2 m CS, (GDJ CS), originate from certain quadratic cosets of ReedMuller (1, m). These can be g ..."
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Cited by 15 (8 self)
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A Golay Complementary Sequence (CS) has PeaktoAveragePowerRatio (PAPR) ≤ 2.0 for its onedimensional continuous Discrete Fourier Transform (DFT) spectrum. Davis and Jedwab showed that all known length 2 m CS, (GDJ CS), originate from certain quadratic cosets of ReedMuller (1, m). These can be generated using the RudinShapiro construction. This paper shows that GDJ CS have PAPR ≤ 2.0 under all unitary transforms whose rows are unimodular linear (Linear Unimodular Unitary Transforms (LUUTs)), including one and multidimensional generalised DFTs. We also propose tensor cosets of GDJ sequences arising from RudinShapiro extensions of nearcomplementary pairs, thereby generating many infinite sequence families with tight low PAPR bounds under LUUTs.
CrossCorrelation Analysis of Cryptographically Useful Boolean Functions and Sboxes
 Theory of Computing Systems
, 2001
"... We use the crosscorrelation function as a fundamental tool to study cryptographic properties of Boolean functions. This provides a unified treatment of a large section of Boolean function literature. In the process we generalize old results and obtain new characterizations of cryptographic properti ..."
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Cited by 15 (6 self)
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We use the crosscorrelation function as a fundamental tool to study cryptographic properties of Boolean functions. This provides a unified treatment of a large section of Boolean function literature. In the process we generalize old results and obtain new characterizations of cryptographic properties. In particular, new characterizations of bent functions and functions satisfying propagation characteristics are obtained in terms of the crosscorrelation and autocorrelation properties of sub functions. The exact relationship between the algebraic structure of the non zeros of the spectrum and the autocorrelation values is obtained for a cryptographically important class of functions. Finally we study the suitability of Sboxes in stream ciphers and conclude that currently known constructions for Sboxes have potential weaknesses for such application.
A Construction for Binary Sequence Sets with Low PeaktoAverage Power Ratio
"... A recursive construction is provided for sequence sets which possess good Hamming Distance and low PeaktoAverage Power Ratio (PAR) under any Local Unitary Unimodular Transform (including all one and multidimensional Discrete Fourier Transforms). An important instance of the construction identifie ..."
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Cited by 14 (9 self)
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A recursive construction is provided for sequence sets which possess good Hamming Distance and low PeaktoAverage Power Ratio (PAR) under any Local Unitary Unimodular Transform (including all one and multidimensional Discrete Fourier Transforms). An important instance of the construction identifies an iteration and specialisation of the MaioranaMcFarland (MM) construction. I.
On cryptographic properties of the cosets of R(1;m
 IEEE Trans. Inf. Theory
, 2001
"... Abstract—We introduce a new approach for the study of weight distributions of cosets of the Reed–Muller code of order 1. Our approach is based on the method introduced by Kasami in [1], using Pless identities. By interpreting some equations, we obtain a necessary condition for a coset to have a “hig ..."
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Abstract—We introduce a new approach for the study of weight distributions of cosets of the Reed–Muller code of order 1. Our approach is based on the method introduced by Kasami in [1], using Pless identities. By interpreting some equations, we obtain a necessary condition for a coset to have a “high ” minimum weight. Most notably, we are able to distinguish such cosets which have three weights only. We then apply our results to the problem of the nonlinearity of Boolean functions. We particularly study the links between this criterion and the propagation characteristics of a function. Index Terms—Boolean function, derivation, nonlinearity, propagation criterion, Reed–Muller codes.
Highly Nonlinear Balanced Boolean Functions with very good Autocorrelation Property
 In Workshop on Coding and Cryptography  WCC 2001, Electronic
, 2000
"... Constructing highly nonlinear balanced Boolean functions with very good autocorrelation property is an interesting open question. In this direction we use the measure f , the highest magnitude of all autocorrelation coecients for a function f . We provide balanced functions f with currently best kn ..."
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Cited by 14 (4 self)
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Constructing highly nonlinear balanced Boolean functions with very good autocorrelation property is an interesting open question. In this direction we use the measure f , the highest magnitude of all autocorrelation coecients for a function f . We provide balanced functions f with currently best known nonlinearity and f values together. We extend the result of Maitra and Sarkar (2000) for 15variable functions which experimentally disprove the conjecture proposed by Zhang and Zheng (1995). We prove it theoretically for dierent ranges of nonlinearity, where our constructions are based on modications of PattersonWiedemann (1983) functions. Also we propose a simple bent based construction technique to get functions with very good f values for odd number of variables. This construction has a root in Kerdock Codes. Moreover, our construction on even number of variables is a recursive one and we conjecture (similar to Dobbertin's conjecture (1994) with respect to nonlinearity) that this provides the minimum possible value of f for a balanced function f on even number of variables. Next we discuss about the autocorrelation values of correlation immune and resilient Boolean functions. We provide new lower bounds and related results on absolute indicator and sum of square indicator (of autocorrelation) for certain orders of correlation immunity and resiliency and clearly show that autocorrelation goes against order of correlation immunity. We also point out the weakness of two recursive construction techniques for resilient functions in terms of autocorrelation values. Key words: Boolean Function, Nonlinearity, Balancedness, Correlation Immunity, Autocorrelation, Propagation Characteristics, Global Avalanche Characteristics. 1
Differential properties of power functions
 Int. J. Inform. and Coding Theory
, 2010
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GolayDavisJedwab Complementary Sequences and RudinShapiro Constructions
 IEEE TRANS. INFORM. THEORY
, 2001
"... A Golay Complementary Sequence (CS) has a PeaktoAveragePowerRatio (PAPR) ≤ 2.0 for its onedimensional continuous Discrete Fourier Transform (DFT) spectrum. Davis and Jedwab showed that all known length 2 m CS, (GDJ CS), originate from certain quadratic cosets of ReedMuller (1, m). These can b ..."
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Cited by 12 (4 self)
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A Golay Complementary Sequence (CS) has a PeaktoAveragePowerRatio (PAPR) ≤ 2.0 for its onedimensional continuous Discrete Fourier Transform (DFT) spectrum. Davis and Jedwab showed that all known length 2 m CS, (GDJ CS), originate from certain quadratic cosets of ReedMuller (1, m). These can be generated using the RudinShapiro construction. This paper shows that GDJ CS have a PAPR ≤ 2.0 under all 2 m ×2 m unitary transforms whose rows are unimodular linear (Linear Unimodular Unitary Transforms (LUUTs)), including one and multidimensional generalised DFTs. In this context we define Constahadamard Transforms (CHTs) and show how all LUUTs can be formed from tensor combinations of CHTs. We also propose tensor cosets of GDJ sequences arising from RudinShapiro extensions of nearcomplementary pairs, thereby generating many more infinite sequence families with tight low PAPR bounds under LUUTs. We m m−⌊ then show that GDJ CS have a PAPR ≤ 2 2 ⌋ under all 2m × 2m unitary transforms whose rows are linear (Linear Unitary Transforms (LUTs)). Finally we present a radix2 tensor decomposition of any 2 m × 2 m LUT.
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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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
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.