How to use this homNAND gate?
What's the point to build a cheap NAND gate ?


Because it's a complete operation; you can build any circuit with a sufficiently large pile of them.


  Maybe to build cheaper AND and OR gates based on this NAND gate and then follow the approach of Regev's type cryptosystems?


You could build AND and OR gates with them (and INVERT; AND and OR by themselves are not complete); however you'd probably be better off designing the larger circuits directly with the NAND gates.