Geodesic
On the construction of $s$-boxes of $4\times4$ size
Prikladnaya Diskretnaya Matematika. Supplement, no. 10 (2017), pp. 104-106

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The paper proposes and implements a method for constructing all $4\times4$ size $s$-boxes, satisfying a number of cryptographic properties: 1) bijectivity; 2) the absence of fixed points; 3) the nonlinearity of all non-trivial linear combinations of the coordinate functions; 4) the value of the difference characteristic $p_s$ of the substitution $s$ is not more than 4/16. A list of all $4\times4$ size $s$-boxes (a total of 383 084 314) with these properties is obtained. The created software is used to determine the specified characteristics of DES algorithm $s$-boxes (32 $s$-boxes of $4\times4$ size using fixations of bits controlled by one of four substitutions of degree 16 of $s$-box) and eight $s$-boxes recommended for use in the GOST 28147-89 algorithm. According to the DES algorithm, it is established that 6 $s$-boxes do not provide the property 2, 16 $s$-boxes have $p_s=8/16$, 14 $s$-boxes have $p_s=6/16$, and there is one $s$-box with $p_s=4/16$ and one $s$-box with $p_s=10/16$. According to the GOST 28147-89 algorithm, it is established that 3 $s$-boxes do not provide the property 2, all $s$-boxes have $p_s=4/16$.
Keywords: $s$-box, differential characteristic. 
@article{PDMA_2017_10_a41,
     author = {V. M. Fomichev and P. V. Ovchinnikov},
     title = {On the construction of $s$-boxes of $4\times4$ size},
     journal = {Prikladnaya Diskretnaya Matematika. Supplement},
     pages = {104--106},
     publisher = {mathdoc},
     number = {10},
     year = {2017},
     language = {ru},
     url = {http://geodesic.mathdoc.fr/item/PDMA_2017_10_a41/}
}
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V. M. Fomichev; P. V. Ovchinnikov. On the construction of $s$-boxes of $4\times4$ size. Prikladnaya Diskretnaya Matematika. Supplement, no. 10 (2017), pp. 104-106. http://geodesic.mathdoc.fr/item/PDMA_2017_10_a41/