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# Highly nonlinear (vectorial) Boolean functions that are symmetric under some permutations

• *Corresponding author: Selçuk Kavut

This work is supported financially by Balıkesir University under grant BAP 2015/23

• We first give a brief survey of the results on highly nonlinear single-output Boolean functions and bijective S-boxes that are symmetric under some permutations. After that, we perform a heuristic search for the symmetric (and involution) S-boxes which are bijective in dimension 8 and identify corresponding permutations yielding rich classes in terms of cryptographically desirable properties.

Mathematics Subject Classification: 11T71, 94A60.

 Citation: • • Table 1.  A summary of the highest nonlinearities for odd $n\ge 9$

 Number of variables ($n$) 9 11 13 15 Bounds Bent concatenation bound 240 992 4032 16256 ($2^{n-1}-2^\frac{n-1}{2}$) Upper bound 244 1000 4050 16292 ($2\left\lfloor 2^{n-2}-2^{\frac{n}{2}-2}\right\rfloor$) Unbalanced nonlinearities  $-$ $-$ $-$ 16276  242 996 4040 $-$ Balanced nonlinearities  $-$ $-$ 4036 $-$  $-$ $-$ $-$ 16272

Table 2.  Best achieved cryptographic properties [nonlinearity, differential uniformity, algebraic degree]

 $\#$ Representativepermutation Space size Best result (for involution S-boxes) Best result 1 $(7,6,2,1,8,5,4,3)$ $2^{147.93}$ $[84,44,7]$ $[84,44,7]$ 2 $(2,3,1,7,4,5,6,8)$ $2^{191.48}$ $[84,52,7]$ $[84,52,7]$ 3 $(6,7,5,8,4,3,1,2)^a$ $2^{208.29}$ $\bf{[106,6,7]}$ $\bf{[106,6,7]}, \bf{[108,8,6]}$ 4 $(4,3,2,5,8,1,7,6)$ $2^{227.35}$ $[0, -, -]$ $[0, -, -]$ 5 $(4,5,3,2,8,1,6,7)$ $2^{243.74}$ $\bf {[106,6,7]}$ $\bf {[106,6,7]}$ 6 $(8,3,4,6,7,1,5,2)$ $2^{277.78}$ $[104,6,7]$ $[104,6,7], {\bf{[106,8,7]}}$ 7 $(8,6,3,5,2,1,7,4)$ $2^{283.02}$ $[104,10,7]$ $\it {[104,8,7]}$ 8 $(4,6,7,5,1,2,3,8)$ $2^{357.97}$ $[84,44,7]$ $[84,44,7]$ 9 $(2,6,3,4,5,8,1,7)$ $2^{358.65}$ $[100,10,7]$ $[100,10,7],\it{[104,20,7]}$ 10 $(7,3,6,1,8,2,4,5)$ $2^{359.22}$ $[0, -, -]$ $[0, -, -]$ 11 $(7,6,1,2,3,8,5,4)^b$ $2^{412.21}$ $[104,6,7]$ $[104,6,7], {\bf{[106,8,7]}}$ 12 $(2,7,4,3,5,6,1,8)$ $2^{431.91}$ $[0, -, -]$ $[0, -, -]$ 13 $(6,4,8,2,1,7,5,3)$ $2^{440.19}$ $[84,22,7]$ $[84,22,7]$ 14 $(1,3,6,7,2,5,4,8)$ $2^{446.24}$ $[84,22,7]$ $[84,22,7]$ 15 $(1,5,6,4,3,2,7,8)$ $2^{476.86}$ $[84,52,7]$ $[84,52,7]$ 16 $(4,3,8,5,1,6,7,2)$ $2^{565.87}$ $[104,6,7]$ $[104,6,7]$ 17 $(1,6,3,4,2,5,7,8)$ $2^{693.43}$ $[84,44,7]$ $[84,44,7]$ 18 $(7,6,5,8,3,2,1,4)^c$ $2^{824.73}$ $[104,6,7]$ $[104,6,7]$ 19 $(1,5,8,4,2,7,6,3)$ $2^{835.24}$ $[104,8,7]$ $[104,8,7]$ 20 $(1,2,7,4,5,8,3,6)$ $2^{890.27}$ $[84,22,7]$ $[84,22,7]$ 21 $(8,2,3,4,5,6,7,1)$ $2^{1076.16}$ $[0, -, -]$ $[0, -, -]$ 22 $(1,2,3,4,5,6,7,8)^d$ $2^{1684}$ $[102,6,7]$ $\it{[104,6,7]}$ $^a:$ Linear equivalet to RSSBs $^b:$ Linear equivalent to 2-RSSBs $^c:$ Linear equivalent to 4-RSSBs $^d:$ The search space of all bijective S-boxes
• Tables(2)

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