Let me be more concrete in implementation details: 

1) burn transaction sends at least n satoshis to an OP_RETURN h, 
2) h mod m matches the bitcoin block hash mod m, for the block the burn 
transaction was mined into.
3) The side chain block header hashes to h and also contains the bitcoin block 
hight.
4) a side chain block releases x new side coins

Since the burn hash does not reveal in advance which side chain it will be used 
for, the Bitcoin miner can not selectively block burn mining. They will include 
loosing bets for the Bitcoin fee. Bitcoin miner have no advantage over 
independent burn miner of the side chain.

Anyone who issues a burn transaction that complies the rules 1-3 has 1/m the 
chance to win the next block on the side chain. This implies a fair exchange 
rate of n*m satoshis = x side coins (at the margin).

Should two burn transactions fulfill the mod m lottery criteria, then we have a 
competing fork on the side chain. Just as for Bitcoin, the next block(s) will 
pick the winner. 

To contain fork rate, the parameter m would have to be adjusted dynamically, 
similar to Bitcoins difficulty. It needs to increase if fork rate increases and 
decrease if no valid block is burned with Bitcoin blocks. Unfortunately SPV can 
only prove the existence of a transaction, but not the non-existence of an 
alternative. Therefore the fork rate within a block cycle can not be evaluated 
with SPV proofs. 

Rational burn miner who frequently faces and loses head-to-head runs with a 
competing forks would increase his bet for the next burn cycle, as increasing 
the individual bet amount has the advantage that if he wins his victory is more 
stable. Remember the side chain trunk is selected as the one with highest 
cumulative burn.

Consequently cumulative burn within an adjustment period (measured in Bitcoin 
blocks) is expected to rise in face of high fork rate. If the sample period 
burn exceeds a target, then it would trigger a rise to the lottery criteria m, 
reducing the fork rate and vs.

Tamas Blummer
Bits of Proof

On Dec 10, 2014, at 8:35 AM, Tamas Blummer <ta...@bitsofproof.com> wrote:

> 
> We spend scarce resources external to the digital realm to create Bitcoin. 
> Real world sacrifice is needed to avoid “nothing at stake”  and sybil 
> attacks. With Bitcoin we now have a scarce resource within the digital realm, 
> so it appeals my intuition to re-use it for sacrifice instead of linking 
> again an external, real world resource. 
> 
> In following I outline a new mining algorithm for side chains, that burn 
> Bitcoins to secure them.
> 
> The side chain block validity rules would require that a transaction on the 
> Bitcoin block chain provably destroys Bitcoins with an OP_RET output, that 
> contains the hash of the block header of the side chain. To also introduce a 
> lottery, the burn transaction’s hash is required to satisfy some function of 
> the block hash it was included in on the Bitcoin block chain. For example 
> modulo m of the burn transaction hash must match modulo m of the block hash, 
> that is not known in advance.
> 
> Those who want to mine the side chain will assemble  side chain block 
> candidates that comply the rules of the side chain, then a Bitcoin 
> transaction burning to the hash of the block candidate and submit it to the 
> Bitcoin network. Should he burn transaction be included into the Bitcoin 
> block chain and the Bitcoin block’s hash satisfy the lottery criteria, then 
> the block candidate can be submitted to extend the side chain.
> 
> A side chain block header sequence would be accepted as side chain trunk if a 
> sequence of Bitcoin SPV proofs for burn transactions prove, that linked 
> blocks have the highest cumulative burn, if compared to alternative 
> sequences. 
> 
> The Bitcoin miner will include burn transactions because they offer Bitcoin 
> fees. Bitcoin miner can not selectively block side chains since the hashes 
> associated with the burn do not disclose which side chain or other project 
> they are for. Here you have a “merged mining” that does not need Bitcoin 
> miner support or even consent.
> 
> Mining difficulty of the side chain could be adjusted by stepping up the 
> required burn and/or hardening the criteria that links a burn proof 
> transaction with the bitcoin block hash it is included in.
> 
> The difficulty to mine with burn would be dynamic and would also imply a 
> floating exchange rate between Bitcoin and the side coin.
> 
> Tamas Blummer
> Bits of Proof
> 
> 00000000000000001172380e63346e3e915b52fcbae838ba958948ac9aa85edd

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