The Existence of Minimal Logarithmic Signatures for some Finite Simple Unitary Groups

A. R. Ashrafi; A. R. Rahimipour

REVIEW 4 major objections 4 minor 31 references

The Existence of Minimal Logarithmic Signatures for some Finite Simple Unitary Groups

T0 review · 4 major / 4 minor · reviewed 2026-08-14 · deepseek-v4-flash

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Pith's one-line read The paper argues that a published proof of the MLS conjecture for unitary groups has a gap, so the conjecture remains open.

desk verdict The paper's positive MLS constructions for special unitary groups are plausible, but the headline gap report against Hong et al. misfires because it relies on a subfield embedding that does not exist. read the letter →

arxiv 1908.04125 v1 pith:MGRBJ2JL submitted 2019-08-12 math.GR

A. R. Rahimipour , A. R. Ashrafi This is my paper · ORCID

classification math.GR MSC 20D0894A60

keywords minimallogarithmicsignaturesMLSconjecturefinitesimplegroupsunitarySingercyclesisotropicpointsspreads

verification ladder T0 review T1 audit T2 compute T3 formal

The pith

A machine-rendered reading of the paper's core claim, the machinery that carries it, and where it could break.

The reading

The paper tries to establish two things: that the earlier proof in [10] of the MLS conjecture for unitary groups has a genuine gap, and that the conjecture is nevertheless true for $PSU_3(q)$ and $PSU_4(q)$ in certain parameter ranges. If the first claim is right, the MLS conjecture—which says every finite simple group has a logarithmic signature of shortest possible length—remains open for unitary groups. This matters because such factorizations are the algebraic core of several non-abelian group cryptosystems. The positive results are obtained through a double-coset decomposition criterion, using pairs of solvable subgroups with coprime orders so that the required transversals split into blocks of prime size.

What carries the argument

The load-bearing object is the minimal logarithmic signature: an ordered tuple of subsets $A_1,\ldots,A_s$ of $G$ such that every element of $G$ factors uniquely as $a_1\cdots a_s$ with $a_i\in A_i$, and whose total size reaches the lower bound $\sum_j \beta_j p_j$ coming from $|G|=\prod_j p_j^{\beta_j}$. The construction engine is a double-coset criterion: if $G=\bigcup H g_i K$, the subgroups $H$ and $K$ have MLS, and the chosen representatives $g_i$ factor into blocks that are cyclic, are subgroups with MLS, or have prime size or size $4$, then $G$ has an MLS. The paper applies this to $PSU_3(q)$ by choosing two solvable subgroups of coprime orders, so their conjugates intersect trivially, with double-coset index equal to $q+1$ or $q^2-q+1$; in the cases considered these indices are prime, so the block condition is satisfied. For the gap report, the key objects are the purported spread $S_3=\{W'_i\}$ and the Singer cycle subgroup (generated by one element of order $q^n-1$) claimed to be sharply transitive on it; the paper checks them against the known count of isotropic points in the projective space.

What would settle it

For $n=3$ and a small prime power $q$, form $W'_0=GF(q^2)$ and $W'_i=GF(q^2)\alpha^{i(q-1)}$ inside $V=GF(q^6)$, with $f(x,y)=\sum_{i=0}^{2}(xy^q)^{q^{2i}}$, and test whether the $W'_i$ are pairwise disjoint and together cover all isotropic points. If they form such a cover for some $q$, the gap report is wrong; if they do not, the criticism is confirmed.

Watch

Extended reading notes

Core claim

The negative discovery is that [10]'s construction does not produce the partition it needs. Working in $V=GF(q^{2n})$ with $n=2m+1$, the paper takes $W'_0=GF(q^{2m})$ and $W'_i=W'_0\alpha^{i(q^m-1)}$, then claims these are totally isotropic and partition the isotropic points. The paper objects that $f(1,1)=n$, so the point $1$ is isotropic only when $p\mid n$; that the $W'_i$ need not intersect trivially; and that the Singer subgroup used has order $(q^n+1)/(q+1)$, not $q^n+1$. The resulting count of isotropic points contradicts the number $(q^n+1)(q^{n-1}-1)/(q^2-1)$ that the unitary group actually has. On the positive side, Theorem 4.2 proves $PSU_3(q)$ has an MLS when $q=2^n>2$ and either $q+1$ or $q^2-q+1$ is prime; Theorem 4.4 covers odd $q$ when $q^2-q+1$ is prime or when $q>5$ and $q+1=2p$ with $p$ prime; and a reduction theorem transfers these conclusions to $PSU_4(q)$.

Load-bearing premise

The gap report assumes the construction in [10] is faithfully described by taking $W'_0=GF(q^{2m})$ inside $V=GF(q^{2n})$ with $n=2m+1$; if the original paper used a different subspace or spread, this objection may miss its target.

Editorial extensions

If this is right

The existence of MLS for $PSU_n(q)$, $SU_n(q)$, and $GU_n(q)$ is not settled by [10]; the conjecture remains open for unitary simple groups.
For $q=2^n>2$, $PSU_3(q)$ has an MLS whenever $q+1$ or $q^2-q+1$ is prime, giving new cases such as $q=2^8,2^{16},2^{32}$.
For odd $q$, $PSU_3(q)$ has an MLS when $q^2-q+1$ is prime, and when $q>5$ with $q+1=2p$ for a prime $p$.
The same conditions yield MLS for $PSU_4(q)$ via the paper's reduction theorem.
Among the twelve unitary simple groups of order at most $10^{12}$, the paper leaves eight with no proved MLS, so those are concrete open cases.

Reading between the lines

Editorial extensions of the paper, not claims the author makes directly.

An independent check of the gap report could be made by computing, for small $q$, the subspaces $W'_i=GF(q^2)\alpha^{i(q-1)}$ inside $V=GF(q^6)$; the paper's point-count argument predicts nontrivial intersections or uncovered isotropic points.
The double-coset recipe used here—two solvable subgroups of coprime orders with prime double-coset index—is a transferable template for other low-rank simple groups of Lie type.
If the gap report stands, earlier computational classifications that relied on [10] for unitary groups would need to be audited; the eight small groups listed in the paper are natural first targets.

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Editorial analysis

A structured set of objections, weighed in public.

Desk editor's note, referee report, and a circularity audit.

Desk Editor's Note

read the letter

Two things to know. The paper does two things: it claims a gap in Hong, Wang, Yang's 2015 proof that unitary groups have MLS, and it proves MLS for PSU_3(q) (and PSU_4(q) via reduction) when q is an even prime power and q+1 or q^2-q+1 is prime, plus some odd-q cases. The positive theorems are worth taking seriously. The gap report is not.

The Section 3 critique centers on W'_0 = GF(q^{2m}) inside V = GF(q^{2n}) with n = 2m+1. That object does not exist: GF(q^{2m}) embeds into GF(q^{2n}) only when 2m | 2n, i.e. m | n, and n=2m+1 fails for m>1. So the point-count 'contradiction' is computed on an ill-defined set. If Hong et al. used something else — the natural subfield, or an m-dimensional GF(q^2)-subspace — the critique may attack a straw man. The paper doesn't quote their definitions verbatim, and the Singer-cycle order issue (q^n+1 vs (q^n+1)/(q+1)) is asserted without pinning down which group and which action [10] used. That is a load-bearing problem: the claim 'MLS conjecture still open' rests entirely on this.

The positive part is better. The double-coset method from their earlier paper is a legitimate transfer tool, not circular. Theorem 4.1 (reducing PSU_{2n} to SU_{2n-1}) is a clean use of a known factorization, and the even-q PSU_3 constructions look like they can be made to work under the stated primality hypotheses. But there are soft spots here too. Theorem 4.4(1) claims (|H|,|K|)=1 for H of order q^3(q-1) and K of order (q+1)^2/d with q odd; both orders are even, so the claim is simply false. The proof needs a different way to get H ∩ K^g = 1 (or a different pair of subgroups). There is also a garbled gcd argument in Theorem 4.2: p | q+1 does not imply p = q+1. The final gcd conclusion may survive under the primality assumptions, but the proof as written is wrong.

Bottom line: this is a paper with one solid-looking special-case theorem and one unsupported negative headline. If the authors drop or repair the Hong et al. critique, the positive results could be a reasonable short note. As it stands, I would not rely on the 'conjecture still open' claim. It deserves a serious referee only if revised to quote [10] faithfully and prove the alleged gap; the current text is not ready. I'd bring it to a reading group to talk about why subfield embeddings matter in spread constructions, but I wouldn't cite it yet.

Referee Report

4 major / 4 minor

Summary. The paper has two goals. First, it claims that the proof of the main theorem of Hong, Wang, and Yang [10] on minimal logarithmic signatures for unitary groups contains a gap, specifically that the spread S3 used there is not a partition of the isotropic points; consequently the MLS conjecture for unitary groups is declared still open. Second, it proves existence of MLS for PSU_3(q) when q is a power of 2 and q+1 or q^2-q+1 is prime (Theorem 4.2), and for some odd q under analogous prime conditions (Theorem 4.4), with PSU_4(q) cases derived through a reduction theorem (Theorems 4.1, 4.3, and 4.5).

Significance. The negative claim, if correct, would change the status of the MLS conjecture for the unitary family, and the positive results would add new unitary groups to the list of groups known to admit MLS. The paper is transparent about using maximal subgroup classifications and standard factorization methods, and the even-q PSU_3 argument is a plausible route. However, as detailed below, the critique of [10] is not anchored to a well-defined object, and the deductions for PSU_4 and for odd q are not supported as written. The overall contribution is therefore not established.

major comments (4)

[Section 3] The alleged counterexample to [10] is built on W'_0 = GF(q^{2m}) inside V = GF(q^{2n}) with n = 2m+1. Since a finite field GF(q^a) embeds in GF(q^b) only when a divides b, and 2m does not divide 2n for n = 2m+1, the set W'_0 is not a well-defined subspace of V for m>1. The computations in items (1)-(3), including f(1,1)=n and the point-count contradiction, are therefore not valid calculations about any object in V. The manuscript does not quote [10]'s definitions verbatim, so the critique may refute a reconstruction rather than the paper itself. Consequently the central claim that the main result of [10] is wrong, and hence that the MLS conjecture for unitary groups is still open, is not established.
[Theorems 4.3 and 4.5] Theorem 4.1 has the hypothesis that SU_{2n-1}(q) has an MLS and concludes that PSU_{2n}(q) has one. Theorems 4.2 and 4.4 prove MLS for PSU_3(q), not for SU_3(q). No lifting theorem from a quotient to a covering group is stated or cited. Therefore the proofs of Theorems 4.3 and 4.5, and the applications to PSU_4(q) in Section 5, do not follow from the quoted results.
[Theorem 4.4(1)] For odd q the asserted coprimality (q^3(q-1), (q+1)^2/d) = 1 fails: for example, when q=7 and d=1, gcd(7^3*6, 8^2) = 2. Thus H and K are not guaranteed to intersect trivially, and the double-coset count q^2-q+1 used in the application of Theorem 2.5 is unjustified. The proof of Theorem 4.4(1) is therefore invalid.
[Theorem 4.1 proof] The proof invokes Theorem 2.6, which requires K = SU_{2n-1}(q) to have an MLS over K∩H. The text only derives that H has an MLS over H∩K from the solvability of H. No argument or reference is given for the required property of K, so the reduction in Theorem 4.1 is incomplete.

minor comments (4)

[Theorem 4.2 statement] The statement 'q = 2 n' should read 'q = 2^n'; the proof uses the latter.
[Section 5] Theorem 4.3 has no numbered parts, but Section 5 refers to 'Theorem 4.3(2)'.
[Theorem 4.1] The displayed factorization 'q^{n^2} : q^{2n-1}/d(q+1)' is ambiguous; the order of the parabolic subgroup should be stated unambiguously.
[Section 3, item (1)] The computation f(1,1)=n should be interpreted as n times the identity in GF(q^2); in any event the argument rests on the nonexistent embedding discussed in the first major comment.

Circularity Check

0 steps flagged · score 0.0 of 10

No significant circularity: the new MLS constructions are self-contained; the self-citations are non-load-bearing transfer lemmas, and the gap report concerns another paper's proof.

full rationale

The authors' positive results are not circular. Theorems 2.4 and 2.5 cite the authors' earlier paper [26] ('See [26, Lemma 1.1]' and 'given by the present authors [26, Theorem 2.1]'), but these are general transfer lemmas about MLS-over-subgroups and double-coset decompositions; their assumptions (solvability, prime or cyclic block sizes, trivial intersection of conjugate subgroups) do not include the existence of an MLS for PSU_3(q) or PSU_4(q). The actual constructions in Theorems 4.2 and 4.4 use independent maximal-subgroup classifications from [6], [12], [24], and [29], and the coprimality and double-coset counts are verified directly in the text. Theorem 4.1 is explicitly conditional ('If SU_{2n-1}(q) has an MLS then P SU_{2n}(q) has an MLS'), so it does not assume the desired conclusion. The Section 3 gap report against [10] is a correctness critique of another paper's construction; even if that reconstruction is disputable, the issue is external correctness, not circular reasoning. No fitted parameter is relabelled as a prediction, and no uniqueness theorem from the authors' prior work is used to force a choice. Thus no circularity is present.

Assumptions & free parameters 0 free parameters · 6 assumptions · 0 invented entities

The positive theorems rely on standard classification and MLS-transfer theorems cited from [4,6,15,24,26,29]; none of these are reduced to the target result. No parameters are fitted to data, and no new entities are introduced.

assumptions (6)

domain assumption For q a prime power and n odd, SU_n(q) has a Singer cycle of order (q^n+1)/(q+1), and the number of isotropic points of PG(n-1,q^2) is (q^n+1)(q^{n-1}-1)/(q^2-1).
Used in Section 3 both to criticize [10] and to frame the transitive actions; these are standard facts in unitary group theory, cited to [1], [31], and [7].
domain assumption The maximal subgroup classification of PSU_3(q) given in [6] and [24] is complete and correct.
Theorems 4.2 and 4.4 invoke this classification to identify subgroups H and F of specified orders; if the classification is inaccurate, the constructions fail.
domain assumption Theorem 2.5, the double coset MLS reduction method stated in [26], is correct.
All positive existence proofs in Section 4 use Theorem 2.5 as the main engine; it is cited from the authors' own previous paper rather than proved here.
domain assumption Every finite solvable group has a minimal logarithmic signature.
The proofs of Theorems 4.2 and 4.4 need H and F to have MLS; solvability of those subgroups is asserted and MLS is attributed to [4, Proposition 3.1].
domain assumption The factorization PSU_{2n}(q) = (q^{n^2} : q^{2n-1}/(d(q+1))) SU_{2n-1}(q) from [15] holds.
Theorem 4.1 relies on this factorization and on the solvability of the first factor to reduce PSU_{2n}(q) to SU_{2n-1}(q).
domain assumption PSU_3(q) is 2-transitive on the q^3+1 isotropic points.
Theorem 4.4(2) uses 2-transitivity to build the double coset representatives A_1 and A_2; this is cited to [29].

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Pith. "Pith review of The Existence of Minimal Logarithmic Signatures for some Finite Simple Unitary Groups." pith.science (2026). https://pith.science/paper/MGRBJ2JL

@misc{pith2026190804125,
  author       = {Pith},
  title        = {Pith review of: The Existence of Minimal Logarithmic Signatures for some Finite Simple Unitary Groups},
  year         = {2026},
  howpublished = {\url{https://pith.science/paper/MGRBJ2JL}},
  note         = {Machine review of arXiv:1908.04125}
}

abstract

The $MLS$ conjecture states that every finite simple group has a minimal logarithmic signature. The aim of this paper is proving the existence of a minimal logarithmic signature for some simple unitary groups $PSU_{n}(q)$. We report a gap in the proof of the main result of [H. Hong, L. Wang, Y. Yang, Minimal logarithmic signatures for the unitary group $U_n(q)$, \textit{Des. Codes Cryptogr.} \textbf{77} (1) (2015) 179--191] and present a new proof in some special cases of this result. As a consequence, the $MLS$ conjecture is still open.

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Reference graph

Works this paper leans on

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Pith tools

Reviewed August 14, 2026 · model on record in the stance chip above.

[1] [10]

H. Hong, L. Wang, Y. Yang, Minimal logarithmic signatures for th e unitary group Un(q), Des. Codes Cryptogr. 77 (1) (2015) 179–191

work page 2015

[2] [1]

Babai, P

L. Babai, P. P. P` alfy and J. Saxl, On the number of p−regular elements in ﬁnite simple groups, LMS J. Comput. Math. 12 (2009) 82–119

work page 2009

[3] [2]

J. D. Dixon and B. Mortimer, Permutation Groups , Graduate Texts in Mathematics 163, Springer-Verlag, New York, 1996

work page 1996

[4] [3]

X. G. Fang, G. Havas and J. Wang, A family of non −quasiprimitive graphs admitting a quasiprimitive 2 −arc transitive group action, Eur. J. Combin. 20 (1999) 551–557

work page 1999

[5] [4]

M. I. Gonz´ alez Vasco, M. R¨ otteler and R. Steinwandt, On minima l length factorizations of ﬁnite groups, Exp. Math. 12 (1) (2003) 1–12

work page 2003

[6] [5]

M. I. Gonz´ alez Vasco and R. Steinwandt, Obstacles in two public k ey cryptosystems based on group factorizations, Tatra Mt. Math. Publ. 25 (2002) 23–37

work page 2002

[7] [6]

R. W. Hartley, Determination of the ternary collineation groups w hose coeﬃcients lie in the GF (2n), Ann. Math. 27 (1926) 140–158

work page 1926

[8] [7]

M. D. Hestenes, Singer groups, Canad. J. Math. 22 (1970) 492–513

work page 1970