Reduced APN permutations and a disproof of the Galois-ring lifting conjecture
Abstract
We construct infinite families of reduced almost perfect nonlinear (APN) permutations of 𝔽2m that admit coefficientwise permutation lifts to GR(2k,m) for every k>1. These families disprove Conjecture~2 of Rønjom and Sandrib (Cryptogr. Commun., 2026), even for reduced polynomial representatives. For every odd m≥5 with 3nmid m, we obtain explicit seven-term polynomials over 𝔽2 with formal derivative identically one. We derive them from a complete criterion for the reduced representative of A(x)2s+1 to have no critical point in 𝔽2m, where A is an invertible linearized polynomial and gcd(s,m)=1. We then apply the cube case to construct a second family of degree 2m−1+2, whose nonconstant derivatives avoid zero by an explicit trace identity. A fixed parameter in 𝔽32 gives this family in every dimension m=5r with r odd, including a degree-$18$ example over 𝔽32. Both constructions remain in Gold linear equivalence classes, so liftability of reduced representatives is not invariant under invertible 𝔽2-linear input changes. We obtain the lifting conclusions from the standard Galois-ring permutation criterion. We also verify a degree-$24$ example supplied by Zhang and Haobo, prove a trace-adjoint criterion for reduced quadratic functions, and interpret the derivative condition through ramification at rational points.
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BibTeX
@misc{bartoli2026reduced,
title = {{Reduced APN permutations and a disproof of the Galois-ring lifting conjecture}},
author = {Daniele Bartoli and Pantelimon Stanica},
howpublished = {arXiv},
year = {2026},
month = oct,
doi = {10.48550/arxiv.2608.30808},
eprint = {2608.30808},
archivePrefix = {arXiv},
url = {https://arxiv.org/abs/2608.30808},
}
RIS
TY - UNPB
TI - Reduced APN permutations and a disproof of the Galois-ring lifting conjecture
AU - Daniele Bartoli
AU - Pantelimon Stanica
PY - 2026
DA - 2026/10/17
PB - arXiv
DO - 10.48550/arxiv.2608.30808
UR - https://arxiv.org/abs/2608.30808
AB - We construct infinite families of reduced almost perfect nonlinear (APN) permutations of 𝔽_2^m that admit coefficientwise permutation lifts to GR(2^k,m) for every k>1. These families disprove Conjecture~2 of Rønjom and Sandrib (Cryptogr. Commun., 2026), even for reduced polynomial representatives. For every odd m≥5 with 3nmid m, we obtain explicit seven-term polynomials over 𝔽_2 with formal derivative identically one. We derive them from a complete criterion for the reduced representative of A(x)^2^s+1 to have no critical point in 𝔽_2^m, where A is an invertible linearized polynomial and gcd(s,m)=1. We then apply the cube case to construct a second family of degree 2^m−1+2, whose nonconstant derivatives avoid zero by an explicit trace identity. A fixed parameter in 𝔽_32 gives this family in every dimension m=5r with r odd, including a degree-$18$ example over 𝔽_32. Both constructions remain in Gold linear equivalence classes, so liftability of reduced representatives is not invariant under invertible 𝔽_2-linear input changes. We obtain the lifting conclusions from the standard Galois-ring permutation criterion. We also verify a degree-$24$ example supplied by Zhang and Haobo, prove a trace-adjoint criterion for reduced quadratic functions, and interpret the derivative condition through ramification at rational points.
ER -