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As special types of factorization of finite groups, logarithmic signatures and covers have been used as the main components of cryptographic keys for secret key cryptosystems such as <inline-formula> <tex-math notation="LaTeX">$PGM$ </tex-math></inline-formula> and public key cryptosystems like <inline-formula> <tex-math notation="LaTeX">$MST_{1}$ </tex-math></inline-formula>, <inline-formula> <tex-math notation="LaTeX">$MST_{2}$ </tex-math></inline-formula>, <inline-formula> <tex-math notation="LaTeX">$MST_{3}$ </tex-math></inline-formula> and <inline-formula> <tex-math notation="LaTeX">$eMST_{3}$ </tex-math></inline-formula>. In particular, as a natural analogue of integer factorization problem (IFP), group factorization problem (GFP) and its hardness assumption over certain factorization basis, referred as logarithmic signature, play a core role in the security arguments for the family of <inline-formula> <tex-math notation="LaTeX">$MST$ </tex-math></inline-formula> cryptosystems. Security is not the unique goal of designing a cryptosystem. Instead, efficiency is also a major issue. In this paper, we design a new secure encryption scheme based on group factorization problem (GFP). Furthermore, we present the security analysis and demonstrate the performance of our scheme. Comparing with <inline-formula> <tex-math notation="LaTeX">$eMST_{3}$ </tex-math></inline-formula>, our scheme is simplified with more efficiency.

A New Secure Encryption Scheme Based on Group Factorization Problem