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Differential uniformity and nonlinearity are two basic properties of S-boxes, which measure the resistance of S-boxes to differential and linear attack respectively. Besides these two properties, the hardware cost of S-boxes is also an important property which should be considered primarily in a limited resource environment. By use of Feistel structure, we investigate the problem of constructing S-boxes with excellent cryptographic properties and low hardware implementation cost in the present paper. Feistel structure is a widely used structure in the design of block ciphers, and it can be implemented easily in hardware. Three-round Feistel structure has been used to construct S-boxes in symmetric algorithms, such as CS-Ciper, CRYPTON and ZUC. In the present paper, we investigate the bounds on differential uniformity and nonlinearity of S-boxes constructed with three-round Feistel structure. By choosing suitable round functions, we show that for odd k, differential 4-uniform S-boxes over <InlineEquation ID=\"IEq1\" <EquationSource Format=\"TEX\"$\\mathbb{F}_{2^{k}}^2$</EquationSource> </InlineEquation> with the best known nonlinearity can be constructed via three-round Feistel structure. Some experiment results are also given which show that optimal 4-bit S-boxes can be constructed with 4 or 5 round unbalanced Feistel structure.

Constructing S-boxes for Lightweight Cryptography with Feistel Structure