Shi Bai, Hansraj Jangir, Tran Ngo, William Youmans
{"title":"An algebraic algorithm for breaking NTRU with multiple keys","authors":"Shi Bai, Hansraj Jangir, Tran Ngo, William Youmans","doi":"10.1007/s10623-024-01473-z","DOIUrl":null,"url":null,"abstract":"<p>We describe a heuristic polynomial-time algorithm for breaking the NTRU problem with multiple keys when given a sufficient number of ring samples. Following the linearization approach of the Arora-Ge algorithm (<i>ICALP ’11</i>), our algorithm constructs a system of linear equations using the public keys. Our main contribution is a kernel reduction technique that extracts the secret vector from a linear space of rank <i>n</i>, where <i>n</i> is the degree of the ring in which NTRU is defined. Compared to the algorithm of Kim-Lee (<i>Designs, Codes and Cryptography, ’23</i>), our algorithm does not require prior knowledge of the Hamming weight of the secret keys. Our algorithm is based on some plausible heuristics. We demonstrate experiments and show that the algorithm works quite well in practice, with close to cryptographic parameters.\n</p>","PeriodicalId":11130,"journal":{"name":"Designs, Codes and Cryptography","volume":"303 1","pages":""},"PeriodicalIF":1.4000,"publicationDate":"2024-08-10","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":"0","resultStr":null,"platform":"Semanticscholar","paperid":null,"PeriodicalName":"Designs, Codes and Cryptography","FirstCategoryId":"100","ListUrlMain":"https://doi.org/10.1007/s10623-024-01473-z","RegionNum":2,"RegionCategory":"数学","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":null,"EPubDate":"","PubModel":"","JCR":"Q3","JCRName":"COMPUTER SCIENCE, THEORY & METHODS","Score":null,"Total":0}
引用次数: 0
Abstract
We describe a heuristic polynomial-time algorithm for breaking the NTRU problem with multiple keys when given a sufficient number of ring samples. Following the linearization approach of the Arora-Ge algorithm (ICALP ’11), our algorithm constructs a system of linear equations using the public keys. Our main contribution is a kernel reduction technique that extracts the secret vector from a linear space of rank n, where n is the degree of the ring in which NTRU is defined. Compared to the algorithm of Kim-Lee (Designs, Codes and Cryptography, ’23), our algorithm does not require prior knowledge of the Hamming weight of the secret keys. Our algorithm is based on some plausible heuristics. We demonstrate experiments and show that the algorithm works quite well in practice, with close to cryptographic parameters.
期刊介绍:
Designs, Codes and Cryptography is an archival peer-reviewed technical journal publishing original research papers in the designated areas. There is a great deal of activity in design theory, coding theory and cryptography, including a substantial amount of research which brings together more than one of the subjects. While many journals exist for each of the individual areas, few encourage the interaction of the disciplines.
The journal was founded to meet the needs of mathematicians, engineers and computer scientists working in these areas, whose interests extend beyond the bounds of any one of the individual disciplines. The journal provides a forum for high quality research in its three areas, with papers touching more than one of the areas especially welcome.
The journal also considers high quality submissions in the closely related areas of finite fields and finite geometries, which provide important tools for both the construction and the actual application of designs, codes and cryptographic systems. In particular, it includes (mostly theoretical) papers on computational aspects of finite fields. It also considers topics in sequence design, which frequently admit equivalent formulations in the journal’s main areas.
Designs, Codes and Cryptography is mathematically oriented, emphasizing the algebraic and geometric aspects of the areas it covers. The journal considers high quality papers of both a theoretical and a practical nature, provided they contain a substantial amount of mathematics.