A Miniature CCA Public Key Encryption Scheme Based on Non-abelian Factorization Problem in Finite Groups of Lie Type

With the development of Lie theory, Lie groups have attained profound significance in several branches of Mathematics and Physics. In Lie theory, the matrix exponential plays a crucial role between Lie groups and Lie algebras. Meanwhile, as the finite analogue of Lie groups, finite groups of Lie type have potential applications in cryptography due to their unique mathematical structures. In this paper, we first put forward a novel idea of designing cryptosystems based on Lie theory. First of all, combing with discrete logarithm problem and group factorization problem, we proposed several new intractable assumptions based on the matrix exponential in finite groups of Lie type. Subsequently, in analog with Boyen's scheme (Asiacrypt 2007), we designed a public-key encryption scheme based on the non-abelian factorization problem in finite groups of Lie type. Finally, our proposal was proved to be indistinguishable against adaptively chosen-ciphertext attack in the random oracle model. It is encouraging that our scheme also has the potential to resist against Shor's quantum algorithm attack.