Optimal Reduced Representatives of Quadratic APN Permutations in Odd Dimension
Every quadratic APN permutation in odd dimension at least 5 has an optimal reduced representative.
The paper proves that every quadratic APN permutation of F_{2^n}, for odd n at least 5, is linearly equivalent to a reduced representative whose derivative has no zeros and whose nonempty fibres each have two elements. The range is exact: the property fails for n=3, and no such permutations exist for even n. For n at least 9 the existence argument is formalized in Lean 4, yielding counterexamples to the reduced critical-point conjecture that lift to permutation polynomials of optimal differential uniformity over every corresponding Galois ring.
- Every quadratic APN permutation in odd dimension n at least 5 has a suitable reduced form.
- The reduced derivative has no zeros and two-element nonempty fibres.
- Result fails for n=3 and even dimensions have no such permutations.
- Existence proof for odd n at least 9 is checked in Lean 4.
From the original article
1 min read at arxiv.orgThe formal derivative of the reduced polynomial representative of a map on a binary finite field depends on the coordinates, although its differential uniformity does not.
By Hiroki Fukui · arxiv.org
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