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amspir
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March 30, 2014, 10:15:09 PM
Last edit: March 30, 2014, 11:21:59 PM by amspir
 #1

If

   H = current network hashrate
   M = cost per TH/s
   C = market cap of bitcoin

The attacker index (my invention) would be:

   A = (H*M)/C

which represents the percentage of the market cap needed to purchase enough hashing power to get 50% control the network.

Using current numbers,

   H = 45,000 TH/s
   M = $3000 per TH/s
   C = $5.7 billion

This number is 2.4%.    This seems to me to be too low, thus too easy to launch a 51% attack.   What would be a comfortable number?


Here is my concern:  to control a corporation, you need to own more than 50% of the shares.   To control bitcoin, you currently need only 2.4% of the bitcoins in existence
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March 30, 2014, 10:52:08 PM
 #2

I don't think a 51% attack will ever happen to Bitcoin, maybe another cryptocurrency, but not Bitcoin. I can see how it's fun to think about the possibility but it wouldn't be  worth the trouble.

If you owned over 50% of the hash rate you would be making 25 BTC (~ $12,000 3/30/14) every 18 minutes. Why would you risk prison and make Bitcoin dive in price overnight to gain bitcoin? You're already gaining LOADS of bitcoin!

I'm sure some conspiracy theorist will say that a government or giant corporation like Visa might want to. But again, if you're controlling half the bitcoin entering the market it would be very easy to just create crazy wild price swings until everyone is running away from investing in it and it dies, and no one risks prison.

By the way - don't let any government agents or finance corporations read this. Don't want to give them any ideas.
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March 30, 2014, 11:07:13 PM
 #3

I don't think a 51% attack will ever happen to Bitcoin, maybe another cryptocurrency, but not Bitcoin. I can see how it's fun to think about the possibility but it wouldn't be  worth the trouble.

I'd just like to see that number get out of the reach of larger governments.

mtGox lost then found 200K btc.  That represents 1.6% of the current market cap, which is uncomfortably close to the Attacker Index.

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March 30, 2014, 11:07:25 PM
 #4

Many currencies have succumbed to hash attacks and other attacks as well.

Any attacker who could attack bitcoin by brute force would have infinite fiat so for btc your index is of less interest than for others.

Stealthy leveraged attacks are another matter.

Give a man a fish and he eats for a day.  Give a man a Poisson distribution and he eats at random times independent of one another, at a constant known rate.
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March 30, 2014, 11:08:40 PM
 #5

For governments consider instead the share of global semi fab reqiired to achieve an attack.

Give a man a fish and he eats for a day.  Give a man a Poisson distribution and he eats at random times independent of one another, at a constant known rate.
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March 30, 2014, 11:27:37 PM
 #6

If

   H = current network hashrate
   M = cost per TH/s
   C = market cap of bitcoin

The attacker index (my invention) would be:

   A = (H*M)/C

which represents the percentage of the market cap needed to purchase enough hashing power to get 50% control the network.

Using current numbers,

   H = 45,000 TH/s
   M = $3000 per TH/s
   C = $5.7 billion

This number is 2.4%.    This seems to me to be too low, thus too easy to launch a 51% attack.   What would be a comfortable number?


One thing to keep in mind is that if someone were to do an operation this big, you would have to factor in more than cost per TH/s to the total cost. You need a physical location to house all of the equipment, cooling equipment to keep all the equipment at a functioning temperature, electricity cost to run everything, probably a significant and expensive amount of bandwidth, shipping cost for all of the equipment, and possibly more (that's just what I thought of off the top of my head). It is interesting to think about how much it would cost, though in reality I think purchasing enough equipment to do this and actually take over the network would be near impossible. If someone were to take over the network, the best way would probably be to use "zombie" computers like in DDoS attacks (except mining bitcoins), and amass an army of "zombie" computers large enough to gather 51% of the mining network, although this is probably impossible too.

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March 30, 2014, 11:38:01 PM
 #7

One thing to keep in mind is that if someone were to do an operation this big, you would have to factor in more than cost per TH/s to the total cost. You need a physical location to house all of the equipment, cooling equipment to keep all the equipment at a functioning temperature, electricity cost to run everything, probably a significant and expensive amount of bandwidth, shipping cost for all of the equipment, and possibly more (that's just what I thought of off the top of my head). It is interesting to think about how much it would cost, though in reality I think purchasing enough equipment to do this and actually take over the network would be near impossible. If someone were to take over the network, the best way would probably be to use "zombie" computers like in DDoS attacks (except mining bitcoins), and amass an army of "zombie" computers large enough to gather 51% of the mining network, although this is probably impossible too.

How it's done wouldn't be that important.  

I've updated my OP with the concern.   To execute a hostile takeover of a corporation, you would need to purchase just over 50% of the company's stock.

To execute a "hostile takeover" of bitcoin, you would only need 2.4% of the bitcoins in existence.

I'm not trying to be alarmist, I would like to see the Attacker Index approach 51%.   

Someone please prove me wrong, so I can sleep better.

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March 30, 2014, 11:53:55 PM
 #8

If

   H = current network hashrate
   M = cost per TH/s
   C = market cap of bitcoin

The attacker index (my invention) would be:

   A = (H*M)/C

which represents the percentage of the market cap needed to purchase enough hashing power to get 50% control the network.

Using current numbers,

   H = 45,000 TH/s
   M = $3000 per TH/s
   C = $5.7 billion

This number is 2.4%.    This seems to me to be too low, thus too easy to launch a 51% attack.   What would be a comfortable number?


Here is my concern:  to control a corporation, you need to own more than 50% of the shares.   To control bitcoin, you currently need only 2.4% of the bitcoins in existence


This is a good start and a useful metric to track. However, I'm sure on a few minutes of thought (and perhaps just by writing this) you realize this is an incomplete model. We will need to make a more complete model before we can attempt to quantify a "comfortable" (safe) value for A.

First, I'm going to be perhaps a pedant and enforce that if we are to deal with formulas, we must not be careless: If you wish for A to be expressed as you have, you must write:

Code:
A = (H*M)(100)/C

Where:
H = current network hashrate
M = cost to buy cheapest hashrate hardware
C = current Bitcoin market cap
A = Attacker index (%)

Next, we must understand that the situation is far from static. This is primarily what must be modelled to correct the formula's lack of descriptive power.

H must be described at the rate that it will be expanding at, because a 51% attack that only lasts for a block is not useful; one must sustain it.

M must be described considering the market effects of attempting to purchase that much hashpower: The nature of supply and demand dictates that you will not be able to buy that much at the 1-unit rate. There may not even be enough, which either means the short-term cost of purchasing units approaches infinity (in the real world the market will not react so fast, but you will spend significantly more and will be left with insufficient hashpower in the case that enough does not exist). This means you must consider the cost of special manufacture, and the cost of doing it in secret. Existing R&D and fabs with hash ASIC experience will not keep this size a secret because it is not in their interest (or, put differently, to motivate them would inflate the cost substantially). Starting from scratch, conversely, is also an incredibly costly endeavor. (ASIC spins at the process geometry we're at -- the one you need to compete assuming you aren't somehow stealing electricity, another cost you have not yet accounted for -- cost $2M+.)

C must be described in motion as well. It will certainly be affected: If you propose (by using A as you do, it seems you propose this; either way, there will be an effect) using X% market cap to buy the ASICs, that amount of market will not have zero effect on the cost, which affects other miners, etc; if you buy with dollars, you are injecting money into the Bitcoin ecosystem which will drive the price likely upward to some extent; etc etc.

C must also be considered in the sense that Bitcoin's C is not isolated from the other coins. Much speculation is done via BTC -> ABC coin. Depending on how things move, they too will react, by re-entering or leaving the Bitcoin market.

There is much more too, as we descend into nth-order effects.

Arbitrarily, I say that we are over-protected by hashpower at present, given the market cap; yet, underprotected IF you assume that a Visa or whoever would find it more valuable to try to kill it than adapt. Many will try to legislate it away, but I do not anticipate a 51% attack. The incentives of it don't add up. Even if Visa wanted to drop tens of millions, it would STILL be more profitable to corner the market in mining cooperatively. I'd entertain some concerns about the potential for cartelization of mining where transactions are not treated on a highest-fee-first basis but based on other dealings (the Bitcoin equivalent of a violation of net neutrality, which has been dead for a while because of peering agreements, which is essentially where Bitcoin will head eventually to some extent).

That combined with some forms of the "selfish miner" attack may warrant some exploration, on how we may disincentivize it.

Beyond that, the primary risk (in that it is non-trivial to stop) is what I call the Joker Attack: Some people just want to see the world burn. If you assume that is true enough, then if a true Joker (who is one that does not behave the way that game theory would describe as "rationally") arose with enough capital to pull off such a feat, we might be in trouble.

But the same is already true for every other system on the planet. All is designed to some degree with the idea that "all" people are rational (in the game theoretic sense). A superhuman (financially or otherwise) Joker is not a threat that can be dealt with. It is a side topic, but I can demonstrate that without a solution to N vs. NP, we cannot know how; and, moreover, depending on the solution, we STILL may not be able to know: We just might know that we cannot know.

-J

Uberlurker. Been here since the Finney transaction. Please consider this before replying; there is a good chance I've heard it before.

-Citizenfive
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March 31, 2014, 12:13:51 AM
 #9

One thing to keep in mind is that if someone were to do an operation this big, you would have to factor in more than cost per TH/s to the total cost. You need a physical location to house all of the equipment, cooling equipment to keep all the equipment at a functioning temperature, electricity cost to run everything, probably a significant and expensive amount of bandwidth, shipping cost for all of the equipment, and possibly more (that's just what I thought of off the top of my head). It is interesting to think about how much it would cost, though in reality I think purchasing enough equipment to do this and actually take over the network would be near impossible. If someone were to take over the network, the best way would probably be to use "zombie" computers like in DDoS attacks (except mining bitcoins), and amass an army of "zombie" computers large enough to gather 51% of the mining network, although this is probably impossible too.

How it's done wouldn't be that important.  

I've updated my OP with the concern.   To execute a hostile takeover of a corporation, you would need to purchase just over 50% of the company's stock.

To execute a "hostile takeover" of bitcoin, you would only need 2.4% of the bitcoins in existence.

I'm not trying to be alarmist, I would like to see the Attacker Index approach 51%.  

Someone please prove me wrong, so I can sleep better.



No sir, that isn't how it works. Around these parts we adhere to strict rules of Socratic dialogue. One need not prove that you are wrong. You must give compelling evidence that you are right. You have not yet done so. You have demonstrated, in a simplified system that does not take into account the dimension of time, with grievous assumptions that there will be no supply-demand market reactions to the actions you propose, that 2.4% of the market cap of Bitcoin could acquire hashpower equivalent to >50% the current network hashrate.

You have far, far, far more to do to even approach an acceptable level of logical support for your premise. Right now, you aren't much better than, "If I could create an airframe with 0.02 Cd, and if I could make it structurally sound and weigh 500 lbs, I could fly from Texas to New York using a propeller fitted to a lawn-trimmer on a single tank. I'm considering being a commercial pilot, but if that's possible, I shouldn't waste my time. Someone prove to me that one cannot make a flying machine from a lawn trimmer as described, or I will stop my flying lessons."

No offense, but I do not care whether you do or do not carry on with your foray into piloting, and it is not my job to prove it profitable against all possible assertions regardless of merit. The only thing I will do, as a participant in this good forum, is engage in a give-and-take logical debate based on the use of well-formed logical structures.

"Given X and Y, if we assume L and M based on supporting evidence Q, R, S, and the logical inferences W and V, I propose C. Discuss."

That is how this works.

Regards,
-J

Uberlurker. Been here since the Finney transaction. Please consider this before replying; there is a good chance I've heard it before.

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March 31, 2014, 12:30:43 AM
 #10

How it's done wouldn't be that important.  

I've updated my OP with the concern.   To execute a hostile takeover of a corporation, you would need to purchase just over 50% of the company's stock.

To execute a "hostile takeover" of bitcoin, you would only need 2.4% of the bitcoins in existence.

I'm not trying to be alarmist, I would like to see the Attacker Index approach 51%.   

Someone please prove me wrong, so I can sleep better.

OHHH I think I misunderstood what you wrote. That is kind of a scary thought I guess. I still think it wouldn't be feasible for a couple reasons though.

1. I think they are a bit more expensive than $3k per TH/s... I didn't look very hard but after a few minutes looking around on google the cheapest I found was $5,500.
2. Obtaining 45,000 of those would be extremely difficult, and by the time you did it, you would probably need a lot more mining power to take over half of the network.

Honestly I think if somebody had the time, money, and resources to do this, they could make a lot more money doing something else like starting a business or something.

Plus even if someone did take over the bitcoin mining network and was able to change the blockchain to give themselves a ton of bitcoins, people would realize this pretty quickly. And as soon as people realized this, bitcoin's value would plummet down to 0 like literally instantly.

Well idk, I hope this helps you sleep lol

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March 31, 2014, 12:38:26 AM
 #11

One thing to keep in mind is that if someone were to do an operation this big, you would have to factor in more than cost per TH/s to the total cost. You need a physical location to house all of the equipment, cooling equipment to keep all the equipment at a functioning temperature, electricity cost to run everything, probably a significant and expensive amount of bandwidth, shipping cost for all of the equipment, and possibly more (that's just what I thought of off the top of my head). It is interesting to think about how much it would cost, though in reality I think purchasing enough equipment to do this and actually take over the network would be near impossible. If someone were to take over the network, the best way would probably be to use "zombie" computers like in DDoS attacks (except mining bitcoins), and amass an army of "zombie" computers large enough to gather 51% of the mining network, although this is probably impossible too.

How it's done wouldn't be that important.   

I've updated my OP with the concern.   To execute a hostile takeover of a corporation, you would need to purchase just over 50% of the company's stock.

To execute a "hostile takeover" of bitcoin, you would only need 2.4% of the bitcoins in existence.

I'm not trying to be alarmist, I would like to see the Attacker Index approach 51%.   

Someone please prove me wrong, so I can sleep better.



No sir, that isn't how it works. Around these parts we adhere to strict rules of Socratic dialogue. One need not prove that you are wrong. You must give compelling evidence that you are right. You have not yet done so. You have demonstrated, in a simplified system that does not take into account the dimension of time, with grievous assumptions that there will be no supply-demand market reactions to the actions you propose, that 2.4% of the market cap of Bitcoin could acquire hashpower equivalent to >50% the current network hashrate.

You have far, far, far more to do to even approach an acceptable level of logical support for your premise. Right now, you aren't much better than, "If I could create an airframe with 0.02 Cd, and if I could make it structurally sound and weigh 500 lbs, I could fly from Texas to New York using a propeller fitted to a lawn-trimmer on a single tank. I'm considering being a commercial pilot, but if that's possible, I shouldn't waste my time. Someone prove to me that one cannot make a flying machine from a lawn trimmer as described, or I will stop my flying lessons."

No offense, but I do not care whether you do or do not carry on with your foray into piloting, and it is not my job to prove it profitable against all possible assertions regardless of merit. The only thing I will do, as a participant in this good forum, is engage in a give-and-take logical debate based on the use of well-formed logical structures.

"Given X and Y, if we assume L and M based on supporting evidence Q, R, S, and the logical inferences W and V, I propose C. Discuss."

That is how this works.

Regards,
-J

Mother of god. I usually just lurk via Zeroblock and read what pops up, but I met Fenix IRL at the texas conference (I was in the circle listening to you and Vitalik chat after the etheruem presentation). I've seen his (FenixRD) posts for a long time, and I just had to register to say: You are fucking scary, man. You're like... idk, maybe Seven from Voyager. Or Spock? One of those fucks with unasailable (afaict) debate skill. But not like "always wins arguments" or something... just "always right" (also afaict lol). I'm glad you're on our side and involved, and thanks for popping into bitcointalk here and there still. I'm roughly trollbox-skill in debate, I didn't know what I thought about ethereum cuz that stuff is more complex than bitcoin which I barely grasp. At least I understand now as much as, it may not work, but it could, and has reason to, and in any case, I can confidently ignore the trollbox etc accusations that ethereum is a scamcoin. (Hopefully it works as designed too!) Anyway, sorry for the bro-lovefest. Hope you stick around for a long time.
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March 31, 2014, 12:42:08 AM
Last edit: March 31, 2014, 01:21:18 AM by amspir
 #12

You have demonstrated, in a simplified system that does not take into account the dimension of time, with grievous assumptions that there will be no supply-demand market reactions to the actions you propose, that 2.4% of the market cap of Bitcoin could acquire hashpower equivalent to >50% the current network hashrate.

That was what I needed, because obviously, as doubling the network's hashrate overnight with a single purchase is not possible, and not assumed in the metric.

I suppose a more sophisticated metric would involve modeling the rate of hashrate/s equipment being produced globally, which could be interpolated from the change in the network hashrate over time?

[edited]
For instance, making a simplified model that the attacker would buy X% of the the globally produced output, increasing M by Y%, which would result in a 2 value answer, the % of market cap and number of weeks needed.

A sustained purchase of over 50% of the world's mining equipment output would be needed to sustain the attack, so X would have to be greater than 50% to gain control.
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March 31, 2014, 01:04:25 AM
 #13

1. I think they are a bit more expensive than $3k per TH/s... I didn't look very hard but after a few minutes looking around on google the cheapest I found was $5,500.

Cointerra IV (2 TH/s for $6000) ready for delivery in June is my basis for that number.

Quote
2. Obtaining 45,000 of those would be extremely difficult, and by the time you did it, you would probably need a lot more mining power to take over half of the network.

Very true.   I am very grateful to have been shown the error of my ways.
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March 31, 2014, 01:16:08 AM
 #14

OHHH I think I misunderstood what you wrote. That is kind of a scary thought I guess. I still think it wouldn't be feasible for a couple reasons though.

1. I think they are a bit more expensive than $3k per TH/s... I didn't look very hard but after a few minutes looking around on google the cheapest I found was $5,500.
2. Obtaining 45,000 of those would be extremely difficult, and by the time you did it, you would probably need a lot more mining power to take over half of the network.

Honestly I think if somebody had the time, money, and resources to do this, they could make a lot more money doing something else like starting a business or something.

Plus even if someone did take over the bitcoin mining network and was able to change the blockchain to give themselves a ton of bitcoins, people would realize this pretty quickly. And as soon as people realized this, bitcoin's value would plummet down to 0 like literally instantly.

Well idk, I hope this helps you sleep lol

Also, there are a couple entities that definitely have that much or more. It's non-trivial, but certainly not difficult, to buy 2.4% of the market cap, even accounting for the bid-ask spread. We've seen 1% sold ON-MARKET in three days and the price only dropped by about as much.

And then, many thousands are aware of BTC and have access to the equivalent to 2.4% in dollars, to say nothing of other currencies, plus corporations etc.

Why has no one yet done this, if it were so simple? Globally, thousands at a minimum have the resources and the awareness of the basics you posted. (Hundreds of thousands have the resources.) Either it is nigh impossible, or it isn't profitable compared with cooperating in any event where it is possible (hint: the game theory consensus is that one).

Mother of god. I usually just lurk via Zeroblock and read what pops up, but I met Fenix IRL at the texas conference (I was in the circle listening to you and Vitalik chat after the etheruem presentation). I've seen his (FenixRD) posts for a long time, and I just had to register to say: You are fucking scary, man. You're like... idk, maybe Seven from Voyager. Or Spock? One of those fucks with unasailable (afaict) debate skill. But not like "always wins arguments" or something... just "always right" (also afaict lol). I'm glad you're on our side and involved, and thanks for popping into bitcointalk here and there still. I'm roughly trollbox-skill in debate, I didn't know what I thought about ethereum cuz that stuff is more complex than bitcoin which I barely grasp. At least I understand now as much as, it may not work, but it could, and has reason to, and in any case, I can confidently ignore the trollbox etc accusations that ethereum is a scamcoin. (Hopefully it works as designed too!) Anyway, sorry for the bro-lovefest. Hope you stick around for a long time.

I'm not good with compliments, so, simply: thanks for the love. I appreciate being appreciated.

That was what I needed, because obviously, as doubling the network's hashrate overnight with a single purchase is not possible, and not assumed in the metric.

I suppose a more sophisticated metric would involve modeling the rate of hashrate/s equipment being produced globally, which could be interpolated from the change in the network hashrate over time?

For instance, ing a simplified model that the attacker would buy 50% of the the globally produced output, increasing M by 200%, which would result in a 2 value answer, the % of market cap and number of weeks needed.

Really, this isn't something that can be done with a few variables and one equation. Either calculus or a very sizable array of variables which affect each other over a large system of equations. (Calculus is easier, probably; self-modifying equations and recursion is half of what it was made for.)

I make the following claim:

It will be far less trivial than 2.4%. Furthermore, while it may hurt your sleep, it will in fact be less than the amount needed to sustain a >50% attack. Perhaps not by a very large amount, but we aren't talking 49.9% either. But that is only before you consider the final piece: The relative cost of pulling it off vs. participating cooperatively.

As this is a claim which I make intuitively and do not intend to support with evidence beyond "it hasn't happened yet, therefore it's simpler (and therefore more mathematically sound) to assume I'm right", I offer the following instead.

I wager 25 BTC for a proof that contradicts this claim.

Specifically, the terms are:

1. Far less trivial than 2.4% but yet not 50%:
I will formalize this further and state that it is between 25% and 40%. I expect it to be somewhere in the vicinity and likely less than the probability of sustained selfish-mining as described in the several analyses about that. (Making the 50% thing already not the target: It's been theoretically demonstrated that proper selfish-mining can achieve sustained probabilistic network control at hashrate percentages in the mid-30s.) For purposes of the bet and economic variability, I formalize, as said, to between 25% and 40% nominally.

2. Nth-order effects:
Regardless of the precise percentage found for (1), it will be found that it is nominally *more profitable* still to cooperatively participate than undertake the "hostile takover", thus again reducing the only unpreventable risk to what I described as the "Wealthy Joker" Attack (which, again, is not an issue for Bitcoin any more than it is for everything else on the planet.)

-J

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-Citizenfive
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March 31, 2014, 01:40:32 AM
 #15

It will be far less trivial than 2.4%. Furthermore, while it may hurt your sleep, it will in fact be less than the amount needed to sustain a >50% attack. Perhaps not by a very large amount, but we aren't talking 49.9% either. But that is only before you consider the final piece: The relative cost of pulling it off vs. participating cooperatively.

My personal concern would be a wealthy government able to "control" the protocol, such that they could enforce sanctions on other nations or actors, where participating cooperatively is not an incentive.

I really hope you keep your 25 BTC.
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March 31, 2014, 02:17:10 AM
Last edit: March 31, 2014, 02:41:33 AM by FenixRD
 #16

It will be far less trivial than 2.4%. Furthermore, while it may hurt your sleep, it will in fact be less than the amount needed to sustain a >50% attack. Perhaps not by a very large amount, but we aren't talking 49.9% either. But that is only before you consider the final piece: The relative cost of pulling it off vs. participating cooperatively.

My personal concern would be a wealthy government able to "control" the protocol, such that they could enforce sanctions on other nations or actors, where participating cooperatively is not an incentive.

I really hope you keep your 25 BTC.


We already know the solution to that. Understanding this was foundational to the creation of the very field known as game theory. See: http://en.wikipedia.org/wiki/Nash_equilibrium

Sleep easy, bud. Wink

To accelerate the creation of a fully-developed equilibrium around Bitcoin (we aren't there yet, you're not wrong to have the idea floating around as a concern) I have a few projects that will help soon. But, this concern is the underlying reason why there almost certainly be only a few "real" (secure and dominant) coins, each with different and incompatible hash algorithms; probably there will be not more than two, perhaps three, algorithms in use (if better is found and it is better by significantly enough, that algorithm will replace another and an older coin will slowly go extinct) by dominant coins; and, it is extremely improbable there will be even two dominant coins using the same algorithm unless they are merge-mined.

EDIT: This applies to pseudonymous or anonymous fully-open-source coin designs. We will surely see specialized designs with some features, but the above does not apply to a coin where, say, verifiable and unspoofable IRL-linked ID is concerned. If Canada created Canadacoin and it was all green addresses from the start and that was effectively enforced, Canadacoin could gain a large userbase and be secured by taxes, or something (less efficient in the end). Probably they would protect this by making the mining algorithms a secret. This only turns it into the existing spy-vs.-spy stuff and loses the advantages of the further economy-of-scale achievable on a global level.

Uberlurker. Been here since the Finney transaction. Please consider this before replying; there is a good chance I've heard it before.

-Citizenfive
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June 15, 2014, 12:14:21 AM
 #17

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Here is my concern:  to control a corporation, you need to own more than 50% of the shares.   To control bitcoin, you currently need only 2.4% of the bitcoins in existence

Your flaw in your concern is what it would cost in terms of fiat to "buy" this attack.

This percentage will likely stay flat over the long term as the price per TH/s (or other unit of hashing) will be valued against the value of the coins a TH/s will be estimated to mine.

What you really want is to have the numerator to be a large as possible (network hashrate times cost per TH/s). 1/2 of this number is the cost to successfully launch a 51% attack (ignoring the fact that when you start to purchase a lot of mining capacity the price of miners will likely increase).
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