Infinite families of APN permutations in constrained trivariate classes over $\mathbb{F}_{2^m}$

• URL: http://arxiv.org/abs/2603.15146v1
• Date: Mon, 16 Mar 2026 11:41:24 GMT
• Title: Infinite families of APN permutations in constrained trivariate classes over $\mathbb{F}_{2^m}$
• Abstract summary: We extend two APN permutation families of Li--Kaleyski.<n>In particular, when $q=2$ and $7nmid m$, every good $aneq 1$ gives APN permutations CCZ-inequivalent to Li--Kaleyski.
• Score: 2.771610203951056
• License: http://creativecommons.org/publicdomain/zero/1.0/
• Abstract: We study trivariate permutation polynomials over $\mathbb{F}_{2^{m}}$ extending two APN permutation families of Li--Kaleyski (IEEE Trans. Inform. Theory, 2024) by allowing the scalar parameter to vary over $\mathbb{F}_{2^m}^*$. For \[ G_a(x,y,z)=(x^{q+1}+ax^qz+yz^q,\; x^qz+y^{q+1},\; xy^q+ay^qz+z^{q+1}), \] where $a\in\mathbb{F}_{2^m}^*$, $q=2^i$, $\gcd(i,m)=1$, and $m$ is odd, we prove that $G_a$ is a permutation if and only if an associated univariate polynomial has no root in $\mathbb{F}_{2^m}^*$, and that this condition is also equivalent to $G_a$ being APN. Hence, writing $d=q^2+q+1$, at least \[ \frac{2^m+1-(d-1)(d-2)2^{m/2}-d}{d} \] values of $a$ yield APN permutations $G_a$. In the binary case $q=2$, we show that $a=1$ is good whenever $7\nmid m$, recovering the Li--Kaleyski family. For the second family \[ H_a(x,y,z)=(x^{q+1}+axy^q+yz^q,\; xy^q+z^{q+1},\; x^qz+y^{q+1}+ay^qz), \] we obtain the same root criterion and prove that its defining polynomial is root-equivalent to that of $G_a$. Thus the same parameters $a$ give APN permutations in both families. We also prove strong inequivalence results. First, $G_a$ (resp.\ $H_a$) is diagonally equivalent to $G_1$ (resp.\ $H_1$) if and only if $a^{q^2+q+1}=1$; moreover, for $m>4$, $m\neq 6$, and $7\nmid m$, diagonal non-equivalence implies CCZ non-equivalence by the monomial restriction theorem of Shi et al.\ (DCC, 2025). In particular, when $q=2$ and $7\nmid m$, every good $a\neq 1$ gives APN permutations CCZ-inequivalent to Li--Kaleyski. Second, for the same range of $m$, no $G_a$ is CCZ-equivalent to any $H_b$. Hence these constructions yield two genuinely new, mutually inequivalent families of APN permutations on $\mathbb{F}_{2^{3m}}$.

Related papers

• SIC dimension towers via cyclotomic polynomials [0.0]
  We prove a structure for the dimension towers$d_k(D)_kgeq0$ which arise in the number-theoretic formulation of Zauner's SIC-POVM conjecture.
  arXiv  Detail & Related papers  (2026-09-03T19:51:19Z)
• The Condition-Number Barrier in Sparse Least Squares [77.64108812086542]
  In [AS21], Axiotis and Sviridenko conjectured that the linear dependence on the restricted condition number in convex optimization cannot be improved by a sparse-time algorithm.<n>We establish their conjectured lower bound for least-squares objectives, conditional on the randomized exact-volume Small-Set Expansion Hypothesis.
  arXiv  Detail & Related papers  (2026-08-03T17:57:01Z)
• Approximating the operator norm of local Hamiltonians via few quantum states [53.16156504455106]
  Consider a Hermitian operator $A$ acting on a complex Hilbert space of $2n$.<n>We show that when $A$ has small degree in the Pauli expansion, or in other words, $A$ is a local $n$-qubit Hamiltonian.<n>We show that whenever $A$ is $d$-local, textiti.e., $deg(A)le d$, we have the following discretization-type inequality.
  arXiv  Detail & Related papers  (2025-09-15T14:26:11Z)
• Towards a universal gateset for $\mathsf{QMA}_1$ [0.0]
  We prove that for all gatesets in the cyclotomic field $mathbbQ(zeta_2k),zeta_2k=e2pi i/2k$, $G_2k$ is universal for all gatesets in $mathbbQ(zeta_2k),zeta_2k=e2pi i/2k$.<n>We also prove that the Gapped Clique Homology problem defined by King
  arXiv  Detail & Related papers  (2024-11-04T23:39:27Z)
• Low-degree approximation of QAC$^0$ circuits [0.0]
  We show that the parity function cannot be computed in QAC$0$. We also show that any QAC circuit of depth $d$ that approximately computes parity on $n$ bits requires $2widetildeOmega(n1/d)$.
  arXiv  Detail & Related papers  (2024-11-01T19:04:13Z)
• The Communication Complexity of Approximating Matrix Rank [50.6867896228563]
  We show that this problem has randomized communication complexity $Omega(frac1kcdot n2log|mathbbF|)$. As an application, we obtain an $Omega(frac1kcdot n2log|mathbbF|)$ space lower bound for any streaming algorithm with $k$ passes.
  arXiv  Detail & Related papers  (2024-10-26T06:21:42Z)
• On estimating the trace of quantum state powers [11.63482880743131]
  We investigate the computational complexity of estimating the trace of quantum state powers $texttr(rhoq)$ for an $n$-qubit mixed quantum state $rho$.<n>Our speedup is achieved by introducing efficiently computable uniform approximations of positive power functions into quantum singular value transformation.
  arXiv  Detail & Related papers  (2024-10-17T13:57:13Z)
• The SUSY partners of the QES sextic potential revisited [0.0]
  We study the SUSY partner Hamiltonians of the quasi- solvable (QES) sextic potential $Vrm qes(x) = nu, x6 + 2, nu, mu,x4 + left[mu2-(4N+3)nu right], x2$, $N in mathbbZ+$.
  arXiv  Detail & Related papers  (2023-11-10T18:38:02Z)
• Algebraic Aspects of Boundaries in the Kitaev Quantum Double Model [77.34726150561087]
  We provide a systematic treatment of boundaries based on subgroups $Ksubseteq G$ with the Kitaev quantum double $D(G)$ model in the bulk. The boundary sites are representations of a $*$-subalgebra $Xisubseteq D(G)$ and we explicate its structure as a strong $*$-quasi-Hopf algebra. As an application of our treatment, we study patches with boundaries based on $K=G$ horizontally and $K=e$ vertically and show how these could be used in a quantum computer
  arXiv  Detail & Related papers  (2022-08-12T15:05:07Z)
• Quantum Complexity of Permutations [0.0]
  We study quantum complexity of permutations in $S_n$ using $sigma, tau, tau-1$ as logic gates. We show that almost all permutations in $S_n$ have quadratic quantum complexity lower bound when $nrightarrow infty$.
  arXiv  Detail & Related papers  (2022-07-21T23:18:54Z)
• Beyond the Berry Phase: Extrinsic Geometry of Quantum States [77.34726150561087]
  We show how all properties of a quantum manifold of states are fully described by a gauge-invariant Bargmann. We show how our results have immediate applications to the modern theory of polarization.
  arXiv  Detail & Related papers  (2022-05-30T18:01:34Z)
• Low-degree learning and the metric entropy of polynomials [44.99833362998488]
  We prove that any (deterministic or randomized) algorithm which learns $mathscrF_nd$ with $L$-accuracy $varepsilon$ requires at least $Omega(sqrtvarepsilon)2dlog n leq log mathsfM(mathscrF_n,d,|cdot|_L,varepsilon) satisfies the two-sided estimate $$c (1-varepsilon)2dlog
  arXiv  Detail & Related papers  (2022-03-17T23:52:08Z)
• Spectral properties of sample covariance matrices arising from random matrices with independent non identically distributed columns [50.053491972003656]
  It was previously shown that the functionals $texttr(AR(z))$, for $R(z) = (frac1nXXT- zI_p)-1$ and $Ain mathcal M_p$ deterministic, have a standard deviation of order $O(|A|_* / sqrt n)$. Here, we show that $|mathbb E[R(z)] - tilde R(z)|_F
  arXiv  Detail & Related papers  (2021-09-06T14:21:43Z)
• Stochastic behavior of outcome of Schur-Weyl duality measurement [45.41082277680607]
  We focus on the measurement defined by the decomposition based on Schur-Weyl duality on $n$ qubits. We derive various types of distribution including a kind of central limit when $n$ goes to infinity.
  arXiv  Detail & Related papers  (2021-04-26T15:03:08Z)

This list is automatically generated from the titles and abstracts of the papers in this site.

This site does not guarantee the quality of this site (including all information) and is not responsible for any consequences.