中文摘要：

对基于单项式的多变量公钥密码算法中的等价密钥问题进行了研究，利用保形变换和有限域方面的知识计算出了其等价密钥的数量。这是一个更一般的结论，是已有结论的推广形式。进一步地，C？， C ？-和Square等算法中等价密钥的数量可以看作是该结论的特殊情形。研究结果表明：选私钥仿射结构具有稀疏特性的矩阵作为等价类的代表元，可以有效地节省存储空间、缩短操作时间。同时，多变量公钥密码算法要想达到预想的安全强度，就必须提高算法参数的大小。

英文摘要：

This paper investigates the problem of equivalent keys in multivariate public key cryptographic schemes based on a monomial, and applies sustaining transformation and finite field theory to compute the number of equivalent keys. This conclusion is more general, and is a generalized form of a conclusion that has been known. Accordingly, the number of equivalent keys of C＊ scheme, C ＊-scheme and square scheme can be regarded as a special case of the result computed above. The results show that the memory space and the operation time can be effectively reduced by choosing the sparse normal form matrix of affine transformations of private keys as the representative ele- ment of the equivalence class. Moreover, to achieve the expected security, larger system parameters should be cho- sen in multivariate public key cryptographic schemes.