立方多变量公钥密码体制的最小秩分析

立方加密体制是经典的多变量公钥密码体制Square的改进方案，其中心映射由平方映射改为了立方映射，由此将公钥多项式从二次提升到三次来抵抗针对二次多变量公钥密码体制的最小秩攻击。针对这种体制，提出一种结合差分的最小秩攻击，旨在恢复它的私钥。首先，分析体制的中心映射差分，并根据差分后的结构来确定它的秩；然后，求解公钥差分，并提取二次项的系数矩阵；接着，由系数矩阵以及确定的秩构造一个最小秩问题；最后，结合扩展的Kipnis-Shamir方法对问题进行求解。实验结果表明，利用最小秩攻击可以恢复立方加密体制的私钥。

MinRank analysis of cubic multivariate public key cryptosystem

The cubic cryptosystem is the improvement of the classical multivariable cryptosystem Square. By increasing the degree of central mapping from square mapping to cubic mapping, the public key polynomial was promoted from quadratic to cubic in order to resist the MinRank attack against the quadratic multivariable public key cryptosystem. Aiming at this system, a MinRank attack combining with difference was proposed to recover its private key. Firstly, the central mapping difference of the system was analyzed, and its rank was determined according to the structure after difference. Then, the difference of the public key was solved and the coefficient matrices of the quadratic term were extracted. After that, a MinRank problem was constructed by the coefficient matrix and the determined rank. Finally, the extended Kipnis-Shamir method was combined to solve the problem. The experimental results show that the private key of cubic cryptosystem can be recovered by using MinRank attack.