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Some properties of the output sequences of combined generator over finite fields

2. Conclusion and Future Work In this work, we have presented a new algorithmic-algebraic scheme based in the Lai — Massey structure for constructing permutations of dimension n = 2k, k > 2. For the common case k = 8, we have obtained new cryptographically strong 8-bit permutations having better resistance to algebraic attacks in comparison with the inversion function in GF(2) which so far has the best-known values for nonlinearity and differential uniformity. Compared to the best nonlinearity (108, for k = 4) offered by the construction presented in [6] and later generalized in [7], the nonlinearity for the permutations obtained by our scheme slightly decrease up to 104, but to the best of our knowledge the schemes presented in [6, 7] can not produce involutions and orthomorphisms with strong cryptographic properties, so we can conclude that the new structure presented in this work is more powerful and attractive due to the diversity of permutations that can be constructed. Interestingly, the involutions and orthomorphisms founded in this work have comparable classical cryptographic properties like those constructed by using spectral-linear and spectraldifference methods [8]. The main advantage of our 8-bit permutations is that they can be constructed using smaller 4-bit components which could be useful for the implementation of the S-Box in hardware or using a bit-sliced approach. We only presented a new scheme that can help to find permutations, involutions and orthomophisms with rather good cryptographic properties. There are several questions (theoretical results, hardware and bit-sliced implementations, efficient methods of masking) about the construction suggested in this work which are left as future work.