I might be wrong here, but it's my understanding that for a program to run on an unmodified XBox, the program has to be 'signed' (which I think is something to with the program (or it's disc) having a numeric value assigned to it as a sort of checksum of the program and it's data (so it's different for every program), and when you try to boot the disc the XBox works out what the correct value is for the program, and if the right number (value) is not present with the program then the XBox knows that the program hasn't been validated by Microsoft, and so the XBox refuses to run the program), and only Microsoft know the algorithm that produces the correct number for the program. Am I right?

If so, then could Microsoft have released details of the algorithm (if they so wished) and then anyone could have burnt discs that had working versions of homebrew on them that would boot and work on unmodified XBoxes, using a program that writes the correct number (deduced from each individual program) on the discs?

Even if it's possible, then there's no legal reason why they should do so, of course, and probably no moral reason why they should do so (they don't owe us the code), but it would have been good if they had done so, as people with unmodded XBoxes could play the emulators and ports of Doom/Duke Nukem 3D/Alien vs. Predator, etc.

Of course, a modded XBox is still much better than an unmodded XBox, even if Microsoft did release the algorithm, so it doesn't really matter to us that we never saw the algoithm, but it would have been good if they'd have released it, I think.

not quite how it works.

could microsoft release their master/private key? sure.

will they ever purposely do so? never.

Oh, I know they won't, I just think it would have been nice for us, and good publicity for them, if they had done so. And after all, since the XBox is commercially dead, then I doubt many software houses would have complained about them releasing it.

But you say "not quite how it works." - do you mean I misunderstand about the key/algorithm/whatever workings? If so, then how does it work, please?

if we had the algorythem then all we would do is sign the softmod installer programs..
We would not bother to sign all the multitude of other stuff excetera.

So really its not going to happen.

The algorithms used are 2048 bit RSA for the signing, and SHA-1 for the hash, both publicly available. It's the signing key that's secret, not the algorithms. This Wikipedia article is a good place to start to understand how it works.