---
title: 'CCZ-Classes: Cryptography & Quantum Codes'
url: https://www.emergentmind.com/topics/ccz-classes
type: topic
---

# CCZ-Classes: Cryptography & Quantum Codes

A CCZ-class is the equivalence class of a function with respect to Carlet–Charpin–Zinoviev (CCZ) equivalence, an affine-graph-based notion vital for the classification of vectorial Boolean and $p$-ary functions in cryptography, combinatorics, and quantum fault tolerance. CCZ-equivalence not only preserves key cryptographic invariants like differential uniformity and nonlinearity, but also defines the essential partition of S-boxes, almost perfect nonlinear (APN) mappings, perfect nonlinear and planar functions, and associated semifields and quantum codes. The structure and size of CCZ-classes underpin both finite field theory and the search for new cryptographically significant functions and codes.

## 1. Foundations of CCZ-Equivalence and CCZ-Classes

For functions $F, G : \mathbb{F}_{q}^n \to \mathbb{F}_{q}^n$, CCZ-equivalence is defined by the existence of an affine permutation $L$ on $\mathbb{F}_{q}^{2n}$ such that $L(\Gamma_F) = \Gamma_G$, where the graph $\Gamma_F = \{ (x, F(x)) : x \in \mathbb{F}_{q}^n \}$ [1709.07664, 2306.13718, 1101.1508]. The resulting CCZ-class consists of all mappings equivalent in this sense. Importantly, CCZ-equivalence strictly generalizes extended-affine (EA) equivalence: EA-equivalent functions are always CCZ-equivalent, but the converse does not hold; CCZ-classes can contain functions of distinct algebraic degree and form.

CCZ-equivalence preserves the following key invariants:
- Differential spectrum (including differential uniformity)
- Walsh spectrum (nonlinearity, propagation characteristics)
- Nonlinearity profiles

For monomial functions $F(x) = x^d$ over finite fields, two functions are CCZ-equivalent if and only if their exponents are in the same orbit under the action generated by Frobenius automorphism ($d \mapsto 2^k d$) and inversion ($d \mapsto d^{-1}$) modulo $2^n-1$ or $p^n-1$ [1101.1508, 2407.03170, 2210.15103].

## 2. CCZ-Classes in APN and Planar Function Classification

CCZ-classes underlie the modern classification of cryptographically optimal mappings, particularly almost perfect nonlinear (APN) and planar (perfect nonlinear) functions. For APN mappings $F: \mathbb{F}_{2^n} \to \mathbb{F}_{2^n}$, the CCZ-class constitutes the central object of interest because APNness is CCZ-invariant. In small dimension ($n=6$), the exhaustive computations confirm exactly fourteen CCZ-classes for 6-bit APNs, with no new classes found in exhaustive searches [2601.11247, 2507.12853]. For $n=6$ to $n=11$, the number of CCZ-classes from infinite APN families remains very small (2 to 13) [1709.07664].

In odd characteristic, for planar functions $F: \mathbb{F}_{p^n} \to \mathbb{F}_{p^n}$ (differential uniformity 1, i.e., perfect nonlinear), CCZ-equivalence provides the primary classification tool, with the further structure from semifield isotopism. Any isotopy class of a commutative presemifield can split into at most two CCZ-classes, as described by the Coulter–Henderson splitting criterion [2407.03170]. For instance, in characteristic three, a full classification up to CCZ-equivalence for $n\le 11$ yields 96 CCZ-classes [2407.03170].

## 3. Structure, Rigidity, and Diversity Within CCZ-Classes

The structure of CCZ-classes can vary substantially:

- **CCZ-class rigidity:** For certain functions, the CCZ-class coincides with the EA-class. For example, the Gold APN function $x^{2^r+1}$ is CCZ-equivalent to another function if and only if they are EA-equivalent [1101.1508]. This is an exceptional rigidity property.

- **CCZ-class richness:** For most quadratics $F$ not within these rigid families (e.g., “good” quadratic APN mappings), the CCZ-class is large and contains many EA-inequivalent members, including those of higher algebraic degree (“twisted” constructions) [2306.13718]. Explicit constructions yield an infinite (exponential-sized) family of CCZ-equivalents that are pairwise EA-inequivalent, highlighting the nontrivial size and diversity of a typical CCZ-class.

- **Sparse classes:** For certain constrained settings (e.g., APN permutations in small even dimension with specific self-equivalence), CCZ-classes may be unique—such as the unique “Dublin” CCZ-class for 6-bit APN permutations with specified spectral moments [2507.12853, 2003.12006].

## 4. CCZ-Classes in Quantum Codes and Quantum Gates

The concept of CCZ-class extends to quantum information, where the "CCZ gate" refers to the three-qubit controlled-controlled-Z gate. Novel families of quantum codes—explicitly, CSS codes—are classified in terms of their ability to support transversal non-Clifford gates such as CCZ. Here, "CCZ-class" denotes families of codes permitting logical CCZ implementation by transversal application of physical CCZ gates [2408.10140, 2510.06798].

The relevant structures are as follows:
- A quantum CSS code $\mathrm{CSS}(C_1, C_2)$ over $\mathbb{F}_q$ admits transversal logical CCZ if $C_1$ satisfies a degree-2 multiplication property ($C_1^{\star 2} \subseteq C_1^\perp$) [2408.10140].
- Explicit constructions yield infinite families of binary quantum codes (CCZ-classes) of linear rate and distance with transversal CCZ, essential for fault-tolerant quantum computation and magic-state distillation with constant space overhead [2408.10140, 2510.06798].
- CCZ-classes of quantum codes are further stratified by parameters such as code distance, locality (parity-check weight), and alphabet size, offering a nuanced landscape for transversal implementation [2510.06798].

## 5. CCZ-Classes and Generalizations: c-CCZ and Partial Equivalences

CCZ-equivalence admits generalizations, most notably $c$-CCZ equivalence, where the equivalence is twisted by a scalar parameter $c$ [2301.09325]. This broader framework includes:
- $c$-CCZ, $c$-EA, and $c^1$-equivalence provide a lattice of equivalence relations capturing “twisted” S-box analysis.
- $c$-CCZ equivalence preserves $cc$-differential uniformity and spectrum, and strictly contains $c$-EA equivalence: there exist function pairs $F,G$ that are $c$-CCZ but not $c$-EA equivalent.
- For perfect nonlinear (PN) functions, $c$-CCZ always coincides with $c$-EA equivalence, reflecting rigidity analogous to certain APN cases.

This generalized notion is necessary to fully classify S-boxes and vectorial Boolean functions up to twist-invariant properties, especially in the context of advanced cryptanalytic criteria.

## 6. Computational Classification and Invariants for CCZ-Classes

Distinguishing CCZ-classes in practice requires efficient invariants and computational methods:
- CCZ-invariants include the full differential spectrum, extended Walsh spectrum, algebraic degree, and structural data from associated codes (e.g., automorphism group order, nuclei in planar case) [1709.07664, 2601.11247, 2407.03170].
- For APN mappings, recent advances exploit spectral moment invariants, component-class invariants, and extendibility criteria to verify completeness of CCZ-classifications; e.g., the confirmation of exactly 14 CCZ-classes of 6-bit APN functions [2601.11247, 2507.12853].
- For monomials, CCZ-inequivalence typically follows from exponent-orbit analysis under Frobenius and inversion, as well as resultant-based elimination for partially APN mappings [2210.15103, 2212.04719].
- For newly constructed APN or 0-APN families, computation of CCZ-invariants such as $\Gamma$-rank or code weight distributions confirms CCZ-inequivalence to prior classes [2105.08464].

## 7. Open Questions and Ongoing Directions

While substantial progress has been made, several open questions regarding CCZ-classes remain:
- The total number and growth rate of CCZ-classes for APN and planar functions as $n$ increases are unknown; current evidence suggests slow growth for moderate $n$ [1709.07664, 2407.03170].
- The existence of new, fundamentally distinct CCZ-classes of APN (or 0-APN) mappings in large $n$, especially outside monomial and existing multivariate families, is an active area of investigation [2210.15103, 2105.08464].
- In the quantum domain, the construction of new CCZ-classes of codes with improved parameters (e.g., locality, distance, alphabet reduction) is ongoing, guided by the algebraic multiplication property and subsystem product frameworks [2408.10140, 2510.06798].
- The possible existence and structure of APN permutations in even dimensions beyond $n=6$ (where only the Dillon permutation is known) remains a major unresolved issue [2507.12853, 2003.12006].

CCZ-classes thus offer the mathematical and computational foundation upon which the classification and discovery of cryptographic and quantum-optimal functions are built, linking field theory, combinatorics, nonlinear analysis, and quantum information in a unified framework.

Source: https://www.emergentmind.com/topics/ccz-classes