An illegal number is a number which is ostensibly illegal to know or distribute under legislation like the DMCA. In this case, the OP is referencing the cryptographic key used for DVDs, which the MPAA licensed to DVD player manufacturers. Since knowledge of it enabled the bypassing of encryption (otherwise known as "decrypting the content with the key"), the MPAA attempted to invoke the anti-circumvention clause of the DMCA to prevent people from publishing it.

Any "Secure Golden Key" would be a number which is the encryption backdoor key. Knowledge of that number would enable you to decrypt any content encrypted with that key, and if someone who were not a government actor were to discover that number, it would doubtless be decried as illegal to know, possess, or publish.

There was a legal battle in the early 2000s about circumventing DVD copy protections. The DVD CSS (content scramble system). The court ruling declared that the program was a 'copyright circumvention device'. It was illegal to distribute the source code. This lead to people creating illegal t-shirts with the source printed on it.

http://en.wikipedia.org/wiki/DeCSS

Later the same thing happened for Blu-ray. Only it was an encryption key that was discovered and subsequently illegal to distribute under the DMCA.

http://en.wikipedia.org/wiki/AACS_encryption_key_controversy