{"title":"更快的截断整数乘法","authors":"David Harvey","doi":"10.1090/mcom/3939","DOIUrl":null,"url":null,"abstract":"<p>We present new algorithms for computing the low <inline-formula content-type=\"math/mathml\"> <mml:math xmlns:mml=\"http://www.w3.org/1998/Math/MathML\" alttext=\"n\"> <mml:semantics> <mml:mi>n</mml:mi> <mml:annotation encoding=\"application/x-tex\">n</mml:annotation> </mml:semantics> </mml:math> </inline-formula> bits or the high <inline-formula content-type=\"math/mathml\"> <mml:math xmlns:mml=\"http://www.w3.org/1998/Math/MathML\" alttext=\"n\"> <mml:semantics> <mml:mi>n</mml:mi> <mml:annotation encoding=\"application/x-tex\">n</mml:annotation> </mml:semantics> </mml:math> </inline-formula> bits of the product of two <inline-formula content-type=\"math/mathml\"> <mml:math xmlns:mml=\"http://www.w3.org/1998/Math/MathML\" alttext=\"n\"> <mml:semantics> <mml:mi>n</mml:mi> <mml:annotation encoding=\"application/x-tex\">n</mml:annotation> </mml:semantics> </mml:math> </inline-formula>-bit integers. We show that these problems may be solved in asymptotically <inline-formula content-type=\"math/mathml\"> <mml:math xmlns:mml=\"http://www.w3.org/1998/Math/MathML\" alttext=\"75\"> <mml:semantics> <mml:mn>75</mml:mn> <mml:annotation encoding=\"application/x-tex\">75%</mml:annotation> </mml:semantics> </mml:math> </inline-formula> of the time required to compute the full <inline-formula content-type=\"math/mathml\"> <mml:math xmlns:mml=\"http://www.w3.org/1998/Math/MathML\" alttext=\"2 n\"> <mml:semantics> <mml:mrow> <mml:mn>2</mml:mn> <mml:mi>n</mml:mi> </mml:mrow> <mml:annotation encoding=\"application/x-tex\">2n</mml:annotation> </mml:semantics> </mml:math> </inline-formula>-bit product, assuming that the underlying integer multiplication algorithm relies on computing cyclic convolutions of sequences of real numbers.</p>","PeriodicalId":2,"journal":{"name":"ACS Applied Bio Materials","volume":null,"pages":null},"PeriodicalIF":4.6000,"publicationDate":"2024-01-10","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":"0","resultStr":"{\"title\":\"Faster truncated integer multiplication\",\"authors\":\"David Harvey\",\"doi\":\"10.1090/mcom/3939\",\"DOIUrl\":null,\"url\":null,\"abstract\":\"<p>We present new algorithms for computing the low <inline-formula content-type=\\\"math/mathml\\\"> <mml:math xmlns:mml=\\\"http://www.w3.org/1998/Math/MathML\\\" alttext=\\\"n\\\"> <mml:semantics> <mml:mi>n</mml:mi> <mml:annotation encoding=\\\"application/x-tex\\\">n</mml:annotation> </mml:semantics> </mml:math> </inline-formula> bits or the high <inline-formula content-type=\\\"math/mathml\\\"> <mml:math xmlns:mml=\\\"http://www.w3.org/1998/Math/MathML\\\" alttext=\\\"n\\\"> <mml:semantics> <mml:mi>n</mml:mi> <mml:annotation encoding=\\\"application/x-tex\\\">n</mml:annotation> </mml:semantics> </mml:math> </inline-formula> bits of the product of two <inline-formula content-type=\\\"math/mathml\\\"> <mml:math xmlns:mml=\\\"http://www.w3.org/1998/Math/MathML\\\" alttext=\\\"n\\\"> <mml:semantics> <mml:mi>n</mml:mi> <mml:annotation encoding=\\\"application/x-tex\\\">n</mml:annotation> </mml:semantics> </mml:math> </inline-formula>-bit integers. We show that these problems may be solved in asymptotically <inline-formula content-type=\\\"math/mathml\\\"> <mml:math xmlns:mml=\\\"http://www.w3.org/1998/Math/MathML\\\" alttext=\\\"75\\\"> <mml:semantics> <mml:mn>75</mml:mn> <mml:annotation encoding=\\\"application/x-tex\\\">75%</mml:annotation> </mml:semantics> </mml:math> </inline-formula> of the time required to compute the full <inline-formula content-type=\\\"math/mathml\\\"> <mml:math xmlns:mml=\\\"http://www.w3.org/1998/Math/MathML\\\" alttext=\\\"2 n\\\"> <mml:semantics> <mml:mrow> <mml:mn>2</mml:mn> <mml:mi>n</mml:mi> </mml:mrow> <mml:annotation encoding=\\\"application/x-tex\\\">2n</mml:annotation> </mml:semantics> </mml:math> </inline-formula>-bit product, assuming that the underlying integer multiplication algorithm relies on computing cyclic convolutions of sequences of real numbers.</p>\",\"PeriodicalId\":2,\"journal\":{\"name\":\"ACS Applied Bio Materials\",\"volume\":null,\"pages\":null},\"PeriodicalIF\":4.6000,\"publicationDate\":\"2024-01-10\",\"publicationTypes\":\"Journal Article\",\"fieldsOfStudy\":null,\"isOpenAccess\":false,\"openAccessPdf\":\"\",\"citationCount\":\"0\",\"resultStr\":null,\"platform\":\"Semanticscholar\",\"paperid\":null,\"PeriodicalName\":\"ACS Applied Bio Materials\",\"FirstCategoryId\":\"100\",\"ListUrlMain\":\"https://doi.org/10.1090/mcom/3939\",\"RegionNum\":0,\"RegionCategory\":null,\"ArticlePicture\":[],\"TitleCN\":null,\"AbstractTextCN\":null,\"PMCID\":null,\"EPubDate\":\"\",\"PubModel\":\"\",\"JCR\":\"Q2\",\"JCRName\":\"MATERIALS SCIENCE, BIOMATERIALS\",\"Score\":null,\"Total\":0}","platform":"Semanticscholar","paperid":null,"PeriodicalName":"ACS Applied Bio Materials","FirstCategoryId":"100","ListUrlMain":"https://doi.org/10.1090/mcom/3939","RegionNum":0,"RegionCategory":null,"ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":null,"EPubDate":"","PubModel":"","JCR":"Q2","JCRName":"MATERIALS SCIENCE, BIOMATERIALS","Score":null,"Total":0}
引用次数: 0
摘要
我们提出了计算两个 n n 位整数乘积的低 n n 位或高 n n 位的新算法。我们的研究表明，假设底层整数乘法算法依赖于计算实数序列的循环卷积，那么这些问题可以在计算全部 2 n 2n 位乘积所需时间的 75% 左右的时间内得到解决。
We present new algorithms for computing the low nn bits or the high nn bits of the product of two nn-bit integers. We show that these problems may be solved in asymptotically 7575% of the time required to compute the full 2n2n-bit product, assuming that the underlying integer multiplication algorithm relies on computing cyclic convolutions of sequences of real numbers.
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