Abstract
In this correspondence, it is shown that the Boolean functions constructed by Pasalic (Cryptogr Commun 4(1):25–45, 2012) do not always have the high degree product of order n − 1 as expected.
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Notes
Here is omitted from [1] that the functions \(f_1^0, f_2^0, f_3^0\) achieve maximum algebraic immunity since it does not influence the \(\mathcal{HDP}\) properties of the constructed functions.
This can be examined in Magma, see also Appendix for the Magma source codes.
This question is suggested by one of the anonymous reviewers.
References
Pasalic, E.: A design of Boolean functions resistant to (fast) algebraic cryptanalysis with efficient implementation. Cryptogr. Commun. 4(1), 25–45 (2012)
Acknowledgements
The authors thank the anonymous referees for their valuable comments and suggestions on improving the manuscript. The authors also thank Tao Shi and Tianze Wang for helpful discussions, and seminar participants at SKLOIS: Shaoyu Du, Lin Jiao, Yao Lu, Wenlun Pan, et. al.
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Supported by the National 973 Program of China under Grant 2011CB302400, the National Natural Science Foundation of China under Grants 10971246, 60970152, and 61173134, the Grand Project of Institute of Software of CAS under Grant YOCX285056.
Appendix: Magma codes
Appendix: Magma codes
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P<[x]>:=PolynomialRing(GF(2),7);
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Q<x1,x2,x3,x4,f1,f2,f3>:=quo<P|[x[i]^2-x[i]:i in [1..7]]> ;
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x:=[x1,x2,x3,x4];
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f:=[f1,f2,f3];
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for i:=1 to 2 do
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n:=2*i;
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tp1:=(f[2]+f[3]+1)*x[n-1]*x[n]
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+(f[1]+f[2])*x[n-1]+x[n]+f[1];
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tp2:=(f[1]+f[3])*x[n-1]*x[n]+(f[2]+f[3]+1)*x[n-1]
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+(f[1]+f[2])*x[n]+f[2];
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tp3:=(f[2]+f[1]+1)*x[n-1]*x[n]+(f[1]+f[3]+1)
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*x[n-1]+(f[2]+f[3]+1)*x[n]+f[3]+1;
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f:=[tp1,tp2,tp3];
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end for;
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(x[1]+x[3])*(x[2]+x[4])*(f2+f[2]);
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x[1]*(x[2]+x[3]+x[4]+1)*(f3+f[2]);
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(x[1]+1)*(x[3]+1)*(f1+f[1]);
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Wang, W., Liu, M. & Zhang, Y. Comments on “A design of Boolean functions resistant to (fast) algebraic cryptanalysis with efficient implementation”. Cryptogr. Commun. 5, 1–6 (2013). https://doi.org/10.1007/s12095-012-0063-9
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DOI: https://doi.org/10.1007/s12095-012-0063-9