Frobenius Additive Fast Fourier Transform

Proceedings of the 2018 ACM International Symposium on Symbolic and Algebraic Computation Pub Date : 2018-02-12 DOI:10.1145/3208976.3208998

Wen-Ding Li, Ming-Shing Chen, Po-Chun Kuo, Chen-Mou Cheng, Bo-Yin Yang

{"title":"Frobenius Additive Fast Fourier Transform","authors":"Wen-Ding Li, Ming-Shing Chen, Po-Chun Kuo, Chen-Mou Cheng, Bo-Yin Yang","doi":"10.1145/3208976.3208998","DOIUrl":null,"url":null,"abstract":"In ISSAC 2017, van der Hoeven and Larrieu showed that evaluating a polynomial P ın Fq [x] of degree <n at all n -th roots of unity in Fqd can essentially be computed d times faster than evaluating Q ın Fqd x at all these roots, assuming Fqd contains a primitive n -th root of unity. Termed the Frobenius FFT, this discovery has a profound impact on polynomial multiplication, especially for multiplying binary polynomials, which finds ample application in coding theory and cryptography. In this paper, we show that the theory of Frobenius FFT beautifully generalizes to a class of additive FFT developed by Cantor and Gao-Mateer. Furthermore, we demonstrate the power of Frobenius additive FFT for q=2: to multiply two binary polynomials whose product is of degree <256, the new technique requires only 29,005 bit operations, while the best result previously reported was 33,397. To the best of our knowledge, this is the first time that FFT-based multiplication outperforms Karatsuba and the like at such a low degree in terms of bit-operation count.","PeriodicalId":105762,"journal":{"name":"Proceedings of the 2018 ACM International Symposium on Symbolic and Algebraic Computation","volume":"31 1","pages":"0"},"PeriodicalIF":0.0000,"publicationDate":"2018-02-12","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":"9","resultStr":null,"platform":"Semanticscholar","paperid":null,"PeriodicalName":"Proceedings of the 2018 ACM International Symposium on Symbolic and Algebraic Computation","FirstCategoryId":"1085","ListUrlMain":"https://doi.org/10.1145/3208976.3208998","RegionNum":0,"RegionCategory":null,"ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":null,"EPubDate":"","PubModel":"","JCR":"","JCRName":"","Score":null,"Total":0}