# American Institute of Mathematical Sciences

doi: 10.3934/amc.2020115
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## A New Construction of odd-variable Rotation symmetric Boolean functions with good cryptographic properties

 School of Mathematics and Statistics, Henan University, Kaifeng, 475004, China

*Corresponding author: Sihong Su (E-mail: sush@henu.edu.cn)

Received  April 2020 Revised  August 2020 Early access October 2020

Fund Project: The second author is supported by the National Natural Science Foundation of China (Grant No. 61502147)

Rotation symmetric Boolean functions constitute a class of cryptographically significant Boolean functions. In this paper, based on the theory of ordered integer partitions, we present a new class of odd-variable rotation symmetric Boolean functions with optimal algebraic immunity by modifying the support of the majority function. Compared with the existing rotation symmetric Boolean functions on odd variables, the newly constructed functions have the highest nonlinearity.

Citation: Bingxin Wang, Sihong Su. A New Construction of odd-variable Rotation symmetric Boolean functions with good cryptographic properties. Advances in Mathematics of Communications, doi: 10.3934/amc.2020115
##### References:

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##### References:
The nonlinearities of the rotation symmetric Boolean functions
 function nonlinearity [14] ${2^{n-1}-{n-1\choose k}}+2$ [17] $2^{n-1}-{n-1\choose k}+2^k-2$ [11] $2^{n-1}-{n-1\choose k}+2^k+2^{k-2}-k$ [21] $2^{n-1}-{n-1\choose k}+2^k+2^{k-1}-2k$ [20] $2^{n-1}-{n-1\choose k}+(k-5)2^{k-1}+2k+2$ [6] $2^{n-1}-{n-1\choose k}+\sum_{h=3}^k(n-2h)|T_h|+L_k$
 function nonlinearity [14] ${2^{n-1}-{n-1\choose k}}+2$ [17] $2^{n-1}-{n-1\choose k}+2^k-2$ [11] $2^{n-1}-{n-1\choose k}+2^k+2^{k-2}-k$ [21] $2^{n-1}-{n-1\choose k}+2^k+2^{k-1}-2k$ [20] $2^{n-1}-{n-1\choose k}+(k-5)2^{k-1}+2k+2$ [6] $2^{n-1}-{n-1\choose k}+\sum_{h=3}^k(n-2h)|T_h|+L_k$
The entries of the vectors in $T$ for $n = 13$
 $\alpha_1$ $\alpha_2$ $\alpha_3$ $\alpha_4$ $\alpha_5$ $\alpha_6$ $\alpha_7$ $\alpha_8$ $\alpha_9$ $\alpha_{10}$ $\alpha_{11}$ $\alpha_{12}$ $\alpha_{13}$ $0$ $1$ $1$ $1$ $1$ $1$ $1$ $1$ $1$ $1$ $1$ $1$ $1$ $1$ $1$ $1$ $1$ $1$ $1$ $1$ $1$ $1$ $1$ $1$ $1$ $1$ $1$ $1$ $2$ $1$ $1$ $1$ $1$ $1$ $1$ $1$ $1$ $1$ $1$ $1$ $1$ $1$ $3$ $1$ $1$ $1$ $1$ $0$ $0$ $0$ $0$ $0$ $0$ $0$ $0$ $1$ $4$ $1$ $0$ $0$ $0$ $0$ $0$ $1$ $0$ $1$ $0$ $1$ $1$ $0$ $5$ $0$ $0$ $0$ $1$ $1$ $1$ $1$ $1$ $0$ $1$ $0$ $0$ $0$ $6$ $0$ $1$ $1$ $0$ $1$ $1$ $0$ $0$ $0$ $0$ $0$ $1$ $1$ $7$ $1$ $1$ $0$ $0$ $1$ $0$ $0$ $1$ $1$ $1$ $1$ $0$ $0$ $8$ $0$ $0$ $1$ $1$ $0$ $1$ $1$ $1$ $1$ $0$ $0$ $0$ $0$ $9$ $0$ $0$ $0$ $0$ $0$ $0$ $0$ $0$ $0$ $1$ $1$ $1$ $1$ $10$ $0$ $0$ $0$ $0$ $0$ $0$ $0$ $0$ $0$ $0$ $0$ $0$ $0$ $11$ $0$ $0$ $0$ $0$ $0$ $0$ $0$ $0$ $0$ $0$ $0$ $0$ $0$ $12$ $0$ $0$ $0$ $0$ $0$ $0$ $0$ $0$ $0$ $0$ $0$ $0$ $0$
 $\alpha_1$ $\alpha_2$ $\alpha_3$ $\alpha_4$ $\alpha_5$ $\alpha_6$ $\alpha_7$ $\alpha_8$ $\alpha_9$ $\alpha_{10}$ $\alpha_{11}$ $\alpha_{12}$ $\alpha_{13}$ $0$ $1$ $1$ $1$ $1$ $1$ $1$ $1$ $1$ $1$ $1$ $1$ $1$ $1$ $1$ $1$ $1$ $1$ $1$ $1$ $1$ $1$ $1$ $1$ $1$ $1$ $1$ $1$ $2$ $1$ $1$ $1$ $1$ $1$ $1$ $1$ $1$ $1$ $1$ $1$ $1$ $1$ $3$ $1$ $1$ $1$ $1$ $0$ $0$ $0$ $0$ $0$ $0$ $0$ $0$ $1$ $4$ $1$ $0$ $0$ $0$ $0$ $0$ $1$ $0$ $1$ $0$ $1$ $1$ $0$ $5$ $0$ $0$ $0$ $1$ $1$ $1$ $1$ $1$ $0$ $1$ $0$ $0$ $0$ $6$ $0$ $1$ $1$ $0$ $1$ $1$ $0$ $0$ $0$ $0$ $0$ $1$ $1$ $7$ $1$ $1$ $0$ $0$ $1$ $0$ $0$ $1$ $1$ $1$ $1$ $0$ $0$ $8$ $0$ $0$ $1$ $1$ $0$ $1$ $1$ $1$ $1$ $0$ $0$ $0$ $0$ $9$ $0$ $0$ $0$ $0$ $0$ $0$ $0$ $0$ $0$ $1$ $1$ $1$ $1$ $10$ $0$ $0$ $0$ $0$ $0$ $0$ $0$ $0$ $0$ $0$ $0$ $0$ $0$ $11$ $0$ $0$ $0$ $0$ $0$ $0$ $0$ $0$ $0$ $0$ $0$ $0$ $0$ $12$ $0$ $0$ $0$ $0$ $0$ $0$ $0$ $0$ $0$ $0$ $0$ $0$ $0$
Comparison of the nonlinearities
 $n$ $9$ 11 13 15 17 19 21 $F(x)$ 186 772 3172 12952 52666 213524 863820 [3] 232 980 3988 16212 65210 261428 1046552 [17] $-$ 802 3234 13078 52920 214034 864842 [21] $-$ 810 3256 13130 53034 214274 865336 [20] $-$ 784 3218 13096 53068 214568 866402 [6] $-$ 794 3230 13098 53044 214486 866294 $f$ in (13) 188 782 3208 13064 52988 214406 866160
 $n$ $9$ 11 13 15 17 19 21 $F(x)$ 186 772 3172 12952 52666 213524 863820 [3] 232 980 3988 16212 65210 261428 1046552 [17] $-$ 802 3234 13078 52920 214034 864842 [21] $-$ 810 3256 13130 53034 214274 865336 [20] $-$ 784 3218 13096 53068 214568 866402 [6] $-$ 794 3230 13098 53044 214486 866294 $f$ in (13) 188 782 3208 13064 52988 214406 866160
Comparison of the nonlinearities
 $n$ 27 37 47 57 $F(x)$ 56708264 59644341436 62135313450064 64408903437167496 [17] 56716454 59644603578 62135321838670 64408903705602950 [21] 56720526 59644734616 62135326032930 64408903839820624 [20] 56741060 59646045410 62135388947584 64408906524175298 [6] 56748298 59648002864 62135605652036 64408924613659456 $f$ in (13) 56747394 59647951550 62135614817362 64408926590774154
 $n$ 27 37 47 57 $F(x)$ 56708264 59644341436 62135313450064 64408903437167496 [17] 56716454 59644603578 62135321838670 64408903705602950 [21] 56720526 59644734616 62135326032930 64408903839820624 [20] 56741060 59646045410 62135388947584 64408906524175298 [6] 56748298 59648002864 62135605652036 64408924613659456 $f$ in (13) 56747394 59647951550 62135614817362 64408926590774154
Comparison of the nonlinearity biases
 $n$ 9 11 13 15 17 19 21 27 37 47 57 $F(x)$ 0.273 0.246 0.226 0.209 0.1964 0.1855 0.1762 0.15498 0.132061 0.11700409 0.10614691 [3] 0.094 0.043 0.026 0.01 0.005 0.0027 0.0019 $-$ $-$ $-$ $-$ [17] $-$ 0.217 0.21 0.202 0.1925 0.1835 0.1752 0.15486 0.132057 0.11700397 0.10614690 [21] $-$ 0.209 0.205 0.199 0.1908 0.1826 0.1748 0.15480 0.132055 0.11700391 0.10614690 [20] $-$ 0.234 0.214 0.201 0.1902 0.1815 0.1737 0.15449 0.132036 0.11700302 0.10614686 [6] $-$ 0.224 0.211 0.201 0.1906 0.1818 0.1738 0.15438 0.132007 0.11699994 0.10614661 $f$ in (13) 0.266 0.236 0.217 0.203 0.1915 0.1821 0.174 0.15440 0.132008 0.11699981 0.10614658
 $n$ 9 11 13 15 17 19 21 27 37 47 57 $F(x)$ 0.273 0.246 0.226 0.209 0.1964 0.1855 0.1762 0.15498 0.132061 0.11700409 0.10614691 [3] 0.094 0.043 0.026 0.01 0.005 0.0027 0.0019 $-$ $-$ $-$ $-$ [17] $-$ 0.217 0.21 0.202 0.1925 0.1835 0.1752 0.15486 0.132057 0.11700397 0.10614690 [21] $-$ 0.209 0.205 0.199 0.1908 0.1826 0.1748 0.15480 0.132055 0.11700391 0.10614690 [20] $-$ 0.234 0.214 0.201 0.1902 0.1815 0.1737 0.15449 0.132036 0.11700302 0.10614686 [6] $-$ 0.224 0.211 0.201 0.1906 0.1818 0.1738 0.15438 0.132007 0.11699994 0.10614661 $f$ in (13) 0.266 0.236 0.217 0.203 0.1915 0.1821 0.174 0.15440 0.132008 0.11699981 0.10614658
Comparison of the fast algebraic immunities
 $n$ $9$ 11 13 15 [3] 8 10 12 14 [21] $-$ 10 12 14 [20] $-$ 10 12 13 [6] $-$ 10 12 14 $f$ in (13) 6 8 10 10
 $n$ $9$ 11 13 15 [3] 8 10 12 14 [21] $-$ 10 12 14 [20] $-$ 10 12 13 [6] $-$ 10 12 14 $f$ in (13) 6 8 10 10
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