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As I understand it (please correct me if I'm wrong) a typical Litecoin derived alt coin has its own blockchain and can be mined entirely separately from any other crypto-currency. Some alt coin miners may choose to mine the coin because they speculate it may increase in value (amongst other reasons).

If I create my own crypto-currency based on the Ethereum Ecosystem, is it intrinsically bound to the Ethereum Blockchain? Therefore mining will occur only by those already mining Ether?

I did see this post where the author suggests:

Alternatively, I could accept gas fees in a currency other than Ether, by statically checking contracts to make sure they are guaranteed to pay me in some alternative token.

I guess that could occur if you reward miners with your own crypto-currency using:

balanceOf[block.coinbase] += 1; 

However, this would require altering the mining software?

Putting all this together, to replicate the concept, withing the Ethereum ecosystem, of an typical Bitcoin derived altcoin (and mining of it). I'd need to create my own token, include the balanceOf[block.coinbase] += 1 within the contract and offer my own mining software (or modification of an existing software) that filters for payments of my token?

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Ethereum-based tokens are intrinsically bound to Ethereum. Any token transaction would be in the form of a valid Ethereum transaction, in a block on the Ethereum chain. (see footnote)

It's entirely possible, as you've seen, to reward miners in a token for including a token-related transaction. This would indeed probably require modifying mining software, although in a future where tokens are common this might become a default setting. Either way, a token miner would also be an Ethereum miner. There'd be no way to mine only the tokens--nor, really, would there be any reason when you could get 5+ ETH per mined block for no extra effort.

At the moment, token transactions are payed for in ETH, which any miner would accept. The problem with paying solely in tokens is that even if someone will include the transaction, no unmodified miner would. That's why, as far as I know, there are no token-based fees currently in common use.

Footnote: An Ethereum smart contract could host a sidechain, and then the sidechain would have its own kind of miner. That said, sidechains have less use in Ethereum, since you can write contracts to have most sidechain features.

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