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On the Derivative Imbalance and Ambiguity of Functions
Fu, Shihui1; Feng, Xiutao1,2; Wang, Qiang3; Carlet, Claude4,5
AbstractIn 2007, Carlet and Ding introduced two parameters, denoted by Nb-F and NBF, quantifying respectively the balancedness of general functions F between finite Abelian groups and the (global) balancedness of their derivatives D-a F(x) = F(x + a) - F(x), a is an element of G \ {0} (providing an indicator of the nonlinearity of the functions). These authors studied the properties and cryptographic significance of these two measures. They provided inequalities relating the nonlinearity N L(F) to NBF for S-box and specifically obtained an upper bound on the nonlinearity that unifies Sidelnikov-Chabaud-Vaudenay's bound and the covering radius bound. At the Workshop WCC 2009 and in its postproceedings in 2011, a further study of these parameters was made; in particular, the first parameter was applied to the functions F + L, where L is affine, providing more nonlinearity parameters. In 2010, motivated by the study of Costas arrays, two parameters called ambiguity and deficiency were introduced by Panario et al. for permutations over finite Abelian groups to measure the injectivity and surjectivity of the derivatives, respectively. These authors also studied some fundamental properties and cryptographic significance of these two measures. Further studies followed without comparing the second pair of parameters to the first one. In this paper, we observe that ambiguity is the same parameter as NBF up to additive and multiplicative constants (i.e., up to rescaling). We perform the necessary work of comparison and unification of the results on NBF and on ambiguity, which have been obtained in the five papers devoted to these parameters. We generalize some known results to any finite Abelian groups. More importantly, we derive many new results on these parameters.
KeywordAmbiguity derivative imbalance differential uniformity Fourier transform nonlinearity
Funding ProjectNational Natural Science Foundation of China[61572491] ; National Natural Science Foundation of China[11688101] ; Science and Technology on Communication Security Laboratory[6142103010701] ; NSERC of Canada
WOS Research AreaComputer Science ; Engineering
WOS SubjectComputer Science, Information Systems ; Engineering, Electrical & Electronic
WOS IDWOS:000481981000037
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Document Type期刊论文
Corresponding AuthorFu, Shihui
Affiliation1.Chinese Acad Sci, Acad Math & Syst Sci, Key Lab Math Mechanizat, Beijing 100190, Peoples R China
2.Sci & Technol Commun Secur Lab, Chengdu 610041, Sichuan, Peoples R China
3.Carleton Univ, Sch Math & Stat, Ottawa, ON K1S 5B6, Canada
4.Univ Paris 08, LAGA, F-93526 St Denis, France
5.Univ Bergen, N-5020 Bergen, Norway
Recommended Citation
GB/T 7714
Fu, Shihui,Feng, Xiutao,Wang, Qiang,et al. On the Derivative Imbalance and Ambiguity of Functions[J]. IEEE TRANSACTIONS ON INFORMATION THEORY,2019,65(9):5833-5845.
APA Fu, Shihui,Feng, Xiutao,Wang, Qiang,&Carlet, Claude.(2019).On the Derivative Imbalance and Ambiguity of Functions.IEEE TRANSACTIONS ON INFORMATION THEORY,65(9),5833-5845.
MLA Fu, Shihui,et al."On the Derivative Imbalance and Ambiguity of Functions".IEEE TRANSACTIONS ON INFORMATION THEORY 65.9(2019):5833-5845.
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