Reading Mifare Classic chips with Samsung Galaxy S4? Searching the web far and wide has me almost out of hope in making Samsung's Galaxy S4 phone read the Mifare CLassic chip through NFC. But does anyone know of any form of work-around, or is this strictly a result of the NFC hardware in this phone?
No, the S4 can't read MIFARE Classic cards. The main problem is that NXP does not license the reader-side of its proprietary MIFARE Classic technology (specifically the use of the (broken) Crypto-1 algorithm) to other chip manufacturers. As a result, the Broadcom NFC controller inside the S4 does not support communication with such cards.

This alone would not be too difficult to circumvent. The Crypto-1 algorithm has been reverse-engineered and published, so it could easily be implemented in software within an Android app. However, the MIFARE Classic protocol does not fully comply to ISO/IEC 14443-3 (NFC-A) in that the authentication command uses a non-standard frame format. As a result, the communication facilities accessible on Android (the closest would be the NfcA tag technology, that requires adherence to ISO/IEC 14443-3 framing) can't be used to communicate with MIFARE Classic tags.