基于Chebyshev多项式的公钥加密和密钥交换方案的改进

构成一种公钥加密方案的关键是找到一个带陷门的单向函数。Kocarev利用Chebyshev映射的半群特性，提出了一种混沌公钥加密方案。但是经分析发现其是不安全的，由于三角函数的固有周期性，使得攻击者很容易破解，攻击者在已知密文的条件下很容易解出明文。基于此，提出了一种改进方案。该方案将Chebyshev多项式的状态空间扩展到实数域，从而利用Chebyshev多项式的半群特性构成具有单向带陷门的公钥加密算法，同时又可以避免上述已知密文的攻击。经过对其性能的研究发现，新方案是安全有效的；同时提出基于这种扩展的Chebyshev多项式的半群特性的密钥交换方案。

Improvement in Public-key Encryption and Key Exchange Scheme Based on Chebyshev Polynomials

It is essential to discover a one-way trap-door function in constructing a public-key encryption scheme. Kocarev invented an encryption scheme based on the semi-group property of Chebyshev mapping. The scheme was found insecure. Due to the inherent periodicity of the trigonometric function , an attacker can easily decode the scheme when some ciphers are given. In the paper we extend the state space of Chebyshev polynomials to the real number domain, employ the semi-group property of Chebyshev polynomials to construct a one- way door-trap function, and avoid the attack . A new scheme based on the extended Chebyshev polynomials is given. Moreover, the relative public-key encryption and key exchange schemes are presented. The encrypt systems show themselves to be practical and secure.