{"id":"d6ee6ffe-d0c6-422e-8030-c3f9460d96ce","arxiv_id":"1908.04125","paper_version":1,"verdict":"REJECT","confidence":"MODERATE","novelty_score":5.0,"correctness_risk":"high","formal_verification":"none","parameter_count":0,"one_line_summary":"A report that the existing proof of MLS for unitary groups has a gap, plus MLS constructions for some projective special unitary groups under primality conditions.","lead":"This paper claims that a 2015 proof of the minimal logarithmic signature conjecture for unitary groups is invalid, and proves the conjecture for some small projective special unitary groups under primality conditions. If the gap report is correct, the conjecture remains open for unitary groups; if not, the paper's central negative finding collapses.","discovery_kind":"extension","skeptic_critique":{"model":"deepseek-v4-flash","headline":"The gap report against [10] is not anchored: the object W'_0 = GF(q^{2m}) cannot be a subfield of GF(q^{2n}) when n=2m+1, so the Section 3 contradiction may refute a straw man, not Hong et al.","rationale":"The reader's REJECT is well-founded: the Section 3 gap report is the paper's headline contribution, and its validity is not established because the reconstructed spread object is not shown to exist in V. My concrete test is deliberately comparative: it verifies whether the description matches [10] before accepting the point-count contradiction. I did not base the verdict on the paper's positive theorems; however, I note that those also contain independent unproven steps, e.g., the coprimality assertion in Theorem 4.4(1) is arithmetically false for q=7, which would further block the odd-q existence proofs. Since at least the central negative claim is unsupported as written, the reader's REJECT should stand unchanged.","tokens_in":10836,"tokens_out":22101,"duration_ms":225482,"concrete_test":"Obtain Hong, Wang, Yang, Des. Codes Cryptogr. 77 (2015) 179–191 and check the exact definition of W'_0 for n=2m+1: is it GF(q^{2m}) as a subfield, GF(q^n) as the classical spread, or an m-dimensional GF(q^2)-subspace? Also check the exponent in W'_i and the group (GU_n(q) or SU_n(q)) for which the Singer cycle order q^n+1 is claimed. If the actual W'_0 is GF(q^n) or an abstract m-space and the correct order is (q^n+1)/(q+1), recompute the product |A|·|B|: if it equals (q^n+1)(q^{n-1}-1)/(q^2-1), the Section 3 contradiction disappears.","verdict_should_be":"UNCHANGED","load_bearing_attack":"The paper's load-bearing negative claim—that [10]'s proof is wrong and the MLS conjecture for unitary groups is still open—depends entirely on the reconstruction in Section 3 of the alleged spread S3. The text states that [10] fixed n=2m+1, set W'_0=GF(q^{2m}) and W'_i=W'_0 α^{i(q^m-1)} inside V=GF(q^{2n}). As a field, GF(q^{2m}) exists abstractly, but it is not a subset of GF(q^{2n}) unless 2m divides 2n; for n=2m+1 this divisibility fails for m>1. Consequently the computations f(1,1)=n, the intersection W'_i∩W'_j, and the point-count contradiction in §3 are not valid calculations on any well-defined subset of V unless W'_0 is additionally specified as a subfield embedding, which is impossible in general. The paper also attributes to [10] the Singer-cycle order q^n+1 without stating whether those orders are in GU_n(q) or SU_n(q), where the correct orders differ by a factor q+1. If [10] in fact used W'_0=GF(q^n) (the natural subfield of V) or an m-dimensional GF(q^2)-subspace, and used the Singer order appropriate to the group it treated, the contradiction dissolves. The paper does not quote [10]'s definitions verbatim, so the central objection may be directed at a misreconstruction.","agreement_with_reader":"agree"},"referee_report":{"model":"deepseek-v4-flash","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).","tokens_in":11119,"tokens_out":14735,"duration_ms":132914,"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":[{"comment":"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.","section":"Section 3"},{"comment":"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.","section":"Theorems 4.3 and 4.5"},{"comment":"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.","section":"Theorem 4.4(1)"},{"comment":"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.","section":"Theorem 4.1 proof"}],"minor_comments":[{"comment":"The statement 'q = 2 n' should read 'q = 2^n'; the proof uses the latter.","section":"Theorem 4.2 statement"},{"comment":"Theorem 4.3 has no numbered parts, but Section 5 refers to 'Theorem 4.3(2)'.","section":"Section 5"},{"comment":"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":"Theorem 4.1"},{"comment":"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.","section":"Section 3, item (1)"}],"recommendation":"reject","confidential_remarks":"The paper's challenge to [10] would be significant if correct, but the reconstruction in Section 3 is internally inconsistent as written. I would encourage the authors to obtain and quote [10]'s actual spread construction before resubmitting. There is also a recurring pattern of relying on the authors' own earlier paper [26] for transfer theorems; this is not circular in the proofs of the new constructions, but independent verification of those transfers would strengthen the paper."},"author_rebuttal":null,"desk_editor":{"model":"deepseek-v4-flash","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.\n\nThe 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.\n\nThe 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.\n\nBottom 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.","headline":"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.","tokens_in":11670,"tokens_out":9752,"would_cite":false,"duration_ms":91936,"reading_group":"maybe","serious_thinker":"no","would_accept_peer_review":true},"rs_alignment":null,"lean_confirmation":null,"pith_extraction":{"msc":["20D08","94A60"],"pacs":[],"model":"deepseek-v4-flash","headline":"The paper argues that a published proof of the MLS conjecture for unitary groups has a gap, so the conjecture remains open.","keywords":["minimal logarithmic signatures","MLS conjecture","finite simple groups","unitary groups","Singer cycles","isotropic points","spreads"],"falsifier":"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.","tokens_in":10600,"feed_emoji":"🧩","tokens_out":18542,"duration_ms":160032,"temperature":0.7,"pith_summary":"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.","feed_headline":"Unitary-group MLS proof has a gap; conjecture stays open","feed_subtitle":"New factorizations cover some PSU(3,q) and PSU(4,q) cases, but the general question remains open.","key_machinery":"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.","core_discovery":"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 $PSU_4(q)$.","pith_inferences":["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."],"forward_implications":["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."],"supporting_citations":[{"why":"This is the paper whose main theorem is challenged; its spread construction and Singer-cycle orders are the objects of the gap report.","marker":"[10]"},{"why":"It supplies the sharp-transitivity criterion used to build a logarithmic signature from a spread and a point stabilizer.","marker":"[27]"},{"why":"It is the source of the unitary Singer-cycle subgroup orders that the paper uses to expose the order mismatch in [10].","marker":"[1]"},{"why":"It is the authors' earlier survey whose unitary-group cases relied on [10] and are now called into question.","marker":"[26]"},{"why":"It provides the fact that solvable groups have MLS, used in the positive constructions for PSU3(q).","marker":"[4]"},{"why":"It supplies the double-coset MLS criterion and previous low-order unitary cases.","marker":"[13]"},{"why":"It gives the factorization of PSU_{2n}(q) used in the reduction to SU_{2n-1}(q).","marker":"[15]"},{"why":"It supplies the subgroup structure of PSU3(q) used to choose the two coprime-order subgroups in the odd case.","marker":"[24]"},{"why":"It shows 2-transitivity of PSU3(q), which supports the transporter blocks when q+1=2p.","marker":"[29]"}],"fun_headline_variants":["Gap exposed in unitary MLS proof; conjecture stays open","Unitary MLS: proof gap blocks full resolution, partial wins","MLS conjecture open: unitary proof flawed, PSU(3,4) cases done","Some PSU groups get MLS, but gap leaves conjecture open"],"cache_read_input_tokens":3200,"weakest_assumption_plain":"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.","fun_headline_variants_meta":{"raw":{"variants":["Gap exposed in unitary MLS proof; conjecture stays open","Unitary MLS: proof gap blocks full resolution, partial wins","MLS conjecture open: unitary proof flawed, PSU(3,4) cases done","Some PSU groups get MLS, but gap leaves conjecture open"]},"model":"deepseek-v4-flash","effort":"low","cost_usd":0.000239,"raw_usage":{"total_tokens":1519,"prompt_tokens":951,"completion_tokens":568,"prompt_tokens_details":{"cached_tokens":384},"prompt_cache_hit_tokens":384,"prompt_cache_miss_tokens":567,"completion_tokens_details":{"reasoning_tokens":492}},"tokens_in":567,"tokens_out":568,"duration_ms":6183,"temperature":1.0,"reasoning_tokens":492,"cache_read_input_tokens":384,"cache_creation_input_tokens":0},"cache_creation_input_tokens":0},"created_at":"2026-08-14T13:52:56.308061+00:00","model_set":{"reader":"deepseek-v4-flash"},"falsifier":"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.","supporting_citations":[{"cited_title":null,"cited_arxiv_id":null,"evidence_quote":"This is the paper whose main theorem is challenged; its spread construction and Singer-cycle orders are the objects of the gap report."},{"cited_title":"Singhi and N","cited_arxiv_id":null,"evidence_quote":"It supplies the sharp-transitivity criterion used to build a logarithmic signature from a spread and a point stabilizer."},{"cited_title":"Babai, P","cited_arxiv_id":null,"evidence_quote":"It is the source of the unitary Singer-cycle subgroup orders that the paper uses to expose the order mismatch in [10]."},{"cited_title":null,"cited_arxiv_id":null,"evidence_quote":"It is the authors' earlier survey whose unitary-group cases relied on [10] and are now called into question."},{"cited_title":null,"cited_arxiv_id":null,"evidence_quote":"It provides the fact that solvable groups have MLS, used in the positive constructions for PSU3(q)."},{"cited_title":"Lempken and T","cited_arxiv_id":null,"evidence_quote":"It supplies the double-coset MLS criterion and previous low-order unitary cases."},{"cited_title":"Factorizations of almost simple groups with a solvable factor, and Cayley graphs of solvable groups","cited_arxiv_id":"1408.0350","evidence_quote":"It gives the factorization of PSU_{2n}(q) used in the reduction to SU_{2n-1}(q)."},{"cited_title":null,"cited_arxiv_id":null,"evidence_quote":"It supplies the subgroup structure of PSU3(q) used to choose the two coprime-order subgroups in the odd case."},{"cited_title":"Suzuki, A characterization of the 3 −dimensional projective unitary group over a finite field of odd characteristic, J","cited_arxiv_id":null,"evidence_quote":"It shows 2-transitivity of PSU3(q), which supports the transporter blocks when q+1=2p."}],"review_version":1}