Cyclic Equalizability: Theory & Applications

• Cyclic equalizability is a collection of problems where cyclic symmetry defines equivalence among structured objects in fields like combinatorics, coding theory, and graphical models.
• It leverages invariants such as Parikh vectors, cyclic shifts, and affine maps to reduce complex isomorphism challenges to more tractable cyclic problems.
• Research in this area drives efficient algorithms and clear criteria for equivalence testing, with significant implications in cryptography, coding theory, and causal graphical models.

Cyclic equalizability is a family of problems in which cyclic symmetry determines when objects can be made cyclically identical or should be treated as equivalent. In combinatorics on words and card-based cryptography, it asks whether equal-length words can be transformed by simultaneous insertion into words that are cyclic shifts of one another, and for two words over an arbitrary finite alphabet this is equivalent to equality of Parikh vectors (Thongjarast et al., 21 Apr 2026). In coding theory, related results reduce equivalence of cyclic and constacyclic codes to structured actions on defining sets, colored Cayley digraphs, or controlled permutation families (Muzychuk, 2011). In the theory of directed cyclic graphical models, the analogous question is Markov equivalence under the $d$-separation criterion (Claassen et al., 2023).

1. Scope and domain-specific meanings

Across these works, the expression appears in several domain-specific senses. This suggests that cyclic equalizability is best understood as a class of cyclic equivalence problems rather than a single canonical definition.

Domain	Objects	Equivalence mechanism
Combinatorics on words	Equal-length words	Simultaneous insertion followed by cyclic shift (Thongjarast et al., 21 Apr 2026)
Coding theory	Cyclic and constacyclic codes	Solving sets, affine maps on defining sets, multipliers, or Sylow-controlled permutations (Muzychuk, 2011)
Directed graphical models	Directed graphs with cycles	Equality of $d$-separation models, i.e. Markov equivalence (Claassen et al., 2023)
Cyclic incidence structures	Balanced configurations	CI-property under automorphisms of the cyclic group (Koike et al., 2015)

What these settings share is the replacement of unrestricted isomorphism by a cyclicly constrained one. In each case, a cyclic action is not merely decorative: it supplies the invariants, admissible transformations, and algorithmic reductions. The common pattern is that a large ambient equivalence problem is replaced by a smaller cyclicly structured one.

2. Equalizability of words under simultaneous insertion

In the word-theoretic setting, let $\Sigma$ be a finite alphabet and let $w=a_0a_1\cdots a_{n-1}\in\Sigma^n$. For $r\in\mathbb Z_n$, the cyclic shift is

$$w^{(r)} = a_r a_{r+1} \cdots a_{n-1} a_0 a_1 \cdots a_{r-1}.$$

Two words $u,v\in\Sigma^*$ of equal length are cyclically equivalent if $v=u^{(r)}$ for some $r\in\mathbb Z_n$. A simultaneous insertion transforms words $w_i=a_{i,0}a_{i,1}\cdots a_{i,n-1}$ into

$d$0

where the same inserted strings $d$1 are used for all words. Words are cyclically equalizable if, after such a simultaneous insertion, the resulting words are cyclically equivalent (Thongjarast et al., 21 Apr 2026).

Shinagawa and Nuida introduced cyclic equalizability in 2025 in the context of card-based cryptography. Their binary theorem states that for $d$2,

$d$3

so equal Hamming weight is both necessary and sufficient in the binary two-word case (Shinagawa et al., 7 Jul 2025). The later generalization proves that for two equal-length words over an arbitrary finite alphabet,

$d$4

where $d$5 is the Parikh vector (Thongjarast et al., 21 Apr 2026). In the binary case, this reduces to Hamming weight because $d$6.

The constructive proof in the general two-word case proceeds by reducing repeated letters to pairwise distinct tagged letters, normalizing to

$d$7

for some permutation $d$8, and then building simultaneous insertions via two auxiliary operations. The first is reading with step size $d$9,

$\Sigma$0

for $\Sigma$1. The second is columnwise interleaving

$\Sigma$2

The proof solves the single-cycle case by constructing words of length $\Sigma$3 using a block/group decomposition with $\Sigma$4, and then handles a general permutation by decomposing $\Sigma$5 into disjoint cycles and interleaving the corresponding constructions (Thongjarast et al., 21 Apr 2026).

The cryptographic motivation is that a random cut of a face-down card sequence acts as an unknown cyclic shift. In the information erasure problem, one inserts cards, performs a random cut, and then opens all cards; information is erased exactly when the possible hidden words can be made cyclically equal by simultaneous insertion. In the binary two-word case, information erasure is therefore possible if and only if the two candidate words have equal Hamming weight. For single-cut full-open protocols, the paper proves the lower bound

$\Sigma$6

where $\Sigma$7 is the number of inputs on which the Boolean function $\Sigma$8 outputs $\Sigma$9 (Shinagawa et al., 7 Jul 2025). The exact characterization for more than two words remains open (Thongjarast et al., 21 Apr 2026).

3. Cyclic and constacyclic code equivalence

For linear codes, the basic equivalence notion is permutation or monomial equivalence. If $w=a_0a_1\cdots a_{n-1}\in\Sigma^n$0 are linear codes and $w=a_0a_1\cdots a_{n-1}\in\Sigma^n$1, then

$w=a_0a_1\cdots a_{n-1}\in\Sigma^n$2

and $w=a_0a_1\cdots a_{n-1}\in\Sigma^n$3 and $w=a_0a_1\cdots a_{n-1}\in\Sigma^n$4 are permutation equivalent if $w=a_0a_1\cdots a_{n-1}\in\Sigma^n$5 for some $w=a_0a_1\cdots a_{n-1}\in\Sigma^n$6. In the broader monomial-semilinear setting, two codes $w=a_0a_1\cdots a_{n-1}\in\Sigma^n$7 are equivalent if $w=a_0a_1\cdots a_{n-1}\in\Sigma^n$8 for some monomial matrix $w=a_0a_1\cdots a_{n-1}\in\Sigma^n$9 and field automorphism $r\in\mathbb Z_n$0; over finite fields, monomial equivalence and isometric equivalence are the same relation (Muzychuk, 2011).

A cyclic code of length $r\in\mathbb Z_n$1 over $r\in\mathbb Z_n$2 can be realized as an ideal in $r\in\mathbb Z_n$3 for a cyclic group $r\in\mathbb Z_n$4. In the semisimple case $r\in\mathbb Z_n$5, a cyclic code $r\in\mathbb Z_n$6 has a unique generating idempotent

$r\in\mathbb Z_n$7

The central theorem in this setting states that a solving set for the colored Cayley digraph $r\in\mathbb Z_n$8 is also a solving set for the code $r\in\mathbb Z_n$9. The partition

$$w^{(r)} = a_r a_{r+1} \cdots a_{n-1} a_0 a_1 \cdots a_{r-1}.$$0

determines a solving set $$w^{(r)} = a_r a_{r+1} \cdots a_{n-1} a_0 a_1 \cdots a_{r-1}.$$1, and once this partition is constructed, $$w^{(r)} = a_r a_{r+1} \cdots a_{n-1} a_0 a_1 \cdots a_{r-1}.$$2 can be built using

$$w^{(r)} = a_r a_{r+1} \cdots a_{n-1} a_0 a_1 \cdots a_{r-1}.$$3

arithmetic operations in $$w^{(r)} = a_r a_{r+1} \cdots a_{n-1} a_0 a_1 \cdots a_{r-1}.$$4. Equivalence testing for semisimple cyclic codes is therefore reduced to checking only permutations in $$w^{(r)} = a_r a_{r+1} \cdots a_{n-1} a_0 a_1 \cdots a_{r-1}.$$5, rather than all of $$w^{(r)} = a_r a_{r+1} \cdots a_{n-1} a_0 a_1 \cdots a_{r-1}.$$6 (Muzychuk, 2011).

A second line of work uses cyclotomic-coset data. In the simple-root case $$w^{(r)} = a_r a_{r+1} \cdots a_{n-1} a_0 a_1 \cdots a_{r-1}.$$7, the $$w^{(r)} = a_r a_{r+1} \cdots a_{n-1} a_0 a_1 \cdots a_{r-1}.$$8-cyclotomic cosets

$$w^{(r)} = a_r a_{r+1} \cdots a_{n-1} a_0 a_1 \cdots a_{r-1}.$$9

parametrize irreducible factors of $u,v\in\Sigma^*$0, so a cyclic code is represented by a defining set consisting of a union of such cosets. The 2021 algorithm compares two cyclic codes by searching for affine maps

$u,v\in\Sigma^*$1

that send one defining-set description, or repeated-root multiset description, to the other. Its stated complexity is

$u,v\in\Sigma^*$2

and in the binary case it is a full equivalence test; in general it is only a sound detection criterion, so a False output does not prove inequivalence (Aydin et al., 2021).

The constacyclic extension replaces $u,v\in\Sigma^*$3 by

$u,v\in\Sigma^*$4

with $u,v\in\Sigma^*$5, $u,v\in\Sigma^*$6, and introduces the order

$u,v\in\Sigma^*$7

The relevant roots are indexed by

$u,v\in\Sigma^*$8

inside the $u,v\in\Sigma^*$9-th roots of unity, and the algorithm again searches for affine maps $v=u^{(r)}$0 between the corresponding cyclotomic-coset multisets. The paper is explicit that this criterion is sufficient but not necessary: it gives a cyclic example with $v=u^{(r)}$1 and $v=u^{(r)}$2 where the algorithm returns False although the codes are actually equivalent (Akre et al., 2021).

A third line of results gives exact algebraic criteria for particular transformations. If $v=u^{(r)}$3 are cyclic codes with defining sets $v=u^{(r)}$4, then isometric equivalence through the shift map $v=u^{(r)}$5 holds if and only if $v=u^{(r)}$6 and

$v=u^{(r)}$7

For $v=u^{(r)}$8-constacyclic codes over $v=u^{(r)}$9, if

$r\in\mathbb Z_n$0

then all permutation equivalent constacyclic codes of length $r\in\mathbb Z_n$1 are given by the action of multipliers (Dastbasteh et al., 2022). Collectively, these results show that cyclic-code equalizability ranges from exact criteria in semisimple or arithmetic special cases to one-way certificates based on affine-coset transport in more general settings.

4. Group-theoretic control of cyclic-object isomorphism

For cyclic codes of length $r\in\mathbb Z_n$2, permutation equivalence can be expressed in terms of Sylow $r\in\mathbb Z_n$3-subgroups of the permutation group. Let $r\in\mathbb Z_n$4 be the full cyclic shift and let $r\in\mathbb Z_n$5 be a Sylow $r\in\mathbb Z_n$6-subgroup of $r\in\mathbb Z_n$7 containing $r\in\mathbb Z_n$8. The decisive set is

$r\in\mathbb Z_n$9

If $w_i=a_{i,0}a_{i,1}\cdots a_{i,n-1}$0 and $w_i=a_{i,0}a_{i,1}\cdots a_{i,n-1}$1 are cyclic codes of length $w_i=a_{i,0}a_{i,1}\cdots a_{i,n-1}$2, then they are equivalent if and only if they are equivalent by an element of $w_i=a_{i,0}a_{i,1}\cdots a_{i,n-1}$3. When $w_i=a_{i,0}a_{i,1}\cdots a_{i,n-1}$4, one has

$w_i=a_{i,0}a_{i,1}\cdots a_{i,n-1}$5

If $w_i=a_{i,0}a_{i,1}\cdots a_{i,n-1}$6 and $w_i=a_{i,0}a_{i,1}\cdots a_{i,n-1}$7, then

$w_i=a_{i,0}a_{i,1}\cdots a_{i,n-1}$8

where $w_i=a_{i,0}a_{i,1}\cdots a_{i,n-1}$9 and $d$00 are explicit polynomial permutation groups modulo $d$01. For quasi-cyclic codes of length $d$02, the analogous necessary search set is

$d$03

and $d$04 (Guenda, 2010). This formulation makes cyclic equalizability a problem of conjugating the distinguished cyclic shift into a controlled $d$05-subgroup.

A broader CI-theoretic formulation replaces codes by cyclic combinatorial objects. For balanced configurations $d$06 of type $d$07, the main theorem is that every finite cyclic group satisfies the CI-property for balanced configurations. Equivalently, if a balanced configuration carries a regular cyclic action, then any isomorphism to another such cyclic object is induced by an automorphism of the underlying cyclic group. In the connected case, if $d$08 is cyclic regular on $d$09 and

$d$10

is $d$11-minimal, then $d$12. The proof uses solvability of $d$13, control of the Fitting subgroup $d$14, and a detailed analysis of possible normal $d$15-subgroups (Koike et al., 2015).

A related coding-theoretic CI consequence appears for semisimple cyclic codes: a cyclic group of square-free or twice square-free order is a CI-group with respect to semisimple cyclic codes, and more generally a solving set for colored Cayley digraphs over a $d$16-group is also a solving set for semisimple group codes over that $d$17-group (Muzychuk, 2011). This group-theoretic perspective shows that many cyclic equivalence problems are controlled not by arbitrary permutations but by regular cyclic subgroups, their normalizers, and the associated Cayley-object machinery.

5. Markov equivalence in directed cyclic graphs

In graphical models, cyclic equalizability takes the form of Markov equivalence for directed graphs that may contain directed cycles. Two such graphs are Markov equivalent when they imply exactly the same $d$18-separations. Richardson’s polynomial-time characterization for directed cyclic graphs shows that Markov equivalence is governed by a richer invariant set than in DAGs: the two graphs must have the same adjacencies, where adjacency means real or virtual adjacency; the same unshielded conductors; the same unshielded perfect non-conductors; the same mutually exclusive conductors on uncovered itineraries; and the same specified ancestor/descendant relations involving imperfect non-conductors and mutually exclusive conductors (Richardson, 2013). This already implies that the DAG criterion “same skeleton + same unshielded colliders” does not survive intact in the cyclic case.

The 2023 reformulation rewrites Richardson’s Cyclic Equivalence Theorem in ancestral language using the cyclic maximal ancestral graph (CMAG). In that formulation, two directed graphs $d$19, with corresponding CMAGs $d$20, are Markov equivalent if and only if $d$21 and $d$22 have the same skeleton, the same v-structures, the same virtual collider triples, and the same corrected ancestry relation

$d$23

whenever $d$24 is a virtual collider triple and $d$25 is a virtual v-structure (Claassen et al., 2023). The reformulation introduces virtual v-structures, u-structures, and virtual collider triples as the cyclic analogues of the familiar collider invariants.

Algorithmically, the ancestral reformulation eliminates direct $d$26-separation testing. The old CPAG-from-Graph procedure had complexity

$d$27

whereas the new Graph-to-CMAG and Graph-to-CPAG pipeline has worst-case complexity

$d$28

The paper reports that for random graphs with $d$29 and density $d$30, the old method took about $d$31 seconds on average, while the new one took about $d$32 seconds (Claassen et al., 2023). In this domain, cyclic equalizability is therefore an efficiently decidable equality of conditional-independence structure, but only after replacing purely local DAG criteria by SCC-mediated and virtual-collider invariants.

6. Related notions, misconceptions, and open problems

Several nearby notions involve cyclic structure but are not equalizability in the strict sense. A digraph with ring structure is called essentially cyclic if its Laplacian spectrum is not completely real. This is a spectral property of directed imbalance, not an equivalence criterion: a pure directed Hamiltonian cycle is essentially cyclic, while the full bidirectional cycle and certain one-missing-arc or two-most-distant-missing-arcs cases are not (0910.3113). Likewise, the problem of describing one cyclic code by another through a non-zero-locator code is a distance-bounding mechanism based on zero-pattern alignment,

$d$33

rather than a notion of code equivalence (Zeh et al., 2012).

Other cyclic questions are adjacent but logically different. For single-orbit cyclic subspace codes, equidistance is a distance-regularity property, not an isomorphism problem. The sharp classification is that if $d$34 has $d$35, then $d$36 is equidistant if and only if

$d$37

so there are only trivial equidistant single-orbit cyclic subspace codes, whereas quasi-cyclic orbit codes admit genuine nontrivial examples, including sunflower and non-sunflower families (Mahak et al., 16 Jan 2025).

A recurring misconception is that cyclicly structured certificates are always complete. In the word-theoretic setting, the Parikh-vector criterion is complete only for two words; the case $d$38 remains open (Thongjarast et al., 21 Apr 2026). In coding theory, affine-coset transport is a sufficient condition in the constacyclic generalization and need not detect all true equivalences (Akre et al., 2021). In cyclic graphical models, the ancestral characterization is complete for the causally sufficient $d$39-separation setting, but the extension to latent confounders is presented as a conjectural direction rather than an established theorem (Claassen et al., 2023). For balanced configurations, the cyclic case is settled, but classifying all finite groups that are CI-groups for balanced configurations remains open (Koike et al., 2015).

Taken together, these literatures show two dominant modes of cyclic equalizability. The first is an exact invariant theory, as in Parikh vectors for two words or CMAG invariants for directed cyclic graphs. The second is a controlled certificate theory, as in solving sets, multipliers, shift divisibility, or affine-coset transport for cyclic codes. The unifying principle is that cyclic symmetry does not eliminate structure; it compresses it into a smaller and often algorithmically tractable equivalence calculus.