### Cryptosystems based on logarithmic signatures and covers of finite groups

#### Abstract

In this paper, we survey a cryptographic direction of post-quantum cryptography based on logarithmic signatures and covers of finite groups. These mathematical structures allows designing cryptosystems with security based on hardness of the factorization problem in finite group. This problem is assumed computationally hard even in post-quantum era. We give basic definitions and functions related to logarithmic signatures and covers of finite groups. Relations between these functions and the factorization problem in finite group are explained. We describe some logarithmic signatures generation methods and consider the hardness of the factorization problem in each case. We give a description of the existing cryptosystems based on logarithmic signatures and covers of finite groups in chronological order. These cryptographic systems applicable for such purposes as data ciphering, digital signing or pseudo random number generation. We mainly focus on cryptosystem MST3 that is the most perspective ciphering system in the direction. Description of Suzuki 2-groups traditionally used as a finite group in cryptographic system MST3 is given. A toy example of MST3 based on Suzuki 2-group is demonstrated. We also consider the main analysis results of existing cryptosystems based on logarithmic signatures and covers of finite groups.

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