Some Representations of the S-Box of Camellia in GF (((2 2 ) 2 ) 2 )

Substitution Box (S-box) is usually the most complex module in some block ciphers. Some prominent ciphers such as AES and Camellia use S-boxes, which are affine equivalents of a multiplicative inverse in small finite fields. This manuscript describes mathematical representations of the Camellia S-box by using composite fields such as polynomial, normal or mixed. An optimized hardware implementation typically aims to reduce the number of gates to be used. Our theoretical design with composite normal bases allows saving gates in the critical path by using 19 XOR gates, 4 AND gates and 2 NOT gates. With composite mixed bases, the critical path has 2 XOR gates more than the representation with composite normal bases. Redundancies found in the affine transformation matrix that form the composite fields were eliminated. For mixed bases, new Algebraic Normal Form identities were obtained to compute the inner composite multiplicative inverse, reducing the critical path of the complete implementation of the Camellia S-box. These constructions were translated into transistor-gate architectures for hardware representations by using Electric VLSI [29] under MOSIS C5 process [17], [18], thus obtaining the corresponding schematic models.