扩展切比雪夫多项式(混沌映射, Chebyshev chaotic map)

        切比雪夫多项式也叫切比雪夫混沌映射,起源于多倍角的余弦函数和正弦函数的展开式,是计算数学中一类特殊的函数。 扩展切比雪夫多项式(Extended Chebyshev Polynomials, ECP),也有叫作扩展切比雪夫混沌映射(Extended Chebyshev chaotic map),是切比雪夫多项式(混沌映射)参数x在\left ( -\infty ,+\infty \right )上的扩展。

        切比雪夫多项式定义:设n,x为变量且满足n \in Z^{*}, \forall x \in[-1,1],n阶切比雪夫多项式T_{n}(x):[-1,1] \rightarrow[-1,1]的余弦式定义为:T_{n}(x)=\cos (n \cdot \arccos (x))。当n\geq 2时,等价的递归迭代定义为:T_{n}(x)=2 x T_{n-1}(x)-T_{n-2}(x),其中T_{0}(x)=1, T_{1}(x)=x

         切比雪夫多项式的半群性质:\forall m, n \in Z^{*}, T_{m}\left(T_{n}(x)\right)=T_{n}\left(T_{m}(x)\right)=T_{m n}(x)

        2008年,Zhang证明了切比雪夫多项T_{n}(x)中变量x的取值范围扩展到区间(-\infty, \infty)后,T_{n}(x)仍然具有半群性质并且可以有效抵抗Bergamo攻击。

        扩展切比雪夫多项式定义:设n,x为变量且满足n \in Z^{*}, \forall x \in(-\infty, \infty),n阶扩展切比雪夫多项式的余弦式定义为:T_{n}(x)=\cos (n \cdot \arccos (x))(\bmod p)。当n\geq 2时,等价的递归迭代定义为:T_{n}(x)=\left(2 x T_{n-1}(x)-T_{n-2}(x)\right) \bmod p。其中T_{0}(x)=1, T_{1}(x)=x,p是一个大素数。

        扩展切比雪夫多项式的半群性质:若已知T_{n}(x), x的值,求解阶数n的问题称为切比雪夫离散对数问题。CDLP属于计算难问题,在常规多项式线性时间内是无法计算出阶数n。

         切比雪夫离散对数问题(Chebyshev Discrete Logarithm Problem,CDLP):若已知T_{n}(x), x的值,求解阶数n的问题称为切比雪夫离散对数问题。CDLP属于计算难问题,在常规多项式线性时间内是无法计算出阶数n。

        切比雪夫迪菲-赫尔曼问题(Chebyshev Diffie-Hellman Problem,CDHP):若已知T_{n}(x), T_{m}(x), x的值,求解T_{m \cdot n}(x)的值的问题称为切比雪夫迪菲-赫尔曼问题。CDHP属于计算难问题,不能在常规的多项式线性时间内得到有效解决。

        研究现状:

        2015年,Lee等人针对智能卡的安全问题提出了一种基于ECP的口令认证密钥协商协议,但该协议不能有效抵御模仿攻击。同年,为保障远程医疗信息系统的安全,Lu等人结合生物特征识别技术提出了一种基于ECP的口令认证密钥协商协议,但Masdari等人在其调研中证明了Lu等人的方案不能抵抗模仿攻击。2016年,Kumari等人在无线传感网领域提出了一种基于ECP的用户友好认证密钥协商协议,该协议具有双向认证性和完美前向安全性,并提供登录时错误标识符检测机制,但在Ferrag等人和El-hajj等人的调查研究中均指出Kumari等人的协议不具备消息完整性,也无法抵抗已知会话特定临时信息攻击和抗模仿攻击。2017年,针对安全读取远程智能电表数据问题,Sha等人设计实现了一种基于ECP的两阶段认证密钥协商协议,第一阶段智能电表读取器和云端电力服务器通过数字签名进行认证,验证通过后获得一次性的对称密钥,该密钥用于第二阶段智能电表读取器和远程电力设备间的认证和密钥协商。2018年,Abbasinezhad-Mood等人在其研究中证明了Sha等人的协议缺乏完美前向安全性,然后提出了一种能高效安全地获取远程电力设备数据的匿名认证密钥协商协议,但该协议无法抵抗内部特权攻击也不具备匿名性。2019年,Pak等人在研究中首先指出Kumari等人的协议无法抵抗已知会话特定临时信息攻击和抗模仿攻击,然后又详细论证了Lu等人所提协议不能满足匿名性以及口令更新阶段的安全性,最后结合用户的生物特征和口令提出了一种基于ECP的智能卡认证密钥协商协议,保障了数据的完整性,但无法保障匿名性,也不能抵抗重放攻击和已知会话特定临时信息攻击。同年,Jabbari等人提出了一种基于ECP的无口令的认证密钥协商协议,但该协议不具备匿名性,也不能抵抗重放攻击和已知会话特定临时信息攻击。

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