{"status":"ok","message-type":"work","message-version":"1.0.0","message":{"indexed":{"date-parts":[[2026,10,6]],"date-time":"2026-10-06T08:03:41Z","timestamp":1791273821946,"version":"4.1.0"},"reference-count":38,"publisher":"Wiley","issue":"A","license":[{"start":{"date-parts":[[2014,8,1]],"date-time":"2014-08-01T00:00:00Z","timestamp":1406851200000},"content-version":"unspecified","delay-in-days":212,"URL":"https:\/\/www.cambridge.org\/core\/terms"}],"content-domain":{"domain":[],"crossmark-restriction":false},"short-container-title":["LMS J. Comput. Math."],"published-print":{"date-parts":[[2014]]},"abstract":"<jats:title>Abstract<\/jats:title><jats:p>This paper presents an algorithm to construct cryptographically strong genus <jats:inline-formula><jats:alternatives><jats:inline-graphic xmlns:xlink=\"http:\/\/www.w3.org\/1999\/xlink\" mime-subtype=\"gif\" xlink:type=\"simple\" xlink:href=\"S1461157014000370_inline1\"\/><jats:tex-math>$\\def \\xmlpi #1{}\\def \\mathsfbi #1{\\boldsymbol {\\mathsf {#1}}}\\let \\le =\\leqslant \\let \\leq =\\leqslant \\let \\ge =\\geqslant \\let \\geq =\\geqslant \\def \\Pr {\\mathit {Pr}}\\def \\Fr {\\mathit {Fr}}\\def \\Rey {\\mathit {Re}}2$<\/jats:tex-math><\/jats:alternatives><\/jats:inline-formula> curves and their Kummer surfaces via Rosenhain invariants and related Kummer parameters. The most common version of the complex multiplication (CM) algorithm for constructing cryptographic curves in genus 2 relies on the well-studied Igusa invariants and Mestre\u2019s algorithm for reconstructing the curve. On the other hand, the Rosenhain invariants typically have much smaller height, so computing them requires less precision, and in addition, the Rosenhain model for the curve can be written down directly given the Rosenhain invariants. Similarly, the parameters for a Kummer surface can be expressed directly in terms of rational functions of theta constants. CM-values of these functions are algebraic numbers, and when computed to high enough precision, LLL can recognize their minimal polynomials. Motivated by fast cryptography on Kummer surfaces, we investigate a variant of the CM method for computing cryptographically strong Rosenhain models of curves (as well as their associated Kummer surfaces) and use it to generate several example curves at different security levels that are suitable for use in cryptography.<\/jats:p>","DOI":"10.1112\/s1461157014000370","type":"journal-article","created":{"date-parts":[[2014,8,5]],"date-time":"2014-08-05T06:34:15Z","timestamp":1407220455000},"page":"157-180","source":"Crossref","is-referenced-by-count":7,"title":["Constructing abelian surfaces for cryptography via Rosenhain invariants"],"prefix":"10.1112","volume":"17","author":[{"given":"Craig","family":"Costello","sequence":"first","affiliation":[],"role":[{"vocabulary":"crossref","role":"author"}]},{"given":"Alyson","family":"Deines-Schartz","sequence":"additional","affiliation":[],"role":[{"vocabulary":"crossref","role":"author"}]},{"given":"Kristin","family":"Lauter","sequence":"additional","affiliation":[],"role":[{"vocabulary":"crossref","role":"author"}]},{"given":"Tonghai","family":"Yang","sequence":"additional","affiliation":[],"role":[{"vocabulary":"crossref","role":"author"}]}],"member":"311","published-online":{"date-parts":[[2014,8,1]]},"reference":[{"key":"S1461157014000370_r38","unstructured":"38. T. Yang , \u2018Rational structure of $X(N)$ over $\\mathbb{Q}$ and explicit Galois action on CM points\u2019, Preprint, 2014."},{"key":"S1461157014000370_r37","doi-asserted-by":"publisher","DOI":"10.4310\/AJM.2013.v17.n2.a4"},{"key":"S1461157014000370_r35","doi-asserted-by":"publisher","DOI":"10.1090\/S0025-5718-99-01020-0"},{"key":"S1461157014000370_r33","unstructured":"33. M. Streng , \u2018An explicit version of Shimura\u2019s reciprocity law for Siegel modular functions\u2019, CoRR, 2012, arXiv:abs\/1201.0020."},{"key":"S1461157014000370_r31","unstructured":"31. A. Spallek , \u2018Kurven vom Geschlecht 2 und ihre Anwendung in public-key-Kryptosystemen\u2019, PhD Thesis, Inst. f\u00fcr Experimentelle Mathematik, 1994."},{"key":"S1461157014000370_r30","doi-asserted-by":"publisher","DOI":"10.1515\/9781400883943"},{"key":"S1461157014000370_r28","first-page":"483","article-title":"Elliptic curves over finite fields and the computation of square roots mod p","volume":"44","author":"Schoof","year":"1985","journal-title":"Math. Comp."},{"key":"S1461157014000370_r24","doi-asserted-by":"publisher","DOI":"10.1007\/978-1-4612-0441-1_21"},{"key":"S1461157014000370_r23","unstructured":"23. D. Lubicz and D. Robert , \u2018A generalisation of Miller\u2019s algorithm and applications to pairing computations on abelian varieties\u2019, Cryptology ePrint Archive, Report 2013\/192, 2013, http:\/\/eprint.iacr.org\/."},{"key":"S1461157014000370_r22","doi-asserted-by":"publisher","DOI":"10.1007\/BF01457454"},{"key":"S1461157014000370_r16","doi-asserted-by":"publisher","DOI":"10.1007\/11935230_8"},{"key":"S1461157014000370_r15","doi-asserted-by":"publisher","DOI":"10.1515\/JMC.2007.012"},{"key":"S1461157014000370_r14","unstructured":"14. A. Enge and E. Thom\u00e9 , \u2018Computing class polynomials for abelian surfaces\u2019, Experiment. Math., to appear, http:\/\/eprint.iacr.org\/2013\/299."},{"key":"S1461157014000370_r12","doi-asserted-by":"publisher","DOI":"10.1090\/S0273-0979-07-01153-6"},{"key":"S1461157014000370_r9","doi-asserted-by":"publisher","DOI":"10.1142\/9789812701640_0006"},{"key":"S1461157014000370_r8","first-page":"1917","article-title":"Special values of Green functions at big CM points","volume":"2012","author":"Bruinier","year":"2012","journal-title":"Int. Math. Res. Not. IMRN"},{"key":"S1461157014000370_r7","doi-asserted-by":"publisher","DOI":"10.2140\/ant.2011.5.495"},{"key":"S1461157014000370_r21","unstructured":"21. K. Lauter and B. Viray , \u2018An arithmetic intersection formula for denominators of Igusa class polynomials\u2019, Preprint, 2012, arXiv:1210.7841."},{"key":"S1461157014000370_r5","doi-asserted-by":"publisher","DOI":"10.1007\/978-3-540-76900-2_3"},{"key":"S1461157014000370_r4","unstructured":"4. D.\u00a0J. Bernstein , C. Chuengsatiansup , T. Lange and P. Schwabe , \u2018Kummer strikes back: new DH speed records\u2019, http:\/\/cr.yp.to\/papers.html#kummer."},{"key":"S1461157014000370_r3","unstructured":"3. D.\u00a0J. Bernstein , \u2018Elliptic vs. hyperelliptic, part I\u2019, Talk at ECC, September 2006."},{"key":"S1461157014000370_r2","unstructured":"2. D.\u00a0J. Bernstein , \u2018A software implementation of NIST P-224\u2019, Talk at ECC, October 2001."},{"key":"S1461157014000370_r1","doi-asserted-by":"publisher","DOI":"10.1090\/S0025-5718-1993-1199989-X"},{"key":"S1461157014000370_r20","doi-asserted-by":"publisher","DOI":"10.2307\/2372812"},{"key":"S1461157014000370_r18","doi-asserted-by":"publisher","DOI":"10.1093\/imrn\/rnr052"},{"key":"S1461157014000370_r13","first-page":"161","volume-title":"Arithmetic, geometry, and coding theory (AGCT 2005)","author":"Eisentraeger","year":"2011"},{"key":"S1461157014000370_r27","doi-asserted-by":"publisher","DOI":"10.1090\/S0025-5718-1990-1035941-X"},{"key":"S1461157014000370_r25","unstructured":"25. J.\u00a0S. Milne , \u2018Class field theory (v4.02)\u2019, 2013, available at www.jmilne.org\/math\/."},{"key":"S1461157014000370_r29","volume-title":"Introduction to the arithmetic theory of automorphic functions, vol. 1","author":"Shimura","year":"1971"},{"key":"S1461157014000370_r32","unstructured":"32. M. Streng , \u2018Complex multiplication of abelian surfaces\u2019, PhD Thesis, Leiden University, June 2010,https:\/\/openaccess.leidenuniv.nl\/handle\/1887\/15572."},{"key":"S1461157014000370_r10","doi-asserted-by":"publisher","DOI":"10.1016\/0196-8858(86)90023-0"},{"key":"S1461157014000370_r11","doi-asserted-by":"publisher","DOI":"10.1090\/S0025-5718-09-02295-9"},{"key":"S1461157014000370_r26","doi-asserted-by":"publisher","DOI":"10.1090\/S0025-5718-1987-0866113-7"},{"key":"S1461157014000370_r6","doi-asserted-by":"publisher","DOI":"10.1007\/978-3-642-38348-9_12"},{"key":"S1461157014000370_r17","doi-asserted-by":"publisher","DOI":"10.1016\/j.jsc.2011.09.003"},{"key":"S1461157014000370_r34","doi-asserted-by":"publisher","DOI":"10.1090\/S0002-9947-98-02056-X"},{"key":"S1461157014000370_r36","doi-asserted-by":"publisher","DOI":"10.1090\/S0025-5718-02-01422-9"},{"key":"S1461157014000370_r19","doi-asserted-by":"publisher","DOI":"10.1090\/conm\/521\/10274"}],"container-title":["LMS Journal of Computation and Mathematics"],"original-title":[],"language":"en","link":[{"URL":"https:\/\/www.cambridge.org\/core\/services\/aop-cambridge-core\/content\/view\/S1461157014000370","content-type":"unspecified","content-version":"vor","intended-application":"similarity-checking"}],"deposited":{"date-parts":[[2019,4,22]],"date-time":"2019-04-22T21:49:30Z","timestamp":1555969770000},"score":1,"resource":{"primary":{"URL":"https:\/\/www.cambridge.org\/core\/product\/identifier\/S1461157014000370\/type\/journal_article"}},"subtitle":[],"short-title":[],"issued":{"date-parts":[[2014]]},"references-count":38,"journal-issue":{"issue":"A","published-print":{"date-parts":[[2014]]}},"alternative-id":["S1461157014000370"],"URL":"https:\/\/doi.org\/10.1112\/s1461157014000370","relation":{},"ISSN":["1461-1570"],"issn-type":[{"value":"1461-1570","type":"electronic"}],"subject":[],"published":{"date-parts":[[2014]]}}}