{"id":"98403e00-946a-47f3-8db1-8b7d8c3783a3","arxiv_id":"2606.11967","paper_version":2,"verdict":"ACCEPT","confidence":"LOW","novelty_score":6.0,"correctness_risk":"unknown","formal_verification":"none","parameter_count":0,"one_line_summary":"Structured Groebner basis search in a 40-dimensional self-equivalence subspace of functions over F_{2^8} yields 566 quadratic APN functions in six classes, four of which are absent from prior databases of millions of instances.","lead":"This paper describes a computational search that found 566 quadratic APN functions over the field with 256 elements, including four new CCZ-equivalence classes. A smart generalist might read it for new candidates that could strengthen cryptographic S-box designs against differential attacks.","discovery_kind":"new_application","skeptic_critique":{"model":"grok-4.3","headline":"No significant objection identified","rationale":"The reader's weakest_assumption correctly isolates the database-signature step, yet that step is internally sound: mismatch of an invariant certifies inequivalence regardless of whether the invariant separates all classes. The low reader confidence appears to stem from reliance on external data, but the paper supplies artifacts and the two recovered Gold classes provide an internal consistency check. No adjustment to ACCEPT is warranted.","tokens_in":1937,"tokens_out":307,"duration_ms":29614,"concrete_test":"Execute the public verification scripts on the released dataset to recompute ortho-derivative signatures for the 500 functions and confirm none match any signature obtained from the Beierle et al. database entries.","verdict_should_be":"UNCHANGED","load_bearing_attack":"The paper's argument for four new CCZ-inequivalence classes rests on finding 500 quadratic APN functions whose ortho-derivative signatures do not appear among those of the 3,775,599 functions in the Beierle et al. 2025 database (or the earlier compilation). Yoshiara's theorem equates CCZ-equivalence with EA-equivalence for quadratic APN functions, and the ortho-derivative is an EA-invariant; therefore a signature mismatch rigorously implies the new functions lie outside all previously enumerated classes. The method recovered the known Gold functions x^3 and x^9, confirming correctness of the enumeration pipeline inside V_A. No gap in the invariant application or database comparison is evident.","agreement_with_reader":"disagree"},"referee_report":{"model":"grok-4.3","summary":"The manuscript describes a two-phase computational search for quadratic APN functions over F_{2^8} inside the 40-dimensional self-equivalence subspace V_A = {F : F ∘ A = A ∘ F} for a fixed linear A of order 5. Phase 1 uses explicit RREF sampling to locate APN centers; phase 2 applies Gröbner-basis enumeration in Magma over 24-dimensional hyperplanes through each center. From 428 hyperplanes (0.65 % of the 65 536 total), 566 quadratic APN functions are obtained and partitioned into six CCZ-classes via the ortho-derivative invariant. Four classes (500 functions) have signatures absent from the Beierle et al. 2025 database (3 775 599 entries) and the pre-2020 compilation (12 921 entries); by Yoshiara’s theorem these are therefore new CCZ-inequivalence classes. The remaining two classes recover the Gold functions x^3 and x^9, confirming correctness of the pipeline. All code, data and verification scripts are public.","tokens_in":2077,"tokens_out":553,"duration_ms":15702,"significance":"If the enumeration and database-comparison claims hold, the work adds four new CCZ-inequivalence classes of quadratic APN functions in dimension 8, a concrete advance in the classification problem that is directly relevant to the design of optimal cryptographic S-boxes. The self-equivalence subspace restriction together with the hybrid sampling-plus-Gröbner strategy is a novel and reproducible method that succeeded where an earlier recursive tree search on the same subspace reported none. Public release of the complete dataset, Magma scripts and verification code constitutes a strong reproducibility asset.","major_comments":[],"minor_comments":[{"comment":"Abstract: the figure 65 536 for the total number of hyperplanes in the 40-dimensional F_2-space V_A is stated without derivation; a one-sentence parenthetical (e.g., 2^{16} because each hyperplane is a codimension-16 affine subspace) would aid readers.","section":"Abstract"},{"comment":"Abstract: the phrase “class index 22 in the taxonomy of Beierle, Brinkmann, and Leander” should be accompanied by the precise bibliographic entry in the reference list.","section":"Abstract"},{"comment":"The manuscript repeatedly uses the abbreviation “CCZ” without an initial expansion; while standard in the field, a parenthetical on first use improves accessibility.","section":null}],"recommendation":"accept","confidential_remarks":null},"author_rebuttal":{"model":"grok-4.3","summary":"We thank the referee for the positive and accurate summary of the manuscript, for highlighting its significance in advancing the classification of quadratic APN functions, and for the recommendation to accept. The recognition of the novel search strategy, the rigorous use of the ortho-derivative invariant together with Yoshiara’s theorem, and the value of the public dataset is appreciated.","responses":[],"tokens_in":1572,"tokens_out":88,"duration_ms":6686,"standing_objections":[]},"desk_editor":{"model":"grok-4.3","letter":"This paper turns up four new CCZ-inequivalence classes of quadratic APN functions over F_2^8. The authors restrict to the 40-dimensional subspace V_A fixed by a specific order-5 linear map, sample to locate APN centers, then run Groebner bases over 428 hyperplanes to list all quadratic APN maps in those slices. They recover the two known Gold functions, which checks the pipeline, and the remaining 500 functions have ortho-derivative signatures absent from both the 3.7-million-entry Beierle et al. 2025 database and the earlier 12k compilation. Yoshiara's theorem plus the invariant makes the mismatch a solid certificate of newness.\n\nThe work is narrow but honest. The sampling covers only 0.65 percent of the hyperplanes, so it is not an exhaustive search of V_A, yet the method is explicit, the code is public, and the new instances are concrete. No free parameters or circular definitions appear; the results rest on direct enumeration and external comparison.\n\nThe soft spot is scope: the subspace was chosen because earlier tree searches had reported it empty, and the paper does not explain why this particular A was picked or whether other subspaces might yield more. That is a minor limitation for a computational note, not a flaw in the claims that are made.\n\nThe paper is for specialists who build or classify APN functions for cryptography. It supplies verifiable new examples and a reusable search template, so it deserves referee time even if the broader impact stays inside the subfield.","headline":"Four new CCZ classes of quadratic APN functions in dimension 8, found via Groebner enumeration in a 40-dimensional self-equivalence subspace that prior searches missed, with public code and database verification.","tokens_in":2562,"tokens_out":407,"would_cite":false,"duration_ms":10204,"reading_group":"maybe","serious_thinker":"yes","would_accept_peer_review":true},"rs_alignment":null,"lean_confirmation":null,"pith_extraction":{"msc":[],"pacs":[],"model":"grok-4.3","headline":"A search restricted to the 40-dimensional space of functions commuting with a fixed order-5 linear map yields four new CCZ-inequivalent quadratic APN classes over F_{2^8}.","keywords":["quadratic APN functions","CCZ-equivalence","self-equivalence subspace","Gröbner basis","dimension 8","ortho-derivative invariant","finite fields","cryptography"],"falsifier":"Finding even one quadratic APN function belonging to one of the four reported classes whose ortho-derivative signature matches an entry already present in the Beierle et al. database or the pre-2020 compilation.","tokens_in":2835,"feed_emoji":"","tokens_out":831,"duration_ms":19142,"temperature":0.7,"pith_summary":"The paper restricts attention to the 40-dimensional F_2-linear space V_A of quadratic functions over F_{2^8} that commute with a chosen linear automorphism A of order 5. Inside this space it samples center functions by random search and then uses Gröbner-basis computations to enumerate all APN functions lying in selected 24-dimensional hyperplanes. The resulting 566 functions fall into six CCZ classes; four of those classes (500 functions) produce an ortho-derivative signature absent from two large prior compilations. For quadratic APN functions, Yoshiara's theorem equates CCZ and EA equivalence, so the signature mismatch constitutes a proof that the four classes are new.","feed_headline":"Search in self-equivalence subspace finds four new quadratic APN classes","feed_subtitle":"Signature mismatch with databases of millions of prior examples proves the classes are CCZ-inequivalent.","key_machinery":"The 40-dimensional self-equivalence subspace V_A = {F : F ∘ A = A ∘ F} for a fixed linear A of order 5, combined with the ortho-derivative invariant to certify CCZ-inequivalence.","core_discovery":"Restricting to the self-equivalence subspace V_A of dimension 40 and enumerating APN members of 428 hyperplanes via Gröbner bases produces 566 quadratic APN functions in six CCZ classes; four classes containing 500 functions are absent from the Beierle et al. 2025 database of 3 775 599 quadratic APN functions and from the pre-2020 list of 12 921 instances, thereby proving CCZ-inequivalence by Yoshiara's theorem together with the ortho-derivative invariant.","pith_inferences":["The same subspace-search technique could be applied to other linear automorphisms to locate still more new classes.","The newly found functions may serve as building blocks for S-boxes with improved differential uniformity in cryptographic designs.","Extending the hyperplane enumeration to additional subspaces might reveal whether the total number of quadratic APN classes in dimension 8 is finite or still growing.","Combining the ortho-derivative invariant with further invariants could produce a complete classification of quadratic APN functions up to dimension 8."],"forward_implications":["Four new CCZ-inequivalence classes of quadratic APN functions in dimension 8 are obtained.","The method recovers the known Gold functions x^3 and x^9, confirming that the search pipeline works.","A signature mismatch with the ortho-derivative invariant supplies a rigorous certificate of CCZ-inequivalence for any quadratic APN function.","The 500 functions in the new classes enlarge the known pool of quadratic APN examples in dimension 8.","All data, code, and verification scripts are released for independent checking."],"fun_headline_variants":["Grobner search in self-equivalence subspace finds four new quadratic APN classes","Four new APN classes in self-equivalence subspace V_A via hyperplane search","Database signature mismatch confirms four new quadratic APN classes","500 quadratic APN functions in four CCZ-inequivalent classes from V_A"],"cache_read_input_tokens":2112,"weakest_assumption_plain":"That the absence of a match against the cited databases, when the ortho-derivative invariant is used, rigorously establishes that the four classes are CCZ-inequivalent to every previously known quadratic APN function.","fun_headline_variants_meta":{"raw":{"variants":["Grobner search in self-equivalence subspace finds four new quadratic APN classes","Four new APN classes in self-equivalence subspace V_A via hyperplane search","Database signature mismatch confirms four new quadratic APN classes","500 quadratic APN functions in four CCZ-inequivalent classes from V_A"]},"model":"grok-4.3","cost_usd":0.009051,"raw_usage":{"total_tokens":4163,"prompt_tokens":870,"num_sources_used":0,"completion_tokens":80,"cost_in_usd_ticks":90512000,"prompt_tokens_details":{"text_tokens":870,"audio_tokens":0,"image_tokens":0,"cached_tokens":256},"completion_tokens_details":{"audio_tokens":0,"reasoning_tokens":3213,"accepted_prediction_tokens":0,"rejected_prediction_tokens":0}},"tokens_in":870,"tokens_out":80,"duration_ms":18958,"temperature":1.0,"reasoning_tokens":3213,"cache_read_input_tokens":256,"cache_creation_input_tokens":0},"cache_creation_input_tokens":0},"created_at":"2026-06-27T09:02:06.348988+00:00","model_set":{"reader":"grok-4.3"},"falsifier":"Finding even one quadratic APN function belonging to one of the four reported classes whose ortho-derivative signature matches an entry already present in the Beierle et al. database or the pre-2020 compilation.","supporting_citations":[],"review_version":1}