The magic box puzzle: How to implement a verifiable self-contained secret system?
HMAC is what you need:


The HMAC of the secret is provided at the start of the round
The box gives out the HMAC key at the end of the round
If someone guessed the word correctly in the round, then the box gives the HMAC key out to all the contestants, so everyone can verify that the box is telling the truth that a contestant was correct
If no-one gets the answer by the end of the round, then the box can share the key, and the contestants can then verify that the secret is in fact what the box claimed it to be, and that the box isn't lying.


HMAC is hash based, and relies on a symmetric key, which satisfies your requirements