--- title: 'Cyclic Equalizability: Theory & Applications' url: https://www.emergentmind.com/topics/cyclic-equalizability type: topic --- # Cyclic Equalizability: Theory & Applications 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 [2604.19504]. 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 [1105.4320]. In the theory of directed cyclic graphical models, the analogous question is Markov equivalence under the \(d\)-separation criterion [2309.03092]. ## 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 [2604.19504] | | Coding theory | Cyclic and constacyclic codes | Solving sets, affine maps on defining sets, multipliers, or Sylow-controlled permutations [1105.4320] | | Directed graphical models | Directed graphs with cycles | Equality of \(d\)-separation models, i.e. Markov equivalence [2309.03092] | | Cyclic incidence structures | Balanced configurations | CI-property under automorphisms of the cyclic group [1511.07285] | 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 \[ u_0\,a_{i,0}\,u_1\,a_{i,1}\cdots u_{n-1}\,a_{i,n-1}\,u_n, \] where the same inserted strings \(u_0,\dots,u_n\in\Sigma^*\) are used for all words. Words are cyclically equalizable if, after such a simultaneous insertion, the resulting words are cyclically equivalent [2604.19504]. Shinagawa and Nuida introduced cyclic equalizability in 2025 in the context of card-based cryptography. Their binary theorem states that for \(w_1,w_2\in\{0,1\}^n\), \[ w_1,w_2 \text{ are cyclically equalizable } \iff wt(w_1)=wt(w_2), \] so equal Hamming weight is both necessary and sufficient in the binary two-word case [2507.04916]. The later generalization proves that for two equal-length words over an arbitrary finite alphabet, \[ u \text{ and } v \text{ are cyclically equalizable } \iff \Psi(u)=\Psi(v), \] where \(\Psi(w)=(|w|_a)_{a\in\Sigma}\) is the Parikh vector [2604.19504]. In the binary case, this reduces to Hamming weight because \(|u|_0=n-|u|_1\). The constructive proof in the general two-word case proceeds by reducing repeated letters to pairwise distinct tagged letters, normalizing to \[ u = 0\,1\,\cdots\,(n-1), \qquad v = \pi(0)\,\pi(1)\,\cdots\,\pi(n-1) \] for some permutation \(\pi\in S_n\), and then building simultaneous insertions via two auxiliary operations. The first is reading with step size \(p\), \[ R_p(w)=a_0\,a_p\,a_{2p}\,\cdots\,a_{(n-1)p}, \] for \(\gcd(p,n)=1\). The second is columnwise interleaving \[ I(w_1,\dots,w_m)=w_{1,0}w_{2,0}\cdots w_{m,0}\,w_{1,1}w_{2,1}\cdots w_{m,1}\cdots. \] The proof solves the single-cycle case by constructing words of length \(n^2\) using a block/group decomposition with \(p=n+1\), and then handles a general permutation by decomposing \(\pi\) into disjoint cycles and interleaving the corresponding constructions [2604.19504]. 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 \[ \text{\#cards} \ge \max(N_0(f),N_1(f)), \] where \(N_b(f)\) is the number of inputs on which the Boolean function \(f\) outputs \(b\) [2507.04916]. The exact characterization for more than two words remains open [2604.19504]. ## 3. Cyclic and constacyclic code equivalence For linear codes, the basic equivalence notion is permutation or monomial equivalence. If \(C,D\le \mathbb F_q^\Omega\) are linear codes and \(g\in \operatorname{Sym}(\Omega)\), then \[ f^g(\omega):=f(\omega^{g^{-1}}) \] and \(C\) and \(D\) are permutation equivalent if \(C^g=D\) for some \(g\in \operatorname{Sym}(\Omega)\). In the broader monomial-semilinear setting, two codes \(C_1,C_2\subseteq \mathbb F_q^n\) are equivalent if \(C_1=C_2M\delta\) for some monomial matrix \(M\) and field automorphism \(\delta\); over finite fields, monomial equivalence and isometric equivalence are the same relation [1105.4320]. A cyclic code of length \(n\) over \(\mathbb F_q\) can be realized as an ideal in \(\mathbb F_q[H]\) for a cyclic group \(H\cong C_n\). In the semisimple case \(\gcd(q,n)=1\), a cyclic code \(I=e\mathbb F_q[H]\) has a unique generating idempotent \[ e=\sum_{h\in H} e_h h,\qquad e^2=e. \] The central theorem in this setting states that a solving set for the colored Cayley digraph \(\operatorname{Cay}(H,e)\) is also a solving set for the code \(I\). The partition \[ a,b\in H \text{ in same class } \iff e_a=e_b \] determines a solving set \(S_e\), and once this partition is constructed, \(S_e\) can be built using \[ O(n^2) \] arithmetic operations in \(\mathbb Z_n\). Equivalence testing for semisimple cyclic codes is therefore reduced to checking only permutations in \(S_e\), rather than all of \(\operatorname{Sym}(H)\) [1105.4320]. A second line of work uses cyclotomic-coset data. In the simple-root case \(\gcd(n,q)=1\), the \(q\)-cyclotomic cosets \[ S_i=\{iq^j \bmod n : j\in \mathbb N\} \] parametrize irreducible factors of \(x^n-1\), 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 \[ z\mapsto ez+b \pmod n \] that send one defining-set description, or repeated-root multiset description, to the other. Its stated complexity is \[ O(n^3), \] 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 [2107.00159]. The constacyclic extension replaces \(x^n-a\) by \[ x^n-a=(x^{n'}-b)^{p^t}, \] with \(n=p^t n'\), \(\gcd(n',p)=1\), and introduces the order \[ r=\operatorname{Ord}_q(a). \] The relevant roots are indexed by \[ 1+ir,\qquad i=0,1,\dots,n'-1, \] inside the \(n'r\)-th roots of unity, and the algorithm again searches for affine maps \(x\mapsto ex+b\) between the corresponding cyclotomic-coset multisets. The paper is explicit that this criterion is sufficient but not necessary: it gives a cyclic example with \(n=32\) and \(a=1\) where the algorithm returns False although the codes are actually equivalent [2108.08619]. A third line of results gives exact algebraic criteria for particular transformations. If \(C_1,C_2\) are cyclic codes with defining sets \(A_1,A_2\), then isometric equivalence through the shift map \(\phi_b(x)=x+b \pmod n\) holds if and only if \(\phi_b(A_1)=A_2\) and \[ n\mid |A_1|b(q-1). \] For \(\omega\)-constacyclic codes over \(\mathbb F_4\), if \[ \gcd(3n,\phi(3n))=1, \] then all permutation equivalent constacyclic codes of length \(n\) are given by the action of multipliers [2211.00897]. 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 \(p^r\), permutation equivalence can be expressed in terms of Sylow \(p\)-subgroups of the permutation group. Let \(T\) be the full cyclic shift and let \(P\) be a Sylow \(p\)-subgroup of \(\operatorname{Per}(C)\) containing \(T\). The decisive set is \[ H(P)=\{\sigma\in S_{p^r}\mid \sigma^{-1}T\sigma\in P\}. \] If \(C\) and \(C'\) are cyclic codes of length \(p^r\), then they are equivalent if and only if they are equivalent by an element of \(H(P)\). When \(|P|=p^r\), one has \[ P=\langle T\rangle,\qquad H(P)=AG(p^r). \] If \(r