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There are no 'OUT' transactions with ether (link), but there are a lot of token transfers (link). In my understanding, most likely no one has a private key for this address, so where do these transactions come from?

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ERC20 token transactions don't have to follow the same rules as Ether transactions.

What the list of token transfers for address 0x00 is showing you is a list of Transfer events that have been raised by contracts that resemble ERC20 tokens. Any token contract can raise a Transfer event with any contents, so it's not necessarily reliable.

Take, for example, this "ChangeBank" transaction. The Event Log shows two events. The first one is a Transfer event, and the from address (topic 1) is indeed 0x00....00.

If you now look at the source code for the ChangeBank token, you can see the following at line 199:

  function mint(address _to, uint256 _amount) onlyOwner canMint returns (bool) {
    totalSupply = totalSupply.add(_amount);
    balances[_to] = balances[_to].add(_amount);
    Transfer(0X0, _to, _amount);
    return true;
  }

The author has coded the minting of new tokens to raise a Transfer event that shows as a transfer from address 0x00. This seems a bit odd to me, but fair enough; it has the desired effect of registering as an incoming transfer in the receiver's account, and who cares if the 0x00 account is inconsistent?

So, no private key for 0x00 is necessary to register apparent token movements out of 0x00.

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