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A widespread security claim of the Bitcoin system, presented in the originalBitcoin white-paper, states that the security of the system is guaranteed aslong as there is no attacker in possession of half or more of the totalcomputational power used to maintain the system. This claim, however, is provedbased on theoretically flawed assumptions.  In the paper we analyze two kinds of attacks based on two theoretical flaws:the Block Discarding Attack and the Difficulty Raising Attack. We argue thatthe current theoretical limit of attacker's fraction of total computationalpower essential for the security of the system is in a sense not $\frac{1}{2}$but a bit less than $\frac{1}{4}$, and outline proposals for protocol changethat can raise this limit to be as close to $\frac{1}{2}$ as we want.  The basic idea of the Block Discarding Attack has been noted as early as2010, and lately was independently though-of and analyzed by both author ofthis paper and authors of a most recently pre-print published paper. We thusfocus on the major differences of our analysis, and try to explain theunfortunate surprising coincidence. To the best of our knowledge, the secondattack is presented here for the first time.

Theoretical Bitcoin Attacks with less than Half of the Computational Power (draft)