{"status":"ok","message-type":"work","message-version":"1.0.0","message":{"indexed":{"date-parts":[[2025,12,31]],"date-time":"2025-12-31T01:01:42Z","timestamp":1767142902235,"version":"build-2238731810"},"reference-count":41,"publisher":"Springer Science and Business Media LLC","issue":"3-4","license":[{"start":{"date-parts":[[2023,9,1]],"date-time":"2023-09-01T00:00:00Z","timestamp":1693526400000},"content-version":"tdm","delay-in-days":0,"URL":"https:\/\/creativecommons.org\/licenses\/by\/4.0"},{"start":{"date-parts":[[2023,9,1]],"date-time":"2023-09-01T00:00:00Z","timestamp":1693526400000},"content-version":"vor","delay-in-days":0,"URL":"https:\/\/creativecommons.org\/licenses\/by\/4.0"}],"content-domain":{"domain":["link.springer.com"],"crossmark-restriction":false},"short-container-title":["Math.Comput.Sci."],"published-print":{"date-parts":[[2023,12]]},"abstract":"Abstract<\/jats:title>\n \n In arithmetic and algebraic geometry, superspecial (s.sp. for short) curves are one of the most important objects to be studied, with applications to cryptography and coding theory. If\n \n \n $$g \\ge 4$$<\/jats:tex-math>\n \n \n g<\/mml:mi>\n \u2265<\/mml:mo>\n 4<\/mml:mn>\n <\/mml:mrow>\n <\/mml:math>\n <\/jats:alternatives>\n <\/jats:inline-formula>\n , it is not even known whether there exists such a curve of genus\n g<\/jats:italic>\n in general characteristic\n \n \n $$p > 0$$<\/jats:tex-math>\n \n \n p<\/mml:mi>\n ><\/mml:mo>\n 0<\/mml:mn>\n <\/mml:mrow>\n <\/mml:math>\n <\/jats:alternatives>\n <\/jats:inline-formula>\n , and in the case of\n \n \n $$g=4$$<\/jats:tex-math>\n \n \n g<\/mml:mi>\n =<\/mml:mo>\n 4<\/mml:mn>\n <\/mml:mrow>\n <\/mml:math>\n <\/jats:alternatives>\n <\/jats:inline-formula>\n , several computational approaches to search for those curves have been proposed. In the genus-4 hyperelliptic case, Kudo-Harashita proposed a generic algorithm to enumerate all s.sp. curves, and recently Ohashi-Kudo-Harashita presented an algorithm specific to the case where automorphism group contains the Klein 4-group as a subgroup. In this paper, we propose an algorithm with complexity\n \n \n $${\\tilde{O}}(p^4)$$<\/jats:tex-math>\n \n \n \n O<\/mml:mi>\n ~<\/mml:mo>\n <\/mml:mover>\n \n (<\/mml:mo>\n \n p<\/mml:mi>\n 4<\/mml:mn>\n <\/mml:msup>\n )<\/mml:mo>\n <\/mml:mrow>\n <\/mml:mrow>\n <\/mml:math>\n <\/jats:alternatives>\n <\/jats:inline-formula>\n in theory but\n \n \n $${\\tilde{O}}(p^3)$$<\/jats:tex-math>\n \n \n \n O<\/mml:mi>\n ~<\/mml:mo>\n <\/mml:mover>\n \n (<\/mml:mo>\n \n p<\/mml:mi>\n 3<\/mml:mn>\n <\/mml:msup>\n )<\/mml:mo>\n <\/mml:mrow>\n <\/mml:mrow>\n <\/mml:math>\n <\/jats:alternatives>\n <\/jats:inline-formula>\n in practice to enumerate s.sp. hyperelliptic curves of genus 4 with automorphism group containing the cyclic group of order 6. By executing the algorithm over Magma, we enumerate those curves for\n p<\/jats:italic>\n up to 1000. We also succeeded in finding a s.sp. hyperelliptic curve of genus 4 in every\n p<\/jats:italic>\n with\n \n \n $$p \\equiv 2 \\pmod {3}$$<\/jats:tex-math>\n \n \n p<\/mml:mi>\n \u2261<\/mml:mo>\n 2<\/mml:mn>\n \n (<\/mml:mo>\n mod<\/mml:mo>\n \n 3<\/mml:mn>\n )<\/mml:mo>\n <\/mml:mrow>\n <\/mml:math>\n <\/jats:alternatives>\n <\/jats:inline-formula>\n .\n <\/jats:p>","DOI":"10.1007\/s11786-023-00571-w","type":"journal-article","created":{"date-parts":[[2023,8,31]],"date-time":"2023-08-31T23:31:22Z","timestamp":1693524682000},"update-policy":"https:\/\/doi.org\/10.1007\/springer_crossmark_policy","source":"Crossref","is-referenced-by-count":2,"title":["Efficient Search for Superspecial Hyperelliptic Curves of Genus Four with Automorphism Group Containing $${\\textbf{C}}_6$$"],"prefix":"10.1007","volume":"17","author":[{"given":"Momonari","family":"Kudo","sequence":"first","affiliation":[]},{"given":"Tasuku","family":"Nakagawa","sequence":"additional","affiliation":[]},{"given":"Tsuyoshi","family":"Takagi","sequence":"additional","affiliation":[]}],"member":"297","published-online":{"date-parts":[[2023,9,1]]},"reference":[{"key":"571_CR1","doi-asserted-by":"publisher","first-page":"235","DOI":"10.1006\/jsco.1996.0125","volume":"24","author":"W Bosma","year":"1997","unstructured":"Bosma, W., Cannon, J., Playoust, C.: The Magma algebra system. I. The user language. J. Symb. Comput. 24, 235\u2013265 (1997)","journal-title":"J. Symb. Comput."},{"key":"571_CR2","doi-asserted-by":"crossref","unstructured":"Bostan, A., Gaudry, P., Schost, \u00c9.: Linear Recurrences with Polynomial Coefficients and Computation of the Cartier-Manin Operator on Hyperelliptic Curves, In: G. L. Mullen, A. Poli and H. Stichtenoth (eds.), Finite Fields and Applications. Fq 2003. LNCS, 2948, Springer, Berlin-Heidelberg (2004)","DOI":"10.1007\/978-3-540-24633-6_4"},{"key":"571_CR3","doi-asserted-by":"crossref","unstructured":"Beauville, A.: Finite subgroups of $$\\rm PGL_2(K)$$, In: Vector bundles and complex geometry, volume 522 of Contemp. Math., pp. 23\u201329, Amer. Math. Soc., Providence, RI (2010)","DOI":"10.1090\/conm\/522\/10289"},{"key":"571_CR4","doi-asserted-by":"publisher","first-page":"177","DOI":"10.1515\/JMC.2009.009","volume":"3","author":"L Bettale","year":"2009","unstructured":"Bettale, L., Faugere, J.-C., Perret, L.: Hybrid approach for solving multivariate systems over finite fields. J. Math. Cryptol. 3, 177\u2013197 (2009)","journal-title":"J. Math. Cryptol."},{"issue":"1","key":"571_CR5","doi-asserted-by":"publisher","first-page":"268","DOI":"10.1515\/jmc-2019-0021","volume":"14","author":"W Castryck","year":"2020","unstructured":"Castryck, W., Decru, T., Smith, B.: Hash functions from superspecial genus 2 curves using Richelot isogenies. J. Math. Cryptol. 14(1), 268\u2013292 (2020)","journal-title":"J. Math. Cryptol."},{"issue":"1","key":"571_CR6","doi-asserted-by":"publisher","first-page":"197","DOI":"10.1007\/BF02940746","volume":"14","author":"M Deuring","year":"1941","unstructured":"Deuring, M.: Die Typen der Multiplikatorenringe elliptischer Funktionenk\u00f6rper. Abh. Math. Sem. Univ. Hamburg 14(1), 197\u2013272 (1941)","journal-title":"Abh. Math. Sem. Univ. Hamburg"},{"key":"571_CR7","doi-asserted-by":"publisher","first-page":"102","DOI":"10.1007\/BF01181088","volume":"43","author":"M Eichler","year":"1938","unstructured":"Eichler, M.: \u00dcber die Idealklassenzahl total definiter Quaternionenalgebren. Math. Z. 43, 102\u2013109 (1938)","journal-title":"Math. Z."},{"key":"571_CR8","doi-asserted-by":"publisher","first-page":"151","DOI":"10.7146\/math.scand.a-12178","volume":"60","author":"T Ekedahl","year":"1987","unstructured":"Ekedahl, T.: On supersingular curves and abelian varieties. Math. Scand. 60, 151\u2013178 (1987)","journal-title":"Math. Scand."},{"key":"571_CR9","doi-asserted-by":"publisher","first-page":"1","DOI":"10.1016\/j.jalgebra.2010.09.046","volume":"327","author":"EA Elkin","year":"2011","unstructured":"Elkin, E.A.: The rank of the Cartier operator on cyclic covers of the projective line. J. Algebra 327, 1\u201312 (2011)","journal-title":"J. Algebra"},{"key":"571_CR10","unstructured":"Faber, X.: Finite $$p$$-irregular subgroups of $$\\rm PGL_2(k)$$, arXiv:1112.1999 (2021)"},{"issue":"4","key":"571_CR11","doi-asserted-by":"publisher","first-page":"36","DOI":"10.1145\/274888.274890","volume":"31","author":"P Flajolet","year":"1997","unstructured":"Flajolet, P., Salvy, B.: The SIGSAM challenges: symbolic asymptotics in practice. SIGSAM Bull. 31(4), 36\u201347 (1997)","journal-title":"SIGSAM Bull."},{"key":"571_CR12","doi-asserted-by":"publisher","first-page":"102","DOI":"10.1112\/S1461157000000917","volume":"8","author":"J Gutierrez","year":"2005","unstructured":"Gutierrez, J., Shaska, T.: Hyperelliptic curves with extra involutions. LMS J. Comput. Math. 8, 102\u2013115 (2005)","journal-title":"LMS J. Comput. Math."},{"key":"571_CR13","doi-asserted-by":"crossref","unstructured":"Gutierrez, J., Sevilla, D., Shaska, T.: Hyperelliptic curves of genus 3 with prescribed automorphism group, In: Computational Aspects of Algebraic Curves, In: Lecture Notes Ser. Comput., vol. 13, pp. 109\u2013123, World Sci. Publ., Hackensack, NJ (2003)","DOI":"10.1142\/9789812701640_0009"},{"key":"571_CR14","doi-asserted-by":"publisher","first-page":"257","DOI":"10.1112\/S1461157014000187","volume":"17","author":"D Harvey","year":"2014","unstructured":"Harvey, D., Sutherland, A.V.: Computing Hasse\u2013Witt matrices of hyperelliptic curves in average polynomial time. LMS J. Comput. Math. 17, 257\u2013273 (2014)","journal-title":"LMS J. Comput. Math."},{"issue":"10","key":"571_CR15","doi-asserted-by":"publisher","first-page":"490","DOI":"10.3792\/pjaa.59.490","volume":"59","author":"K Hashimoto","year":"1983","unstructured":"Hashimoto, K.: Class numbers of positive definite ternary quaternion Hermitian forms. Proc. Jpn. Acad. Ser. A Math. Sci. 59(10), 490\u2013493 (1983)","journal-title":"Proc. Jpn. Acad. Ser. A Math. Sci."},{"issue":"3","key":"571_CR16","first-page":"695","volume":"28","author":"K Hashimoto","year":"1982","unstructured":"Hashimoto, K., Ibukiyama, T.: On class numbers of positive definite binary quaternion Hermitian forms II. J. Fac. Sci. Univ. Tokyo Sect. IA Math. 28(3), 695\u2013699 (1982)","journal-title":"J. Fac. Sci. Univ. Tokyo Sect. IA Math."},{"key":"571_CR17","unstructured":"https:\/\/sites.google.com\/view\/m-kudo-official-website\/english\/code\/hyp"},{"key":"571_CR18","first-page":"127","volume":"57","author":"T Ibukiyama","year":"1986","unstructured":"Ibukiyama, T., Katsura, T., Oort, F.: Supersingular curves of genus two and class numbers. Compos. Math. 57, 127\u2013152 (1986)","journal-title":"Compos. Math."},{"key":"571_CR19","doi-asserted-by":"publisher","first-page":"612","DOI":"10.2307\/1970233","volume":"72","author":"J Igusa","year":"1960","unstructured":"Igusa, J.: Arithmetic variety of moduli for genus two. Ann. Math. 72, 612\u2013649 (1960)","journal-title":"Ann. Math."},{"key":"571_CR20","doi-asserted-by":"publisher","first-page":"17","DOI":"10.14495\/jsiaml.2.17","volume":"2","author":"H Komoto","year":"2010","unstructured":"Komoto, H., Kozaki, S., Matsuo, K.: Improvements in the computation of the Hasse\u2013Witt matrix. JSIAM Lett. 2, 17\u201320 (2010)","journal-title":"JSIAM Lett."},{"key":"571_CR21","doi-asserted-by":"publisher","first-page":"131","DOI":"10.1016\/j.ffa.2016.12.001","volume":"45","author":"M Kudo","year":"2017","unstructured":"Kudo, M., Harashita, S.: Superspecial curves of genus $$4$$ in small characteristic. Finite Fields Appl. 45, 131\u2013169 (2017)","journal-title":"Finite Fields Appl."},{"key":"571_CR22","first-page":"58","volume-title":"Arithmetic of Finite Fields WAIFI 2018, LNCS, 11321.","author":"M Kudo","year":"2018","unstructured":"Kudo, M., Harashita, S.: Superspecial Hyperelliptic Curves of Genus 4 over Small Finite Fields. In: Budaghyan, L., Rodriguez-Henriquez, F. (eds.) Arithmetic of Finite Fields WAIFI 2018, LNCS, 11321., pp. 58\u201373. Springer, Cham (2018)"},{"key":"571_CR23","unstructured":"Kudo, M., Harashita, S.: Algorithmic study of superspecial hyperelliptic curves over finite fields, Commentarii Mathematici Universitatis Sancti Pauli, Vol. 70, 49\u201364 (2022)"},{"issue":"1","key":"571_CR24","doi-asserted-by":"publisher","first-page":"259","DOI":"10.3836\/tjm\/1502179310","volume":"43","author":"M Kudo","year":"2020","unstructured":"Kudo, M., Harashita, S.: Computational approach to enumerate non-hyperelliptic superspecial curves of genus 4. Tokyo J. Math. 43(1), 259\u2013278 (2020)","journal-title":"Tokyo J. Math."},{"key":"571_CR25","doi-asserted-by":"crossref","unstructured":"Kudo, M., Harashita, S., Howe, E. W.: Algorithms to enumerate superspecial Howe curves of genus four, In: Proceedings of Fourteenth Algorithmic Number Theory Symposium (ANTS-XIV), Open Book Series, Vol. 4, No. 1, 301\u2013316 (2020)","DOI":"10.2140\/obs.2020.4.301"},{"key":"571_CR26","doi-asserted-by":"crossref","unstructured":"Kudo, M.: Appendix A. Automorphism groups of hyperelliptic curves of genus four, In: Kudo, M., Nakagawa, T., and Takagi, T.: Efficient search for superspecial hyperelliptic curves of genus four with automorphism group containing $${\\mathbb{Z}}_6$$, arXiv: 2210.14822 [math.AG], 2022 (the preprint version of this article)","DOI":"10.1007\/s11786-023-00571-w"},{"key":"571_CR27","unstructured":"Lachaud, G.: Sommes d\u2019Eisenstein et nombre de points de certaines courbes alg\u2019ebriques sur les corps finis, C.R. Acad. Sci. Paris 305, S\u2019erie I (1987), 729\u2013732"},{"key":"571_CR28","doi-asserted-by":"publisher","first-page":"595","DOI":"10.1016\/j.jalgebra.2012.07.054","volume":"372","author":"R Lercier","year":"2012","unstructured":"Lercier, R., Ritzenthaler, C.: Hyperelliptic curves and their invariants: geometric, arithmetic and algorithmic aspects. J. Algebra 372, 595\u2013636 (2012)","journal-title":"J. Algebra"},{"key":"571_CR29","doi-asserted-by":"publisher","first-page":"128","DOI":"10.1112\/S146115701400031X","volume":"17","author":"R Lercier","year":"2014","unstructured":"Lercier, R., Ritzenthaler, C., Rovetta, F., Sijsling, J.: Parametrizing the moduli space of curves and applications to smooth plane quartics over finite fields. LMS J. Comput. Math. 17, 128\u2013147 (2014)","journal-title":"LMS J. Comput. Math."},{"key":"571_CR30","doi-asserted-by":"crossref","unstructured":"Lercier, R., Ritzenthaler, C., Sijsling, J.: Fast computation of isomorphisms of hyperelliptic curves and explicit Galois descent. In: Proceedings of Fourteenth Algorithmic Number Theory Symposium (ANTS-X), Open Book Series, Vol. 1(1), pp. 463\u2013486 (2013)","DOI":"10.2140\/obs.2013.1.463"},{"key":"571_CR31","unstructured":"Lercier, R., Sijsling, J.,Ritzenthaler, C.: Functionalities for genus $$2$$ and $$3$$ curves, MEGA (2021), arXiv:2102.04372"},{"key":"571_CR32","unstructured":"Moriya, T., Kudo, M.: Some explicit arithmetics on curves of genus three and their applications, arXiv: 2209.02926 [math.AG] (2022)"},{"issue":"14","key":"571_CR33","doi-asserted-by":"publisher","first-page":"369","DOI":"10.24033\/asens.1411","volume":"4","author":"NO Nygaard","year":"1981","unstructured":"Nygaard, N.O.: Slopes of powers of Frobenius on crystalline cohomology. Ann. Sci. \u00c9c. Norm. Sup\u00e9r. 4(14), 369\u2013401 (1981)","journal-title":"Ann. Sci. \u00c9c. Norm. Sup\u00e9r."},{"key":"571_CR34","doi-asserted-by":"crossref","unstructured":"Ohashi, R., Kudo, M., Harashita, S.: Fast enumeration of superspecial hyperelliptic curves of genus 4 with automorphism group $$V_4$$, to appear in Proceedings of WAIFI2022, (2022)","DOI":"10.1007\/978-3-031-22944-2_6"},{"key":"571_CR35","doi-asserted-by":"publisher","first-page":"157","DOI":"10.1007\/BF01109838","volume":"117","author":"P Roquette","year":"1970","unstructured":"Roquette, P.: Absch\u00e4tzung der Automorphismenanzahl von Funktionenk\u00f6rpern bei Primzahlcharakteristik. Math. Z. 117, 157\u2013163 (1970)","journal-title":"Math. Z."},{"key":"571_CR36","doi-asserted-by":"crossref","unstructured":"Shaska, T.: Determining the automorphism group of a hyperelliptic curve, In: Proceedings of the 2003 International Symposium on Symbolic and Algebraic Computation (ISSAC\u201903), August (2003), pp. 248\u2013254","DOI":"10.1145\/860854.860904"},{"issue":"1","key":"571_CR37","doi-asserted-by":"publisher","first-page":"75","DOI":"10.1142\/S0219498804000745","volume":"3","author":"T Shaska","year":"2004","unstructured":"Shaska, T.: Some special families of hyperelliptic curves. J. Algebra Appl. 3(1), 75\u201389 (2004)","journal-title":"J. Algebra Appl."},{"key":"571_CR38","doi-asserted-by":"publisher","first-page":"527","DOI":"10.1007\/BF01228251","volume":"24","author":"H Stichtenoth","year":"1973","unstructured":"Stichtenoth, H.: \u00dcber die Automorphismengruppe eines algebraischen Funktionenk\u00f6rpers von Primzahlcharakteristik. I. Eine Absch\u00e4tzung der Ordnung der Automorphismengruppe. Arch. Math. 24, 527\u2013544 (1973)","journal-title":"Arch. Math."},{"key":"571_CR39","doi-asserted-by":"publisher","first-page":"1013","DOI":"10.1016\/j.ffa.2012.07.002","volume":"18","author":"S Tafazolian","year":"2012","unstructured":"Tafazolian, S.: A note on certain maximal hyperelliptic curves. Finite Fields Their Appl. 18, 1013\u20131016 (2012)","journal-title":"Finite Fields Their Appl."},{"key":"571_CR40","doi-asserted-by":"publisher","first-page":"101499","DOI":"10.1016\/j.jco.2020.101499","volume":"62","author":"J van der Hoeven","year":"2021","unstructured":"van der Hoeven, J., Lecerf, G.: Fast computation of generic bivariate resultants. J. Complex. 62, 101499 (2021)","journal-title":"J. Complex."},{"key":"571_CR41","doi-asserted-by":"publisher","first-page":"378","DOI":"10.1016\/0021-8693(78)90247-8","volume":"52","author":"N Yui","year":"1978","unstructured":"Yui, N.: On the Jacobian varieties of hyperelliptic curves over fields of characteristic $$p \\,>\\, 2$$. J. Algebra 52, 378\u2013410 (1978)","journal-title":"J. Algebra"}],"updated-by":[{"DOI":"10.1007\/s11786-023-00573-8","type":"correction","label":"Correction","source":"publisher","updated":{"date-parts":[[2023,11,24]],"date-time":"2023-11-24T00:00:00Z","timestamp":1700784000000}}],"container-title":["Mathematics in Computer Science"],"original-title":[],"language":"en","link":[{"URL":"https:\/\/link.springer.com\/content\/pdf\/10.1007\/s11786-023-00571-w.pdf","content-type":"application\/pdf","content-version":"vor","intended-application":"text-mining"},{"URL":"https:\/\/link.springer.com\/article\/10.1007\/s11786-023-00571-w\/fulltext.html","content-type":"text\/html","content-version":"vor","intended-application":"text-mining"},{"URL":"https:\/\/link.springer.com\/content\/pdf\/10.1007\/s11786-023-00571-w.pdf","content-type":"application\/pdf","content-version":"vor","intended-application":"similarity-checking"}],"deposited":{"date-parts":[[2024,10,27]],"date-time":"2024-10-27T05:01:35Z","timestamp":1730005295000},"score":1,"resource":{"primary":{"URL":"https:\/\/link.springer.com\/10.1007\/s11786-023-00571-w"}},"subtitle":[],"short-title":[],"issued":{"date-parts":[[2023,9,1]]},"references-count":41,"journal-issue":{"issue":"3-4","published-print":{"date-parts":[[2023,12]]}},"alternative-id":["571"],"URL":"https:\/\/doi.org\/10.1007\/s11786-023-00571-w","relation":{},"ISSN":["1661-8270","1661-8289"],"issn-type":[{"value":"1661-8270","type":"print"},{"value":"1661-8289","type":"electronic"}],"subject":[],"published":{"date-parts":[[2023,9,1]]},"assertion":[{"value":"13 October 2022","order":1,"name":"received","label":"Received","group":{"name":"ArticleHistory","label":"Article History"}},{"value":"10 May 2023","order":2,"name":"accepted","label":"Accepted","group":{"name":"ArticleHistory","label":"Article History"}},{"value":"1 September 2023","order":3,"name":"first_online","label":"First Online","group":{"name":"ArticleHistory","label":"Article History"}},{"value":"25 November 2023","order":4,"name":"change_date","label":"Change Date","group":{"name":"ArticleHistory","label":"Article History"}},{"value":"Update","order":5,"name":"change_type","label":"Change Type","group":{"name":"ArticleHistory","label":"Article History"}},{"value":"The original online version of this article was revised: to update the additional corrections.","order":6,"name":"change_details","label":"Change Details","group":{"name":"ArticleHistory","label":"Article History"}},{"value":"24 November 2023","order":7,"name":"change_date","label":"Change Date","group":{"name":"ArticleHistory","label":"Article History"}},{"value":"Correction","order":8,"name":"change_type","label":"Change Type","group":{"name":"ArticleHistory","label":"Article History"}},{"value":"A Correction to this paper has been published:","order":9,"name":"change_details","label":"Change Details","group":{"name":"ArticleHistory","label":"Article History"}},{"value":"https:\/\/doi.org\/10.1007\/s11786-023-00573-8","URL":"https:\/\/doi.org\/10.1007\/s11786-023-00573-8","order":10,"name":"change_details","label":"Change Details","group":{"name":"ArticleHistory","label":"Article History"}}],"article-number":"21"}}