There's a good chance of a historical link from the Slovak word "tuhý" to the English "tough", pronounced (taf), through the German "zäh". The basic idea is something hard or difficult, so a tough battle is one that is hard to win, tough meat is for example a steak that is difficult to chew, and a tough decision is one that is hard to make.

Doing the Low Tatras ridge from Donovaly to Telgart in four days would be a tough hike, physically and mentally demanding (náročné). In bad weather it would be "tough going" ("išlo by to ťažko"), requiring some "tough guys" ("drsní chlapi") to do it. To be "tough on someone" means to be extemely strict ("prísny", even "neúprosný"). Bad luck can be expressed as hard luck, or "tough luck", so the Slovak colloquial expression "máš smolu" can be translated into English with just one word: "Tough!".