Let's imagine that we have some symmetrical encryption algorithm, that has only two functions: Encrypt(Data, Key) and Decrypt(Data, Key). Let's say that the algorithm is strong and quick, but we don't know how it works. Is it possible only using those two functions to create a usable asymmetrical encryption algorithm? If not, is there any way to add anything that will help us create one?
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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?
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$\begingroup$ We can build signature schemes out of arbitrary one-way functions. $\endgroup$– forest ♦Jul 16, 2018 at 23:04
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$\begingroup$ @forest It could be argued that signature schemes are related to public-key cryptosystems only in a superficial way, and that they fundamentally belong to private-key cryptography, precisely because they can be constructed from OWFs. $\endgroup$– fkraiemJul 16, 2018 at 23:58