Explicit reduced APN permutations: ramification and optimal Galois-ring lifts
Paper constructs explicit reduced APN permutations and describes their optimal Galois-ring lifts.
The paper gives explicit reduced APN permutation polynomials P_{m,r} over F_{2^m} for odd m at least 5 and admissible odd r. Each polynomial has coefficients in F2, exactly 3r-2 terms, and a formal derivative with no zeros on the field. It determines wild ramification of the projective-line map and exact differential rows of coefficient lifts to Galois rings GR(2^k,m), and proves the Li-Wang criterion for a Gold function plus a linearised polynomial.
- Explicit reduced APN permutation polynomials over binary fields of odd dimension.
- Polynomials have 3r-2 terms and coefficients in F2.
- Ramification and differential rows of Galois-ring lifts are fully determined.
- Includes a self-contained proof of the Li-Wang criterion.
From the original article
1 min read at arxiv.orgLet $m\ge5$ be odd and $q=2^m$. For every odd $r$ with $3\le r<m$ and $\gcd(r,m)=1$ we give the reduced polynomial $P_{m,r}=\mathrm{red}((T_r^3)^{2^{m-2}})$, $T_r=\sum_{i<r}X^{2^i}$, explicitly: it has coefficients in $\mathbb{F}_2$ and exactly $3r-2$ terms, it is an APN permutation of $\mathbb{F}_q$, and its formal derivative $1+X^{2^{r-2}}+X^{2^{m-2}}$ has no zero on $\mathbb{F}_q$ and is two-to-one.
By Hiroki Fukui · arxiv.org
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