关于3阶Carmichael数

朱文余和孙琦（见《数学进展》，2004，33（4）：505-507）提出了关于3阶Carmichael数的三个问题，我们（见《四川大学学报（自然科学版）》，2006，43（6）：1197-1201）肯定地回答了问题1．本文模仿Howe的寻找严格2阶Carmichael数（见Mathematics of Computation，2000，69（232）：1711—1719）的方法，提出寻找满足某种条件的3阶Carmichael数的方法，并用这种方法确实找到了几百个这样的数，因而完全肯定地回答了问题2．     

物联网中适用于野外战场的基于口令的安全通信方案

Mobile Ad hoc NETwork (MANET) is a part of the Internet of Things (IoT). In battlefield communication systems, ground soldiers, tanks, and unmanned aerial vehicles comprise a heterogeneous MANET. In 2006, Byun et al. proposed the first constant-round password-based group key exchange with different passwords for such networks. In 2008, Nam et al. discovered the shortcomings of the scheme, and modified it. But the works only provide the group key. In this paper, we propose a password-based secure communication scheme for the IoT, which could be applied in the battlefield communication systems and support dynamic group, in which the nodes join or leave. By performing the scheme, the nodes in the heterogeneous MANET can realize secure broadcast, secure unicast, and secure direct communication across realms. After the analyses, we demonstrate that the scheme is secure and efficient.     

适用于移动应用的口令认证多密钥协商协议

To achieve privacy and authentication simultaneously in mobile applications, various Three-party Password-authenticated key exchange (3PAKE) protocols have been proposed. However, some of these protocols are vulnerable to conventional attacks or have low efficiency so that they cannot be applied to mobile applications . In this paper, we proposed a password-authenticated multiple key exchange protocol for mobile applications using elliptic curve cryptosystem. The proposed protocol can achieve efficiency, reliability, flexibility and scalability at the same time. Compared with related works, the proposed protocol is more suitable and practical for mobile applications .     

关于整数集分拆的一个猜测的证明

设n为正整数,记rn=m ax{正整数m:可将集合{1,2,…,m}分为n个子集,使得在每一子集中方程xy=z(x>1,y>1)均无解}.高楠和刘红艳(数学的实践与认识,2005,35(5):151—152)给出了rn的一个下界估计rn n9,并猜测对任意给定的正整数k,当n充分大时有rn nk.本文对此猜测给以肯定回答,并证明了如下更强的结论:对任意给定的正整数k 4,当n>3k时有rn n2k+1.