Can you create asymmetrical encryption algorithm from symmetrical, without knowing how the algorithm works?
It is not known whether it is possible to construct secure public-key encryption schemes from private-key ones (and by extension, from one-way functions). From Barak's recent survey:


  One could ask if there really is an inherent difference between public-key and private-key cryptography or maybe this is simply a reflection of our ignorance.  That is, is it is not possible to build a public-key cryptosystem out  of an arbitrary one-way function and hence base it on the same assumptions as private-key encryption? The answer is that we do not know


In fact, this is one of the most important open questions in cryptography, along with the question of whether one-way functions exist. So, in practice you can consider that it is not possible, and consult the above for a survey of the known methods to construct public-key cryptosystems.

See also Impagliazzo's "Five Worlds"; your question can be reworded as: is it possible that we live in Minicrypt, the world where one-way functions exist but public-key cryptography does not?