is it possible for the cryptographic function to break?
It is possible for mathematicians to find weaknesses in the SHA-2, RIPEMD-160, and ECDSA algorithms that bitcoin uses. It is extremely unlikely (for all intents and purposes you'd be safe to consider it impossible) that significant enough weaknesses will be secretly and suddenly discovered in all 3 algorithms simultaneously by someone who intends to use the information for fraud and theft.
As such, bitcoin won't "die out" just because someone finds some weaknesses in some of the algorithms.
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i assume you mean that an attacker has fully mapped SHA-2 so that he can determine your private key and spend your bitcoins.
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Your private key for any address that has only received bitcoins and never sent them is protected by all three algorithms (ECDSA, SHA-2, and RIPEMD-160). A SHA-2 algorithm that is broken will allow miners to mine much faster, but it won't break bitcoin and it won't allow anyone to determine your private key. Once you send bitcoins from an address, and future bitcoins that are sent to that same address are only protected by ECDSA. As such a broken SHA-2 will have no effect at all on those addresses since the public key is already known to the world.
that's interesting ive been wondering that for a while now, you seem like you know quite a bit about bitcoins so I have another question, when a block is fully mined [releasing 25 coins] does that mean that the other people have to start on a totally different block, that is if three pools are working on a block and pool #1 solves block does that set pool 2 and 3 back?
You need to do some reading on how the mining process works before you start making guesses at the effects of a solved block. You have some very common assumptions about the mining process that are incorrect. It is much like rolling dice.
Hand 10 six sided dice to 100 people and ask them to start rolling those dice all together as fast as they can. Any time someone rolls 8 sixes in a single roll they are rewarded with a "block", and then the rolling continues. When one person rolls 8 sixes, have they set back the other people at all?
In the case of mining, the miners are each essentially rolling a single 2
256 sided die. Whenever a miner rolls a number that is lower than the current target difficulty they are allowed to publish the block that they are attempting to create, and the rolling continues.
that's interesting ive been wondering that for a while now, you seem like you know quite a bit about bitcoins so I have another question, when a block is fully mined [releasing 25 coins] does that mean that the other people have to start on a totally different block, that is if three pools are working on a block and pool #1 solves block does that set pool 2 and 3 back?
yes, that is correct.
No, you are giving out bad information.