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Constructions of Almost Optimal Resilient Boolean Functions on Large Even Number of Variables

WeiGuo Zhang, GuoZhen Xiao
2009 IEEE Transactions on Information Theory  
By using several sets of disjoint spectra functions on a small number of variables, an almost optimal resilient function on a large even number of variables can be constructed.  ...  A large class of highly nonlinear resilient functions which were not known are obtained. Then one method to optimize the degree of the constructed functions is proposed.  ...  A method for constructing resilient functions on large even number of input variables is proposed. We show that all the constructed functions are almost optimal.  ... 
doi:10.1109/tit.2009.2032736 fatcat:df4ly4vvbjgc7i3n5fipvmhqdq

1-Resilient Boolean Functions on Even Variables with Almost Perfect Algebraic Immunity

Gang Han, Yu Yu, Xiangxue Li, Qifeng Zhou, Dong Zheng, Hui Li
2017 Security and Communication Networks  
This paper uses bivariate representation of Boolean function and theory of finite field to construct a generalized and new class of Boolean functions on even variables by extending the Carlet-Feng functions  ...  Compared experimentally with Carlet-Feng functions and the functions constructed by the method of first-order concatenation existing in the literature on even (from 6 to 16) variables, these functions  ...  [19] presented a construction for a class of 1-resilient Boolean functions with optimal algebraic immunity on an even number of variables by dividing them into two correlation classes, that is, equivalence  ... 
doi:10.1155/2017/6268230 fatcat:xrqvbci5uffkfexma5ohzkw34q

Construction of 1-Resilient Boolean Functions with Optimal Algebraic Immunity and Good Nonlinearity

Sen-Shan Pan, Xiao-Tong Fu, Wei-Guo Zhang
2011 Journal of Computer Science and Technology  
This paper presents a construction for a class of 1-resilient Boolean functions with optimal algebraic immunity on an even number of variables by dividing them into two correlation classes, i.e. equivalence  ...  Furthermore, a class of 1-resilient functions on any number n > 2 of variables with at least sub-optimal algebraic immunity is provided.  ...  In this paper we propose a construction method to design 1-resilient Boolean functions on even number variables (n ≥ 3), which retain properties of the maximum degree and optimal algebraic immunity as  ... 
doi:10.1007/s11390-011-9433-6 fatcat:wixo5sppdrfnjdt77rzyi5gbd4

Quantum computing cryptography: Unveiling cryptographic Boolean functions with quantum annealing [article]

Feng Hu, Lucas Lamata, Mikel Sanz, Xi Chen, Xingyuan Chen, Chao Wang,, Enrique Solano
2018 arXiv   pre-print
However, the search of n-variable Boolean functions fulfilling global cryptographic constraints is computationally hard due to the super-exponential size O(2^2^n) of the space.  ...  Additionally, we benchmark small n cases in a D-Wave machine, showing its capacity of devising bent functions, the most relevant set of cryptographic Boolean functions.  ...  ACKNOWLEDGEMENTS We acknowledge support from National Natural Science Foundation of China (NSFC) grants 61332019, 61572304, 61272096, and 11474193, Shuguang (14SG35), the program of Shanghai Municipal  ... 
arXiv:1806.08706v2 fatcat:japqew5hyfgchjpfb56ujrqacm

Quantum computing cryptography: Finding cryptographic Boolean functions with quantum annealing by a 2000 qubit D-wave quantum computer

Feng Hu, Lucas Lamata, Mikel Sanz, Xi Chen, Xingyuan Chen, Chao Wang, Enrique Solano
2020 Physics Letters A  
However, the search of nvariable Boolean functions fulfilling global cryptographic constraints is computationally hard due to the superexponential size O(2 2 n ) of the space.  ...  Additionally, we benchmark small n cases in a D-Wave machine, showing its capacity of devising cryptographic Boolean functions with certain relevant properties.  ...  The "Initialized" column gives the number of mresilient Boolean functions given by the D-Wave device, and the "Optimized" column gives the number of m-resilient functions after repairing the chain via  ... 
doi:10.1016/j.physleta.2019.126214 fatcat:cxjyikex5jdfnmitslxilaqw6u

Generalized Maiorana-McFarland Constructions for Almost Optimal Resilient Functions [article]

WeiGuo Zhang, GuoZhen Xiao
2010 arXiv   pre-print
In a recent paper Zhang-Xiao, Zhang and Xiao describe a technique on constructing almost optimal resilient functions on even number of variables.  ...  In this paper, we will present an extensive study of the constructions of almost optimal resilient functions by using the generalized Maiorana-McFarland (GMM) construction technique.  ...  By Lemma 1, f ′′ is an m-resilient function. 4 Construction of almost optimal m-resilient functions on n variables (n odd) with nonlinearity > 2 n−1 − 2 (n−1)/2 For odd n, 15-variable Boolean functions  ... 
arXiv:1003.3492v1 fatcat:okszokdf7nh3rm3gw7mrobbaji

Maiorana–McFarland Class: Degree Optimization and Algebraic Properties

E. Pasalic
2006 IEEE Transactions on Information Theory  
In this paper, we consider a subclass of the Maiorana-McFarland class used in the design of resilient nonlinear Boolean functions.  ...  We show that these functions allow a simple modification so that resilient Boolean functions of maximum algebraic degree may be generated instead of suboptimized degree in the original class.  ...  More precisely, to construct a -resilient degree optimized variable function (a high nonlinearity is in particular achieved when is odd) the in number -variable -resilient linear functions are selected  ... 
doi:10.1109/tit.2006.881721 fatcat:mljx67x4xrgy5o6jr7p3glsauy

Improving the lower bound on the maximum nonlinearity of 1-resilient Boolean functions and designing functions satisfying all cryptographic criteria

WeiGuo Zhang, Enes Pasalic
2017 Information Sciences  
In this paper, we improve the lower bound on the maximum nonlinearity of 1resilient Boolean functions, for n even, by proposing a method of constructing this class of functions attaining the best nonlinearity  ...  This weakness is repaired by a suitable modification of the original functions giving a class of balanced functions with almost optimal resistance to FAA whose nonlinearity is better than the nonlinearity  ...  Boolean functions Let B n denote the set of Boolean functions in n variables.  ... 
doi:10.1016/j.ins.2016.10.001 fatcat:xttyrajfkzb2tg2xwibzkrq45e

A Maiorana–McFarland type construction for resilient Boolean functions on n variables (n even) with nonlinearity >2n-1-2n/2+2n/2-2

Subhamoy Maitra, Enes Pasalic
2006 Discrete Applied Mathematics  
The construction only considers functions in even number of variables n. So far the maximum nonlinearity attainable by resilient functions was 2 n−1 − 2 n/2 + 2 n/2−2 .  ...  In this paper, we present a construction method of m-resilient Boolean functions with very high nonlinearity for low values of m.  ...  Claude Carlet during WCC 2003 which helped in presenting and analysing the main construction of the paper in a more generalized framework.  ... 
doi:10.1016/j.dam.2005.03.014 fatcat:5aijo4u4qje2rkbcv24h3uczu4

Constructions of Resilient S-Boxes With Strictly Almost Optimal Nonlinearity Through Disjoint Linear Codes

Wei-Guo Zhang, Enes Pasalic
2014 IEEE Transactions on Information Theory  
Using such sets of disjoint linear codes, not necessarily of the same length, we have been able to provide a construction technique of t-resilient S-boxes F : F n 2 → F m 2 (n even, 1 < m ≤ n/4 ) with  ...  Actually, the nonlinearity of our functions is in many cases equal to the best known nonlinearity of balanced Boolean functions.  ...  X n ∈ GF (2) n ) even though a certain number of input variables is kept fixed.  ... 
doi:10.1109/tit.2014.2300067 fatcat:wqwvvk3pufb75hjkyzznjgzemu

Generalized Maiorana–McFarland Construction of Resilient Boolean Functions With High Nonlinearity and Good Algebraic Properties

Wei-Guo Zhang, Enes Pasalic
2014 IEEE Transactions on Information Theory  
It is shown that for any given m, this technique can be used to construct a large class of n-variable (n both even and odd) mresilient degree-optimized Boolean functions with currently best known nonlinearity  ...  A new framework concerning the construction of resilient Boolean functions whose nonlinearity is strictly greater than 2 n−1 − 2 n/2 is given.  ...  ACKNOWLEDGMENT The first author would like to dedicate this paper to Professor Guo-Zhen Xiao ( I ) on the occasion of his 80th birthday.  ... 
doi:10.1109/tit.2014.2345772 fatcat:6jkyppcu3ndglmb7c5zeo2g234

Highly Nonlinear Resilient Functions Optimizing Siegenthaler's Inequality [chapter]

Subhamoy Maitra, Palash Sarkar
1999 Lecture Notes in Computer Science  
Our technique can be used to construct functions on large number of input variables with simple hardware implementation.  ...  We provide a new construction method using a small set of recursive operations for a large class of highly nonlinear, resilient Boolean functions optimizing Siegenthaler's inequality m + d = n − 1.  ...  The recursive method proposed here can be used effectively to construct functions with large number of variables.  ... 
doi:10.1007/3-540-48405-1_13 fatcat:mo65n4xuzvcmfp4p7et4gskvwy

Normal Boolean functions

Pascale Charpin
2004 Journal of Complexity  
We later focus on some highly linear functions, bent functions and almost optimal functions. We point out that normality is a property for which these two classes are strongly connected.  ...  introduced the normality of bent functions. His work strengthened the interest for the study of the restrictions of Boolean functions on kdimensional flats providing the concept of k-normality.  ...  Since this paper was mainly devoted to the construction of new bent functions, normality was defined for Boolean functions with an even number m of variables: such a function is normal if it is constant  ... 
doi:10.1016/j.jco.2003.08.010 fatcat:hnvroexyujgvhcofgiu46oub5m

Symmetric Boolean Functions

A. Canteaut, M. Videau
2005 IEEE Transactions on Information Theory  
We finally detail the characteristics of the symmetric functions of degree at most 7, for any number of variables.  ...  For instance, it leads to a new general bound on the order of resiliency of symmetric functions, which improves Siegenthaler's bound.  ...  A straightforward method for constructing -resilient symmetric functions of an even number of variables might be to start from trivial balanced restrictions.  ... 
doi:10.1109/tit.2005.851743 fatcat:gm7vvfxzrrbbrj4mteagriz7jy

Highly nonlinear balanced Boolean functions with a good correlation-immunity [chapter]

Eric Filiol, Caroline Fontaine
1998 Lecture Notes in Computer Science  
We study a corpus of particular Boolean functions: the idempotents.  ...  They enable us to construct functions which achieve the best possible tradeoffs between the cryptographic fundamental properties: balancedness, correlation-immunity, a high degree and a high nonlinearity  ...  CANTEAUT for her motivating discussions on Boolean functions and N. SENDRIER for valuable improvements concerning the implementation.  ... 
doi:10.1007/bfb0054147 fatcat:qedtoovwkzd3nhkyiqore43u44
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