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Perfect Algebraic Immune Functions [chapter]

Meicheng Liu, Yin Zhang, Dongdai Lin
2012 Lecture Notes in Computer Science  
A perfect algebraic immune function is a Boolean function with perfect immunity against algebraic and fast algebraic attacks.  ...  The main results are that for a perfect algebraic immune balanced function the number of input variables is one more than a power of two; for a perfect algebraic immune unbalanced function the number of  ...  The first author would like to thank Dingyi Pei for his enlightening conversations on the resistance of Boolean functions against algebraic attacks.  ... 
doi:10.1007/978-3-642-34961-4_12 fatcat:f25zqpqnczbfvbyh6azklosdt4

Almost perfect algebraic immune functions with good nonlinearity

Meicheng Liu, Dongdai Lin
2014 2014 IEEE International Symposium on Information Theory  
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.  ...  [IEEE TIT 59(1): 653-664, 2013], are almost perfect algebraic immune for any integer k ≥ 3.  ...  Acknowledgment The authors would like to thank Yin Zhang for his helpful discussions on the immunity to fast algebraic attacks, thank Shaoyu Du for her helpful discussions on the nonlinearity, and thank  ... 
doi:10.1109/isit.2014.6875151 dblp:conf/isit/LiuL14a fatcat:rv2apospand4dda7hqe3cwikti

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  
A Boolean function is perfect algebraic immune if it is with perfect immunity against algebraic and fast algebraic attacks.  ...  Implementation results also show that they are almost perfect algebraic immune functions.  ...  [20] initiated perfect algebraic immune (PAI) functions, Boolean functions with perfect immunity against algebraic and fast algebraic attacks.  ... 
doi:10.1155/2017/6268230 fatcat:xrqvbci5uffkfexma5ohzkw34q

Fast algebraic immunity of Boolean functions and LCD codes [article]

Sihem Mesnager, Chunming Tang
2020 arXiv   pre-print
Further, we evaluate our parameter for two secondary constructions of Boolean functions. Moreover, A coding-theory approach to the characterization of perfect algebraic immune functions is presented.  ...  Via this characterization, infinite families of binary linear complementary dual codes (or LCD codes for short) are obtained from perfect algebraic immune functions.  ...  Let f be a perfect algebraic immune function. The perfect algebraic immune function f has degree at least n − 1 (see [36] ). By Theorem 2 and F AI(f ) = n, RM(e, n) D is LCD for any 1 ≤ e ≤ n.  ... 
arXiv:2006.08372v1 fatcat:5vgxykilvjbelc63eidathbwmq

Cardinality spectra of components of correlation immune functions, bent functions, perfect colorings, and codes

V. N. Potapov
2012 Problems of Information Transmission  
We study cardinalities of components of perfect codes and colorings, correlation immune functions, and bent function (sets of ones of these functions).  ...  For perfect colorings with certain parameters and for correlation immune functions, we find components of some of the above-given cardinalities.  ...  ALGEBRAIC DEGREE OF PERFECT COLORINGS AND CORRELATION IMMUNE FUNCTIONS Each Boolean function f : E n → E can be represented as a Zhegalkin polynomial (in algebraic normal form) f (x 1 , . . . , x n ) =  ... 
doi:10.1134/s003294601201005x fatcat:smqtsua4drhnnbcppcna2tj3xq

Strengthening Crypto-1 Cipher Against Algebraic Attacks

Farah Afianti, Ari M. Barmawi
2015 Journal of ICT Research and Applications  
In addition, another modified Crypto-1, using the modified feedback function and a modified filter function, had the lowest percentage of revealed variables.  ...  Experimental testing showed that the amount of memory and CPU time needed were highest when attacking the modified Crypto-1 using the modified feedback function and the original filter function.  ...  with perfect algebraic immunity is a power of two (2 s ) [6] .  ... 
doi:10.5614/itbj.ict.res.appl.2015.9.1.5 fatcat:cxu5k3oaufdvtbo2yh7en5jube

Constructions of vectorial Boolean functions with good cryptographic properties

Luyang Li, Weiguo Zhang
2016 Science China Information Sciences  
Unfortunately, perfect nonlinear functions cannot be used directly because they are not balanced or correlation immune.  ...  In addition, to resist Berlekamp-Massay attack, the functions we used should have high algebraic degree.  ...  Unfortunately, perfect nonlinear functions cannot be used directly because they are not balanced or correlation immune.  ... 
doi:10.1007/s11432-015-0863-3 fatcat:uylw4di2vfeejmrowpso2kryoq

A longtime season of friendship and scientific collaboration

R. Grigolia
2022 Journal of Algebraic Hyperstructures and Logical Algebras  
In many ways the immune system is a black box; although many of its inputs and outputs are known, exactly how the system achieves its function is the subject of many investigations.  ...  Perfect M V -algebras form a full subcategory of the category of all M V -algebras. We denote the category of perfect M V -algebras by Perfect.  ...  It is introduced an immune dynamic n-valued Lukasiewicz logic ID L n on the base of n-valued Lukasiewicz logic L n and corresponding to it immune dynamic M V n -algebra (IDL n -algebra), 1 < n < ω, which  ... 
doi:10.52547/hatef.jahla.3.1.5 fatcat:gtraqnyjtvcenj3mwg7fvgp55q

A remark on algebraic immunity of Boolean functions [article]

Baofeng Wu, Jia Zheng
2013 arXiv   pre-print
In this correspondence, an equivalent definition of algebraic immunity of Boolean functions is posed, which can clear up the confusion caused by the proof of optimal algebraic immunity of the Carlet-Feng  ...  function and some other functions constructed by virtue of Carlet and Feng's idea.  ...  For an n-variable Boolean function, the algebraic immunity of it is upper bounded by ⌈ n 2 ⌉ [3] , and when this upper bound is attained, it is often known as an algebraic immunity optimal function, or  ... 
arXiv:1305.5919v1 fatcat:ilnkzeternegdj24k5d5nknlhi

High Security Frequency/Time Hopping Sequence Generators

Yun-Pyo Hong, Seok-Yong Jin, Hong-Yeop Song
2007 2007 3rd International Workshop on Signal Design and Its Applications in Communications  
functions are correlation-immune by the Fourier transform • X.  ...  N • g: p-ary bent function, i.e. perfect nonlinear function, with N arguments Theorem 4 Let a p-ary function f with N + 2 arguments be given by f (X 1 , X 2 , . . . , X N +2 ) = a 1 X 1 + a 2 X 2 + a  ...  Other attacks • Attacker may try an algebraic attack by multiplying the combinatorial function f by a well-chosen multivariate polynomial ⇒ By increasing the order of F p , the monomials of linear equations  ... 
doi:10.1109/iwsda.2007.4408415 fatcat:ram5dal7onfjjaqxcazsjjgkre

Page 23 of Mathematical Reviews Vol. , Issue 83a [page]

1983 Mathematical Reviews  
A sample result (Theorem 2.3) is that every recursive Boolean algebra with infinitely many atoms is isomorphic to a recursive Boolean algebra in which the set of atoms is immune.  ...  Section four uses Sacks forcing with perfect trees to produce a minimal E,-degree.  ... 

Construction of correlation immune Boolean functions

Chuan-Kun Wu, Ed Dawson
2000 The Australasian Journal of Combinatorics  
Constructions of higher order correlation immune functions as well as algebraic non-degenerate correlation immune functions are discussed in particular.  ...  It is shown in this paper that every correlation immune Boolean function of n variables can be written as f(x) = g(xG T ), where 9 is an algebraic non-degenerate Boolean function of k (k :::; n) variables  ...  In this paper we investigate the inherent structure of correlation immune functions in terms of algebraic degeneration and subsequently the constructions of functions with concrete correlation immunity  ... 
dblp:journals/ajc/WuD00 fatcat:jow7v6lmprfhfknychvx65irse

An Algebraic Solution for the Kermack-McKendrick Model [article]

Alexsandro M. Carvalho, Sebastian Gonçalves
2016 arXiv   pre-print
We present an algebraic solution for the Susceptible-Infective-Removed (SIR) model originally presented by Kermack-McKendrick in 1927.  ...  Then, using algebraic techniques and some well justified numerical assumptions we obtain an analytic solution for the integral.  ...  The basic reproductive number R 0 = βτ , the infection curve (that is the asymptotic number of removed as function of R 0 ) are concepts that found in the SIR model the basement that provide the perfect  ... 
arXiv:1609.09313v2 fatcat:7ir54bbs6bayrk6atsf7whh264

On Exact Algebraic [Non-]Immunity of S-Boxes Based on Power Functions [chapter]

Nicolas T. Courtois, Blandine Debraize, Eric Garrido
2006 Lecture Notes in Computer Science  
In this paper we are interested in algebraic immunity of several well known highly-nonlinear vectorial Boolean functions (or Sboxes), designed for block and stream ciphers.  ...  Instead of complex and "explicit" Boolean functions we have then simple and "implicit" algebraic relations that can be combined to fully describe the secret key of the system.  ...  . 10 Which S-box has the Lowest Algebraic Immunity ?  ... 
doi:10.1007/11780656_7 fatcat:aiqmjpgmcrawdkjjie4n745scq

Page 625 of Mathematical Reviews Vol. , Issue 96b [page]

1996 Mathematical Reviews  
@ with a Noetherian congruence lattice such that the word problem on @ is undecidable. (3) Fi- nally, they construct a finitely generated r.e. perfect algebra with immune as well as nonimmune transversals  ...  A (total) recursive function f admits 9 if 7 is a congruence on the algebra (w, f). Let F be a finite set of recursive functions such that f admits 7 for each f € F. Define an algebra ¥% = (w/y,F).  ... 
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