  Since 1870s, scientists have been taking deep insight into Lie groups and Lie
algebras. With the development of Lie theory, Lie groups have got profound
significance in many branches of mathematics and physics. In Lie theory,
exponential mapping between Lie groups and Lie algebras plays a crucial role.
Exponential mapping is the mechanism for passing information from Lie algebras
to Lie groups. Since many computations are performed much more easily by
employing Lie algebras, exponential mapping is indispensable while studying Lie
groups. In this paper, we first put forward a novel idea of designing
cryptosystem based on Lie groups and Lie algebras. Besides, combing with
discrete logarithm problem(DLP) and factorization problem(FP), we propose some
new intractable assumptions based on exponential mapping. Moreover, in analog
with Boyen's sceme(AsiaCrypt 2007), we disign a public key encryption scheme
based on non-Abelian factorization problems in Lie Groups. Finally, our
proposal is proved to be IND-CCA2 secure in the random oracle model.
