{"title":"计算库默线之间的 2-isogenies","authors":"Damien Robert, Nicolas Sarkis","doi":"10.62056/abvua69p1","DOIUrl":null,"url":null,"abstract":"<jats:p> We use theta groups to study <mml:math xmlns:mml=\"http://www.w3.org/1998/Math/MathML\">\n <mml:mrow>\n <mml:mn>2</mml:mn>\n </mml:mrow>\n </mml:math>-isogenies between Kummer lines, with a particular focus on the Montgomery model. This allows us to recover known formulas, along with more efficient forms for translated isogenies, which require only <mml:math xmlns:mml=\"http://www.w3.org/1998/Math/MathML\">\n <mml:mrow>\n <mml:mn>2</mml:mn>\n <mml:mi>S</mml:mi>\n <mml:mo>+</mml:mo>\n <mml:mn>2</mml:mn>\n <mml:msub>\n <mml:mi>m</mml:mi>\n <mml:mn>0</mml:mn>\n </mml:msub>\n </mml:mrow>\n </mml:math> for evaluation. We leverage these translated isogenies to build a hybrid ladder for scalar multiplication on Montgomery curves with rational <mml:math xmlns:mml=\"http://www.w3.org/1998/Math/MathML\">\n <mml:mrow>\n <mml:mn>2</mml:mn>\n </mml:mrow>\n </mml:math>-torsion, which cost <mml:math xmlns:mml=\"http://www.w3.org/1998/Math/MathML\">\n <mml:mrow>\n <mml:mn>3</mml:mn>\n <mml:mi>M</mml:mi>\n <mml:mo>+</mml:mo>\n <mml:mn>6</mml:mn>\n <mml:mi>S</mml:mi>\n <mml:mo>+</mml:mo>\n <mml:mn>2</mml:mn>\n <mml:msub>\n <mml:mi>m</mml:mi>\n <mml:mn>0</mml:mn>\n </mml:msub>\n </mml:mrow>\n </mml:math> per bit, compared to <mml:math xmlns:mml=\"http://www.w3.org/1998/Math/MathML\">\n <mml:mrow>\n <mml:mn>5</mml:mn>\n <mml:mi>M</mml:mi>\n <mml:mo>+</mml:mo>\n <mml:mn>4</mml:mn>\n <mml:mi>S</mml:mi>\n <mml:mo>+</mml:mo>\n <mml:mn>1</mml:mn>\n <mml:msub>\n <mml:mi>m</mml:mi>\n <mml:mn>0</mml:mn>\n </mml:msub>\n </mml:mrow>\n </mml:math> for the standard Montgomery ladder. </jats:p>","PeriodicalId":508905,"journal":{"name":"IACR Cryptol. ePrint Arch.","volume":"7 2","pages":"37"},"PeriodicalIF":0.0000,"publicationDate":"2024-04-09","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":"0","resultStr":"{\"title\":\"Computing 2-isogenies between Kummer lines\",\"authors\":\"Damien Robert, Nicolas Sarkis\",\"doi\":\"10.62056/abvua69p1\",\"DOIUrl\":null,\"url\":null,\"abstract\":\"<jats:p> We use theta groups to study <mml:math xmlns:mml=\\\"http://www.w3.org/1998/Math/MathML\\\">\\n <mml:mrow>\\n <mml:mn>2</mml:mn>\\n </mml:mrow>\\n </mml:math>-isogenies between Kummer lines, with a particular focus on the Montgomery model. This allows us to recover known formulas, along with more efficient forms for translated isogenies, which require only <mml:math xmlns:mml=\\\"http://www.w3.org/1998/Math/MathML\\\">\\n <mml:mrow>\\n <mml:mn>2</mml:mn>\\n <mml:mi>S</mml:mi>\\n <mml:mo>+</mml:mo>\\n <mml:mn>2</mml:mn>\\n <mml:msub>\\n <mml:mi>m</mml:mi>\\n <mml:mn>0</mml:mn>\\n </mml:msub>\\n </mml:mrow>\\n </mml:math> for evaluation. We leverage these translated isogenies to build a hybrid ladder for scalar multiplication on Montgomery curves with rational <mml:math xmlns:mml=\\\"http://www.w3.org/1998/Math/MathML\\\">\\n <mml:mrow>\\n <mml:mn>2</mml:mn>\\n </mml:mrow>\\n </mml:math>-torsion, which cost <mml:math xmlns:mml=\\\"http://www.w3.org/1998/Math/MathML\\\">\\n <mml:mrow>\\n <mml:mn>3</mml:mn>\\n <mml:mi>M</mml:mi>\\n <mml:mo>+</mml:mo>\\n <mml:mn>6</mml:mn>\\n <mml:mi>S</mml:mi>\\n <mml:mo>+</mml:mo>\\n <mml:mn>2</mml:mn>\\n <mml:msub>\\n <mml:mi>m</mml:mi>\\n <mml:mn>0</mml:mn>\\n </mml:msub>\\n </mml:mrow>\\n </mml:math> per bit, compared to <mml:math xmlns:mml=\\\"http://www.w3.org/1998/Math/MathML\\\">\\n <mml:mrow>\\n <mml:mn>5</mml:mn>\\n <mml:mi>M</mml:mi>\\n <mml:mo>+</mml:mo>\\n <mml:mn>4</mml:mn>\\n <mml:mi>S</mml:mi>\\n <mml:mo>+</mml:mo>\\n <mml:mn>1</mml:mn>\\n <mml:msub>\\n <mml:mi>m</mml:mi>\\n <mml:mn>0</mml:mn>\\n </mml:msub>\\n </mml:mrow>\\n </mml:math> for the standard Montgomery ladder. </jats:p>\",\"PeriodicalId\":508905,\"journal\":{\"name\":\"IACR Cryptol. ePrint Arch.\",\"volume\":\"7 2\",\"pages\":\"37\"},\"PeriodicalIF\":0.0000,\"publicationDate\":\"2024-04-09\",\"publicationTypes\":\"Journal Article\",\"fieldsOfStudy\":null,\"isOpenAccess\":false,\"openAccessPdf\":\"\",\"citationCount\":\"0\",\"resultStr\":null,\"platform\":\"Semanticscholar\",\"paperid\":null,\"PeriodicalName\":\"IACR Cryptol. ePrint Arch.\",\"FirstCategoryId\":\"1085\",\"ListUrlMain\":\"https://doi.org/10.62056/abvua69p1\",\"RegionNum\":0,\"RegionCategory\":null,\"ArticlePicture\":[],\"TitleCN\":null,\"AbstractTextCN\":null,\"PMCID\":null,\"EPubDate\":\"\",\"PubModel\":\"\",\"JCR\":\"\",\"JCRName\":\"\",\"Score\":null,\"Total\":0}","platform":"Semanticscholar","paperid":null,"PeriodicalName":"IACR Cryptol. ePrint Arch.","FirstCategoryId":"1085","ListUrlMain":"https://doi.org/10.62056/abvua69p1","RegionNum":0,"RegionCategory":null,"ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":null,"EPubDate":"","PubModel":"","JCR":"","JCRName":"","Score":null,"Total":0}
引用次数: 0
摘要
我们使用 Theta 群来研究库默尔线间的 2 -同源关系，尤其侧重于蒙哥马利模型。这使我们能够恢复已知公式，以及更有效的翻译同源形式，只需 2 S + 2 m 0 即可进行评估。我们利用这些翻译同源建立了一个混合梯子，用于具有有理 2 -扭转的蒙哥马利曲线上的标量乘法，每比特的成本为 3 M + 6 S + 2 m 0，而标准蒙哥马利梯子的成本为 5 M + 4 S + 1 m 0。
We use theta groups to study 2-isogenies between Kummer lines, with a particular focus on the Montgomery model. This allows us to recover known formulas, along with more efficient forms for translated isogenies, which require only 2S+2m0 for evaluation. We leverage these translated isogenies to build a hybrid ladder for scalar multiplication on Montgomery curves with rational 2-torsion, which cost 3M+6S+2m0 per bit, compared to 5M+4S+1m0 for the standard Montgomery ladder.
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