主办单位：数学与统计学学院

报告专家：郑群雄

专家所在单位：战略支援部队信息工程大学

报告时间：2020年6月25日15:00

报告地点：腾讯会议（会议ID：204 639 583）

专家简介：郑群雄，战略支援部队信息工程大学网络空间安全学院副教授。2013年毕业于解放军信息工程大学，获密码学博士学位。硕士学位论文和博士学位论文分别于2010年和2014年获评全军优秀学位论文。2016年入选中国科协青年人才托举工程和人社部首届全国博士后创新人才计划。主要从事对称密码设计与分析方面的研究工作，先后在《IEEE Transactions on Information Theory》、《Designs、Codes and Cryptography》、《Finite Fields and Their Applications》等发表多篇学术论文。担任《IEEE Transactions on Information Theory》、《Designs、Codes and Cryptography》、《Cryptography and Communications》等刊物的评审。主持国家自然科学基金面上项目、青年基金项目、国家重点实验室开放基金课题等项目多项，参研国家自然科学基金项目、863计划项目、国家密码管理局项目等共计10余项。

报告摘要： Let 𝑞 be a prime power and 𝔽𝑞 the finite field with 𝑞 elements. A periodic sequence over 𝔽𝑞 with period 𝑞𝑛−1 is called a modified de Bruijn sequence of order 𝑛 if each nonzero 𝑛-tuple over 𝔽𝑞 occurs exactly once in every period. For a modified de Bruijn sequence of order 𝑛 over 𝔽𝑞, it is known that its minimal polynomial is divided by an irreducible polynomial of degree 𝑛 over 𝔽𝑞, and so the minimal polynomial is of the form 𝑓(𝑥)𝑔(𝑥), where 𝑓(𝑥) is an irreducible polynomial of degree 𝑛 over 𝔽𝑞. In this report, we show that 𝑔(𝑥) cannot be any polynomial of degree 𝑘 over 𝔽𝑞 if 𝑛≥4𝑘. Our main contributions are as follows. First, the results obtained by Kyureghyan (2008, Discrete Appl. Math.) and recently by Dong et al (2019, Finite Fields Appl.) are extended from 𝔽2 to 𝔽𝑞. Second, compared with the result obtained by Dong et al, 𝑔(𝑥) is not necessary primitive and the restriction on the degree of 𝑔(𝑥) is relaxed from 𝑛≥8𝑘 to 𝑛≥4𝑘. Finally, based on this new result, a non-trivial lower bound 5𝑛/4 is obtained for modified de Bruijn sequences of order 𝑛 over 𝔽𝑞, which is the first time that a lower bound larger than 𝑛 could be proved with no restrictions on 𝑛.

邀请人：孙志敏