"lie on the basis of" versus "lie at the basis of"
Solution 1:
The phrase "X lies at the basis of Y", while arguably clumsy, is used reasonably often in English. On the other hand, I do not believe that "X lies on the basis of Y" is something a native English speaker would say. Consider the following Google Ngram :
Solution 2:
Both are clumsy. Simply "X is the basis of Y" would be better, so for example "The rule of law is the basis of our freedom".