Unicyclic Strong Permutations
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For positive integers n and k such that 0≤ k≤ n-1, we study some properties of a certain kind of permutations σ_k over F_2^n. The properties that hold simultaneously are related to: (1) the algebraic degree of the boolean output bits: all of the σ_k's considered have algebraic degree n-1; (2) the cycle structure: the permutations σ_k are unicyclic; (3) the average number of terms to describe the algebraic normal form of the boolean output bits: the average number of terms for each σ_k is 2^n-1. We also study the composition σ_n-1...σ_0=σ and notice a dichotomy about the cycle structure of σ between odd and even values of n; while there are unicyclic permutations for odd n, we could not find any unicyclic permutation for even values of n≤ 30. For the composition σ, we also study empirically the differential uniformity for all values of n≤ 16 and notice that it never exceeds 6. For some specific cases of n=17 and n=19, we report counts of the number of equal entries of their difference table and linear approximation table. The number of bits to describe either the σ_k's or the composition σ is only O(n), and outputs can be generated on the fly. The analysis of the cycle structure bears some analogies with the analysis of continued fraction over finite fields.
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