EDIT: this comment was mistaken, bitcoin uses elliptic curve based key pairs, as hackcasual points out below.
One thing to keep in mind:
Bitcoin wallets are implemented with public/private key pairs. If you believed that you had a method to crack that, well you probably couldn't just take all the bitcoin (people would notice and the market value would evaporate), but you could probably figure out a way to make at least 1% (a couple billion). So if it can be broken with a group of smart people thinking hard, that sounds like a startup opportunity.
Bitcoin signatures rely on the difficulty of elliptic logarithms, not factoring, and only publish hashes of the public keys until they spend an address, meaning the vulnerable window is quite short as long as they never reuse private keys.
The papers claim, that np but very likely not np-hard problems are likely to be in p is applicable though to breaking ECC
Not if you did it in a smart way. Let's say you could arbitrarily make transactions. Probably the best way to do it would be to steal all the coins from a particular exchange, like Coinbase. Then everyone would think Coinbase pwnd but bitcoin is fine. Rinse and repeat with other exchanges once a year and you can make a hefty profit.
One thing to keep in mind:
Bitcoin wallets are implemented with public/private key pairs. If you believed that you had a method to crack that, well you probably couldn't just take all the bitcoin (people would notice and the market value would evaporate), but you could probably figure out a way to make at least 1% (a couple billion). So if it can be broken with a group of smart people thinking hard, that sounds like a startup opportunity.