Learn Polish: Nie sposób złamać tego matematycznego szyfru bez klucza. - It is impossible to break this mathematical cipher without a key.

Nie sposób is an impersonal, fixed phrase meaning 'it is impossible' or 'there is no way.' It is not a verb itself (the noun sposób means 'method' or 'way'). To say 'it is impossible to do something,' you simply combine nie sposób directly with an infinitive, like złamać in this sentence.

Why is it tego matematycznego szyfru (Genitive) instead of ten matematyczny szyfr (Accusative)?

This is the Genitive of Negation in action. Even though the direct object is attached to the infinitive złamać, the negation from the main phrase nie sposób overrides the case. Because the entire idea is negated, the object switches from the Accusative to the Genitive.

Could I just use nie można instead of nie sposób?

Yes, but nie sposób carries a stronger, more absolute feeling of impossibility. While nie można can sometimes just mean 'it is not permitted' or 'one shouldn't', nie sposób strictly means 'it is physically or logically impossible' or 'there is no conceivable way'.

Why use the perfective verb złamać here instead of the imperfective łamać?

We use the perfective złamać because the sentence focuses on the complete, successful result of the action. If you used the imperfective łamać, it would mean 'it is impossible to be in the process of breaking' or 'impossible to try breaking,' which doesn't fit the context.