A trapdoor function isn't the same as asymmetric cryptography. Trapdoor functions can be used to build asymmetric cryptography, i.e. public-key encryption and

A trapdoor function isn't the same as asymmetric cryptography. Trapdoor functions can be used to build asymmetric cryptography, i.e. public-key encryption and signing. In general, hash-based asymmetric cryptography can do it with any hash you consider strong enough, such as SHA-256. Unfortunately the keys and signatures tend to be large, which is the problem the article is talking about. That's why they are not in common use. For example, although SHA-256 is 256 bits, which is considered small, the keys and signatures of asymmetric cryptography built with SHA-256 are very much larger. Making the keys and signatures smaller, yet secure and fast enough, is currently state of the art research.