Majority is not Enough:Bitcoin Mining is Vulnerable∗
Ittay Eyal and Emin Gün Sirer
Department of Computer Science, Cornell University
ittay.eyal@cornell.edu, egs@systems.cs.cornell.edu
Abstract. The Bitcoin cryptocurrency records its transactions in a public log called the blockchain. Its security rests critically on the distributed
protocol that maintains the blockchain, run by participants called miners. Conventional wisdom asserts that the mining protocol is incentivecompatible and secure against colluding minority groups, that is, it incentivizes miners to follow the protocol as prescribed.
We show that the Bitcoin mining protocol is not incentive-compatible.
We present an attack with which colluding miners obtain a revenue larger
than their fair share. This attack can have significant consequences for
Bitcoin: Rational miners will prefer to join the selfish miners, and thecolluding group will increase in size until it becomes a majority. At thispoint, the Bitcoin system ceases to be a decentralized currency.
Unless certain assumptions are made, selfish mining may be feasible forany group size of colluding miners. We propose a practical modification to
the Bitcoin protocol that protects Bitcoin in the general case. It prohibits
selfish mining by pools that command less than 1/4 of the resources. This
threshold is lower than the wrongly assumed 1/2 bound, but better thanthe current reality where a group of any size can compromise the system.

1

Introduction

Bitcoin [23] is a cryptocurrency that has recently emerged as a popular medium
of exchange, with a rich and extensive ecosystem. The Bitcoin network runs atover 42×1018 FLOPS [9], with a total market capitalization around 12 billion US
Dollars as of January 2014 [10]. Central to Bitcoin’s operation is a global, publiclog, called the blockchain, that records all transactions between Bitcoin clients.
The security of the blockchain is established by a chain of cryptographic puzzles,
solved by a loosely-organized network of participants called miners. Each minerthat successfully solves a cryptopuzzle is allowed to record a set of transactions,and to collect a reward in Bitcoins. The more mining power (resources) a miner
applies, the better are its chances to solve the puzzle first. This reward structure
provides an incentive for miners to contribute their resources to the system, andis essential to the currency’s decentralized nature.
The Bitcoin protocol requires a majority of the miners to be honest; thatis, follow the Bitcoin protocol as prescribed. By construction, if a set of colluding miners comes to command a majority of the mining power in the network,∗

This research was supported by the NSF Trust STC and by DARPA

the currency stops being decentralized and becomes controlled by the colludinggroup. Such a group can, for example, prohibit certain transactions, or all ofthem. It is, therefore, critical that the protocol be designed such that miners
have no incentive to form such large colluding groups.
Empirical evidence shows that Bitcoin miners behave strategically and form
pools. Specifically, because rewards are distributed at infrequent, random intervals, miners form mining pools in order to decrease the variance of their income
rate. Within such pools, all members contribute to the solution of each cryptopuzzle, and share the rewards proportionally to their contributions. To the bestof our knowledge, such pools have been benign and followed the protocol so far.
Indeed, conventional wisdom has long asserted that the Bitcoin mining protocol is equitable to its participants and secure against malfeasance by a nonmajority attacker (Section 7). Barring recently-explored Sybil attacks on transaction propagation [4], there were no known techniques by which a minority
of colluding miners could earn disproportionate benefits by deviating from theprotocol. Because the protocol was believed to reward miners strictly in proportion to the ratio of the overall mining power they control, a miner in a largepool was believed to earn the same revenue as it would in a small pool. Consequently, if we ignore the fixed cost of pool operation and potential economies ofscale, there is no advantage for colluding miners to organize into ever-increasing
pools. Therefore, pool formation by honest rational miners poses no threat tothe system.
In this paper, we show that the conventional wisdom is wrong: the Bitcoin
mining protocol, as prescribed and implemented, is not incentive-compatible. We
describe a strategy that can be used by a minority pool to obtain more revenue
than the pool’s fair share, that is, more than its ratio of the total mining power.
The key idea behind this strategy, called Selfish Mining, is for a pool tokeep its discovered blocks private, thereby intentionally forking the chain. The
honest nodes continue to mine on the public chain, while the pool mines on its
own private branch. If the pool discovers more blocks, it develops a longer leadon the public chain, and continues to keep these new blocks private. When the
public branch approaches the pool’s private branch in length, the selfish miners
reveal blocks from their private chain to the public.
This strategy leads honest miners that follow the Bitcoin protocol to waste
resources on mining cryptopuzzles that end up serving no purpose. Our analysis
demonstrates that, while both honest and selfish parties waste some resources,
the honest miners waste proportionally more, and the selfish pool’s rewards
exceed its share of the network’s mining power, conferring it a competitive advantage and incentivizing rational miners to join the selfish mining pool.
We show that, above a certain threshold size, the revenue of a selfish poolrises superlinearly with pool size above its revenue with the honest strategy.
This fact has critical implications for the resulting system dynamics. Once aselfish mining pool reaches the threshold, rational miners will preferentially join
selfish miners to reap the higher revenues compared to other pools. Such a selfish
mining pool can quickly grow towards a majority. If the pool tips the majoritythreshold (due to the addition of malicious actors aimed at undermining thesystem, rational actors wishing to usurp the currency, perhaps covertly, or due
to momentum in pool popularity), it can switch to a modified protocol that
ignores blocks generated outside the pool, to become the only creator of blocks
and reap all the mining revenue. A majority pool wishing to remain covert may
remain a benign monopolist, accepting blocks from third-parties on occasion toprovide the illusion of decentralization, while retaining the ability to reap fullrevenue when needed, as well as the ability to launch double-expenditure attacks
against merchants. Either way, the decentralized nature of the currency will havecollapsed, and a single entity, the selfish pool manager, will control the system.
Since a selfish mining pool that exceeds threshold size poses a threat to the
Bitcoin system, we characterize how the threshold varies as a function of message
propagation speed in the network. We show that, for a mining pool with highconnectivity and good control on information flow, the threshold is close to zero.
This implies that, if less than 100% of the miners are honest, the system may not
be incentive compatible: The first selfish miner will earn proportionally higher
revenues than its honest counterparts, and the revenue of the selfish mining pool
will increase superlinearly with pool size.
We further show that the Bitcoin mining protocol will never be safe against
attacks by a selfish mining pool that commands more than 1/3 of the totalmining power of the network. Such a pool will always be able to collect mining
rewards that exceed its proportion of mining power, even if it loses every single
block race in the network. The resulting bound of 2/3 for the fraction of Bitcoin
mining power that needs to follow the honest protocol to ensure that the protocol
remains resistant to being gamed is substantially lower than the 50% figure
currently assumed, and difficult to achieve in practice. Finally, we suggest asimple modification to the Bitcoin protocol that achieves a threshold of 1/4.
This change is backwards-compatible and progressive; that is, it can be adopted
by current clients with modest changes, does not require full adoption to provide
a benefit, and partial adoption will proportionally increase the threshold.
In summary, the contributions of this work are:
1. Introduction of the Selfish-Mine strategy, which demonstrates that Bitcoin
mining is not incentive compatible (Section 3).
2. Analysis of Selfish-Mine, and when it can benefit a pool (Section 4).
3. Analysis of majority-pool formation in face of selfish mining (Section 5).
4. A simple backward-compatible progressive modification to the Bitcoin protocol that would raise the threshold from zero to 1/4 (Section 6).
We are unaware of previous work that addresses the security of theblockchain. We provide an overview of related work in Section 7, and discuss
the implications of our results in Section 8.

2

Preliminaries

Bitcoin is a distributed, decentralized crypto-currency [8,7,23,6]. The users of
Bitcoin are called clients, each of whom can command accounts, known as addresses. A client can send Bitcoins to another client by forming a transaction andcommitting it into a global append-only log called the blockchain. The blockchain
is maintained by a network of miners, which are compensated for their effort in
Bitcoins. Bitcoin transactions are protected with cryptographic techniques that
ensure only the rightful owner of a Bitcoin address can transfer funds from it.
The miners are in charge of recording the transactions in the blockchain,
which determines the ownership of Bitcoins. A client owns x Bitcoins at time t
if, in the prefix of the blockchain up to time t, the aggregate of transactions
involving that client’s address amounts to x. Miners only accept transactions if
their inputs are unspent.

2.1

Blockchain and Mining

The blockchain records the transactions in units of blocks. Each block includes
a unique ID, and the ID of the preceding block. The first block, dubbed thegenesis block, is defined as part of the protocol. A valid block contains a solution
to a cryptopuzzle involving the hash of the previous block, the hash of thetransactions in the current block, and a Bitcoin address which is to be creditedwith a reward for solving the cryptopuzzle. This process is called Bitcoin mining,
and, by slight abuse of terminology, we refer to the creation of blocks as block
mining. The specific cryptopuzzle is a double-hash whose result has to be smaller
than a set value. The problem difficulty, set by this value, is dynamically adjusted
such that blocks are generated at an average rate of one every ten minutes.
Any miner may add a valid block to the chain by simply publishing it overan overlay network to all other miners. If two miners create two blocks withthe same preceding block, the chain is forked into two branches, forming a tree.
Other miners may subsequently add new valid blocks to either branch. When aminer tries to add a new block after an existing block, we say it mines on theexisting block. This existing block may be the head of a branch, in which case
we say the miner mines on the head of the branch, or simply on the branch.
The formation of branches is undesirable since the miners have to maintain a
globally-agreed totally ordered set of transactions. To resolve forks, the protocolprescribes miners to adopt and mine on the longest chain.1 All miners add blocks
to the longest chain they know of, or the first one they heard of if there are
branches of equal length. This causes forked branches to be pruned; transactionsin pruned blocks are ignored, and may be resubmitted by clients.
We note that block dissemination over the overlay network takes seconds,
whereas the average mining interval is 10 minutes. Accidental bifurcation istherefore rare, and occurs on average once about every 60 blocks [12].
When a miner creates a block, it is compensated for its efforts with Bitcoins.
This compensation includes a per-transaction fee paid by the users whose trans1

The criterion is actually the most difficult chain in the block tree, i.e., the one that
required (in expectancy) the most mining power to create. To simplify presentation,and because it is usually the case, we assume the set difficulty at the different
branches is the same, and so the longest chain is also the most difficult one.

actions are included, as well as an amount of new Bitcoins that did not exist
before.2
2.2 Pool formation
The probability of mining a block is proportional to the computational resources
used for solving the associated cryptopuzzle. Due the nature of the mining process, the interval between mining events exhibits high variance from the point ofview of a single miner. A single home miner using a dedicated ASIC is unlikely tomine a block for years [31]. Consequently, miners typically organize themselves
into mining pools. All members of a pool work together to mine each block, andshare their revenues when one of them successfully mines a block. While joining
a pool does not change a miner’s expected revenue, it decreases the variance andmakes the monthly revenues more predictable.

3

The Selfish-Mine Strategy

First, we formalize a model that captures the essentials of Bitcoin mining behavior and introduces notation for relevant system parameters. Then we detailthe selfish mining algorithm.
3.1

Modeling Miners and Pools

The system is comprised
Pn of a set of miners 1, . . . , n. Each miner i has mining
power mi , such that i=1 mi = 1. Each miner chooses a chain head to mine, andfinds a subsequent block for that head after a time interval that is exponentiallydistributed with mean m−1
i . We assume that miners are rational; that is, they
try to maximize their revenue, and may deviate from the protocol to do so.
A group of miners can form a pool that behaves as single agent with acentralized coordinator, following some strategy. The mining power of a poolis the sum of mining power of its members, and its revenue is divided amongits members according to their relative mining power [30]. The expected relative
revenue, or simply the revenue of a pool is the expected fraction of blocks thatwere mined by that pool out of the total number of blocks in the longest chain.
3.2

Selfish-Mine

We now describe our strategy, called Selfish-Mine. As we show in Section 4,
Selfish-Mine allows a pool of sufficient size to obtain a revenue larger than its
ratio of mining power. For simplicity, and without loss of generality, we assume
that miners are divided into two groups, a colluding minority pool that follows
the selfish mining strategy, and a majority that follows the honest mining strategy (others). It is immaterial whether the honest miners operate as a single
group, as a collection of groups, or individually.
2

The rate at which the new Bitcoins are generated is designed to slowly decrease
towards zero, and will reach zero when almost 21 million Bitcoins are created. Then,the miners’ revenue will be only from transaction fees.

Algorithm 1: Selfish-Mine
1
2
3
4
5
6
7
8
9
10
11
12
13
14
15
16
17
18
19
20
21
22
23
24
25
26
27

on Initpublic chain ← publicly known blocks
private chain ← publicly known blocks
privateBranchLen ← 0Mine at the head of the private chain.
on My pool found a block∆prev ← length(private chain) − length(public chain)append new block to private chain
privateBranchLen ← privateBranchLen + 1
if ∆prev = 0 and privateBranchLen = 2 thenpublish all of the private chain
privateBranchLen ← 0Mine at the new head of the private chain.
on Others found a block∆prev ← length(private chain) − length(public chain)append new block to public chain
if ∆prev = 0 thenprivate chain ← public chain
privateBranchLen ← 0else if ∆prev = 1 thenpublish last block of the private chain
else if ∆prev = 2 thenpublish all of the private chain
privateBranchLen ← 0
else
publish first unpublished block in private block.
Mine at the head of the private chain.

(Was tie with branch of 1)(Pool wins due to the lead of 1)(they win)

(Now same length. Try our luck)(Pool wins due to the lead of 1)(∆prev > 2)

The key insight behind the selfish mining strategy is to force the honest miners into performing wasted computations on the stale public branch. Specifically,
selfish mining forces the honest miners to spend their cycles on blocks that aredestined to not be part of the blockchain.
Selfish miners achieve this goal by selectively revealing their mined blocks toinvalidate the honest miners’ work. Approximately speaking, the selfish mining
pool keeps its mined blocks private, secretly bifurcating the blockchain and creating a private branch. Meanwhile, the honest miners continue mining on the
shorter, public branch. Because the selfish miners command a relatively small
portion of the total mining power, their private branch will not remain ahead of
the public branch indefinitely. Consequently, selfish mining judiciously reveals
blocks from the private branch to the public, such that the honest miners willswitch to the recently revealed blocks, abandoning the shorter public branch.
This renders their previous effort spent on the shorter public branch wasted,and enables the selfish pool to collect higher revenues by incorporating a higher
fraction of its blocks into the blockchain.
Armed with this intuition, we can fully specify the selfish mining strategy,
shown in Algorithm 1. The strategy is driven by mining events by the selfish
pool or by the others. Its decisions depend only on the relative lengths of theselfish pool’s private branch versus the public branch. It is best to illustrate
the operation of the selfish mining strategy by going through sample scenarios
involving different public and private chain lengths.

When the public branch is longer than the private branch, the selfish mining
pool is behind the public branch. Because of the power differential between theselfish miners and the others, the chances of the selfish miners mining on theirown private branch and overtaking the main branch are small. Consequently, theselfish miner pool simply adopts the main branch whenever its private branchfalls behind. As others find new blocks and publish them, the pool updates andmines at the current public head.
When the selfish miner pool finds a block, it is in an advantageous position
with a single block lead on the public branch on which the honest miners operate.
Instead of naively publishing this private block and notifying the rest of theminers of the newly discovered block, selfish miners keep this block private to
the pool. There are two outcomes possible at this point: either the honest miners
discover a new block on the public branch, nullifying the pool’s lead, or else thepool mines a second block and extends its lead on the honest miners.
In the first scenario where the honest nodes succeed in finding a block on the
public branch, nullifying the selfish pool’s lead, the pool immediately publishes
its private branch (of length 1). This yields a toss-up where either branch may
win. The selfish miners unanimously adopt and extend the previously private
branch, while the honest miners will choose to mine on either branch, dependingon the propagation of the notifications. If the selfish pool manages to minea subsequent block ahead of the honest miners that did not adopt the pool’s
recently revealed block, it publishes immediately to enjoy the revenue of boththe first and the second blocks of its branch. If the honest miners mine a block
after the pool’s revealed block, the pool enjoys the revenue of its block, while
the others get the revenue from their block. Finally, if the honest miners minea block after their own block, they enjoy the revenue of their two blocks while
the pool gets nothing.
In the second scenario, where the selfish pool succeeds in finding a second
block, it develops a comfortable lead of two blocks that provide it with somecushion against discoveries by the honest miners. Once the pool reaches thispoint, it continues to mine at the head of its private branch. It publishes one
block from its private branch for every block the others find. Since the selfish poolis a minority, its lead will, with high probability, eventually reduce to a single
block. At this point, the pool publishes its private branch. Since the private
branch is longer than the public branch by one block, it is adopted by all miners
as the main branch, and the pool enjoys the revenue of all its blocks. This brings
the system back to a state where there is just a single branch until the poolbifurcates it again.

4

Analysis

We can now analyze the expected rewards for a system where the selfish poolhas mining power of α and the others of (1 − α).
Figure 1 illustrates the progress of the system as a state machine. The statesof the system represent the lead of the selfish pool; that is, the difference between

Fig. 1: State machine with transition frequencies.

the number of unpublished blocks in the pool’s private branch and the lengthof the public branch. Zero lead is separated to states 0 and 0’. State 0 is thestate where there are no branches; that is, there is only a single, global, public
longest chain. State 0’ is the state where there are two public branches of lengthone: the main branch, and the branch that was private to the selfish miners, andpublished to match the main branch. The transitions in the figure correspondto mining events, either by the selfish pool or by the others. Recall that these
events occur at exponential intervals with an average frequency of α and (1 − α),respectively.
We can analyze the expected rewards from selfish mining by taking into account the frequencies associated with each state transition of the state machine,and calculating the corresponding rewards. Let us go through the various casesand describe the associated events that trigger state transitions.
If the pool has a private branch of length 1 and the others mine one block,the pool publishes its branch immediately, which results in two public branchesof length 1. Miners in the selfish pool all mine on the pool’s branch, because asubsequent block discovery on this branch will yield a reward for the pool. The
honest miners, following the standard Bitcoin protocol implementation, mine on
the branch they heard of first. We denote by γ the ratio of honest miners thatchoose to mine on the pool’s block, and the other (1 − γ) of the non-pool miners
mine on the other branch.
For state s = 0, 1, 2, . . . , with frequency α, the pool mines a block and thelead increases by one to s + 1. In states s = 3, 4, . . . , with frequency (1 − α),the honest miners mine a block and the lead decreases by one to s − 1. If theothers mine a block when the lead is two, the pool publishes its private branch,and the system drops to a lead of 0. If the others mine a block with the lead
is 1, we arrive at the aforementioned state 0’. From 0’, there are three possible
transitions, all leading to state 0 with total frequency 1: (1) the pool mines a
block on its previously private branch (frequency α), (2) the others mine a blockon the previously private branch (frequency γ(1 − α)), and (3) the others minea block on the public branch (frequency (1 − γ)(1 − α)).

4.1State Probabilities

We analyze this state machine to calculate its probability distribution over thestate space. We obtain the following equations:
αp0 = (1 − α)p1 + (1 − α)p2 p00 = (1 − α)p1
αp1 = (1 − α)p2
(1)


∀k
≥
2
:
αp
=
(1
−
α)p

k
k+1

 P∞
0
k=0 pk + p0 = 1
Solving (1) (See our full report for details [14]), we get:α − 2α2α(2α3 − 4α2 + 1)
(1 − α)(α − 2α2 )p00 =1 − 4α2 + 2α3
α − 2α2p1 =
2α3 − 4α2 + 1
k−1αα − 2α2∀k ≥ 2 : pk =1−α2α3 − 4α2 + 1
p0 =

4.2

(2)
(3)
(4)
(5)

Revenue

The probability distribution over the state space provides the foundation foranalyzing the revenue obtained by the selfish pool and by the honest miners.
The revenue for finding a block belongs to its miner only if this block ends up
in the main chain. We detail the revenues on each event below.
(a) Any state but two branches of length 1, pools finds a block. The pool appends
one block to its private branch, increasing its lead on the public branch byone. The revenue from this block will be determined later.

=⇒
or

=⇒
(b) Was two branches of length 1, pools finds a block. The pool publishes its
secret branch of length two, thus obtaining a revenue of two.

=⇒=⇒(c) Was two branches of length 1, others find a block after pool head. The pooland the others obtain a revenue of one each — the others for the new head,
the pool for its predecessor.

=⇒(d) Was two branches of length 1, others find a block after others’ head. The
others obtain a revenue of two.

=⇒(e) No private branch, others find a block. The others obtain a revenue of one,and both the pool and the others start mining on the new head.
(f) Lead was 1, others find a block. Now there are two branches of length one,and the pool publishes its single secret block. The pool tries to mine on its
previously private head, and the others split between the two heads. Denoteby γ the ratio of others that choose the non-pool block.
The revenue from this block cannot be determined yet, because it depends
on which branch will win. It will be counted later.

=⇒=⇒(g) Lead was 2, others find a block. The others almost close the gap as the leaddrops to 1. The pool publishes its secret blocks, causing everybody to startmining at the head of the previously private branch, since it is longer. The
pool obtains a revenue of two.

=⇒=⇒(h) Lead was more than 2, others win. The others decrease the lead, which
remains at least two. The new block (say with number i) will end outside
the chain once the pool publishes its entire branch, therefore the others
obtain nothing. However, the pool now reveals its i’th block, and obtains arevenue of one.

=⇒=⇒

We calculate the revenue of the pool and of the others from the state probabilities and transition frequencies:
Case (c)Case (d)Case (e)

z
}|
{ z
}|
{ z
}|
{
rothers = p00 · γ(1 − α) · 1 + p00 · (1 − γ)(1 − α) · 2 + p0 · (1 − α) · 1
Case (b)Case (c)Case (g)(6)

Case (h)

}|
{ z
}|
{ z
}|
{
z }| { zrpool = p00 · α · 2 + p00 · γ(1 − α) · 1 + p2 · (1 − α) · 2 + P [i > 2](1 − α) · 1(7)As expected, the intentional branching brought on by selfish mining leads
the honest miners to work on blocks that end up outside the blockchain. This, inturn, leads to a drop in the total block generation rate with rpool + rothers < 1.
The protocol will adapt the mining difficulty such that the mining rate at themain chain becomes one block per 10 minutes on average. Therefore, the actual
revenue rate of each agent is the revenue rate ratio; that is, the ratio of its
blocks out of the blocks in the main chain. We substitute the probabilities from(2)-(5) in the revenue expressions of (6)-(7) to calculate the pool’s revenue for0 ≤ α ≤ 21 :rpoolα(1 − α)2 (4α + γ(1 − 2α)) − α3
= ··· =.
(8)
Rpool =
rpool + rothers
1 − α(1 + (2 − α)α)4.3

Simulation

To validate our theoretical analysis, we compare its result with a Bitcoin protocol simulator. The simulator is constructed to capture all the salient Bitcoin
mining protocol details described in previous sections, except for the cryptopuzzle module that has been replaced by a Monte Carlo simulator that simulates
block discovery without actually computing a cryptopuzzle. In this experiment,
we use the simulator to simulate 1000 miners mining at identical rates. A subset
of 1000α miners form a pool running the Selfish-Mine algorithm. The other miners follow the Bitcoin protocol. We assume block propagation time is negligible
compared to mining time, as is the case in reality. In the case of two branchesof the same length, we artificially divide the non-pool miners such that a ratioof γ of them mine on the pool’s branch and the rest mine on the other branch.
Figure 2 shows that the simulation results match the theoretical analysis.
4.4

The Effect of α and γ

When the pool’s revenue given in Equation 8 is larger than α, the pool will earnmore than its relative size by using the Selfish-Mine strategy. Its miners will
therefore earn more than their relative mining power. Recall that the expression
is valid only for 0 ≤ α ≤ 21 . We solve this inequality and phrase the result in thefollowing observation:
Observation 1 For a given γ, a pool of size α obtains a revenue larger than its
relative size for α in the following range:
1−γ
1
<α<
.
3 − 2γ2(9)

Fig. 2: Pool revenue using the Selfish-Mine strategy for different propagation factors γ, comparedto the honest Bitcoin mining protocol. Simulation
matches the theoreticalanalysis, and both showthat Selfish-Mine resultsin higher revenues thanthe honest protocol abovea threshold, which depends on γ.

Fig. 3: For a given γ, the threshold α shows the minimum power
selfish mining pool that will trump
the honest protocol. The current
Bitcoin protocol allows γ = 1,
where Selfish-Mine is always superior. Even under unrealistically favorable assumptions, the thresholdis never below 1/3.

We illustrate this in Figure 2, where we see the pool’s revenue for different γ
values with pool size ranging from 0 (very small pool) to 0.5 (half of the miners).
Note that the pool is only at risk when it holds exactly one block secret, andthe honest miners might publish a block that would compete with it. For γ = 1,
the pool can quickly propagate its one-block branch if the others find their own
branch, so all honest miners would still mine on the pool’s block. In this case,the pool takes no risk when following the Selfish-Mine strategy and its revenue
is always better than when following the honest algorithm. The threshold istherefore zero, and a pool of any size can benefit by following Selfish-Mine. In
the other extreme, γ = 0, the honest miners always publish and propagate theirblock first, and the threshold is at 1/3. With γ = 1/2 the threshold is at 1/4.
Figure 3 shows the threshold as a function of γ.
We also note that the slope of the pool revenue, Rpool , as a function of
the pool size is larger than one above the threshold. This implies the following
observation:
Observation 2 For a pool running the Selfish-Mine strategy, the revenue ofeach pool member increases with pool size for pools larger than the threshold.

5

Pool Formation

We have shown that once a selfish pool’s mining power exceeds the threshold,it can increase its revenue by running Selfish-Mine (Theorem 1). At this point,
rational miners will preferentially join the selfish pool to increase their revenues.
Moreover, the pool’s members will want to accept new members, as this would
increase their own revenue (Observation 2). The selfish pool would therefore
increase in size, unopposed by any mechanism, towards a majority. Once a minerpool, selfish or otherwise, reaches a majority, it controls the blockchain. The
Selfish-Mine strategy then becomes unnecessary, since the others are no longer
faster than the pool. Instead, a majority pool can collect all the system’s revenue
by switching to a modified Bitcoin protocol that ignores blocks generated outside
the pool; it also has no motivation to accept new members. At this point, thecurrency is not a decentralized currency as originally envisioned.

6

Hardening the Bitcoin Protocol

Ideally, a robust currency system would be designed to resist attacks by groups ofcolluding miners. Since selfish mining attacks yield positive outcomes for groupsizes above the threshold, the protocol should be amended to set the thresholdas high as possible. In this section, we argue that the current Bitcoin protocol
has no measures to guarantee a low γ. This implies that the threshold may
be as low as zero, and a pool of any size can benefit by running Selfish-Mine.
We suggest a simple change to the protocol that, if adopted by all non-selfish
miners, sets γ to 1/2, and therefore the threshold to 1/4. This change is backward
compatible; that is, any subset of the miners can adopt it without hindering theprotocol. Moreover, it is progressive; that is, any ratio of the miners that adopts
it decreases γ, and therefore increases the threshold.
6.1 Problem
The Bitcoin protocol prescribes that when a miner knows of multiple branches ofthe same length, it mines and propagates only the first branch it received. Recall
that a pool that runs the Selfish-Mine strategy and has a lead of 1 publishes its
secret block P once it hears of a competing block X found by a non-pool block.
If block P reaches a non-pool miner before block X, that miner will mine on P .
Because selfish mining is reactive, and it springs into action only after thehonest nodes have discovered a block X, it may seem to be at a disadvantage.
But a savvy pool operator can perform a sybil attack on honest miners by adding
a significant number of zero-power miners to the Bitcoin miner network. These
virtual miners act as advance sensors by participating in data dissemination,
but do not mine new blocks. (Babaioff et al. also acknowledge the feasibility ofsuch a sybil attack [4]). The virtual miners are managed by the pool, and once
they hear of block X, they ignore it and start propagating block P . The random
peer-to-peer structure of the Bitcoin overlay network will eventually propagate
X to all miners, but the propagation of X under these conditions will be strictlyslower than that of block P . By adding enough virtual nodes, the pool operator
can thus increase γ. The result, as shown in Equation 9, is a threshold close tozero.
6.2 Solution
We propose a simple, backwards-compatible change to the Bitcoin protocol toaddress this problem and raise the threshold. Specifically, when a miner learnsof competing branches of the same length, it should propagate all of them, andchoose which one to mine on uniformly at random. In the case of two branchesof length 1, as discussed in Section 4, this would result in half the nodes (inexpectancy) mining on the pool’s branch and the other half mining on the otherbranch. This yields γ = 1/2, which in turn yields a threshold of 1/4.
Each miner implementing our change decreases the selfish pool’s ability toincrease γ through control of data propagation. This improvement is independentof the adoption of the change at other miners, therefore it does not require a
hard fork. This change to the protocol does not introduce new vulnerabilities tothe protocol: Currently, when there are two branches of equal length, the choiceof each miner is arbitrary, effectively determined by the network topology andlatency. Our change explicitly randomizes this arbitrary choice, and therefore
does not introduce new vulnerabilities.

7

Related Work

Decentralized digital currencies have been proposed before Bitcoin, startingwith [11] and followed by peer-to-peer currencies [32,34]; see [22,5] for short surveys. None of these are centered around a global log; therefore, their techniquesand challenges are unrelated to this work.
Several dozen cryptocurrencies have followed Bitcoin’s success [18,17,33],
most prominently Litecoin [21]. These currencies are based on a global log, whichis extended by the users’ efforts. We conjecture that the essential technique of
withholding blocks for selfish mining can be directly applied to all such systems.
It was commonly believed that the Bitcoin system is sound as long as a majority of the participants honestly follow the protocol, and the “51% attack” wasthe chief concern [23,1,20]. The notion of soundness for a nascent, distributed,
Internet-wide, decentralized system implies the presence of incentives for adoption of the prescribed protocol, for such incentives ensure a robust system comprised of participants other than enthusiastic and altruistic early adopters. Felten [15] notes that “there was a folk theorem that the Bitcoin system was stable,in the sense that if everyone acted according to their incentives, the inevitable result would be that everyone followed the rules of Bitcoin as written.” Others [25]have claimed that “the well-known argument – never proven, but taken on intuitive faith – that a minority of miners can’t control the network is a special case ofa more general assumption: that a coalition of miners with X% of the network’s
hash power can make no more than X% of total mining revenues.” A survey [5]on the technical features responsible for Bitcoin’s success notes that the Bitcoin

design “addresses the incentive problems most expeditiously,” while Bitcoin tutorials for the general public hint at incentives designed to align participants’and the system’s goals [27]. More formally, Kroll, Davey and Felten’s work [19]provides a game-theoretic analysis of Bitcoin, without taking into account block
withholding attacks such as selfish mining, and argues that the honest strategy
constitutes a Nash equilibrium, implying incentive-compatibility.
Our work shows that the real Bitcoin protocol, which permits block withholding and thereby enables selfish mining-style attacks, does not constitute anequilibrium. It demonstrates that the Bitcoin mining system is not incentive
compatible even in the presence of an honest majority. Over 2/3 of the participants need to be honest to protect against selfish mining, under the most
optimistic of assumptions.
A distinct exception from this common wisdom is a discussion of maintaining
a secret fork in the Bitcoin forums, mostly by users3 btchris, ByteCoin, mtgox,
and RHorning [28]. The approach, dubbed the Mining Cartel Attack, is inferior
to selfish mining in that the cartel publishes two blocks for every block publishedby the honest nodes. This discussion does not include an analysis of the attack(apart from a brief note on simulation results), does not explore the danger ofthe resulting pool dynamics, and does not suggest a solution to the problem.
The influential work of Rosenfeld [30] addresses the behavior of miners in thepresence of different pools with different rewards. Although it addresses revenue
maximization for miners with a set mining power, this work is orthogonal tothe discussion of Selfish Mining, as it centers around the pool reward system.
Both selfish pools and honest pools should carefully choose their reward method.
Since a large-enough selfish pool would earn more than its mining power, any
fair reward method would provide more reward to its miners, so rational miners
would choose it over an honest pool.
Recent work [4] addresses the lack of incentives for disseminating transactions
between miners, since each of them prefers to collect the transaction fee himself.
This is unrelated to the mining incentive mechanism we discuss.
A widely cited study [29] examines the Bitcoin transaction graph to analyze
client behavior. The analysis of client behavior is not directly related to our work.
The Bitcoin blockchain had one significant bifurcation in March 2013 due toa bug [2]. It was solved when the two largest pools at the time manually pruned
one branch. This bug-induced fork, and the one-off mechanism used to resolve it,
are fundamentally different from the intentional forks that Selfish-Mine exploits.
In a block withholding attack, a pool member decreases the pool revenue by
never publishing blocks it finds. Although it sounds similar to the strategy of
Selfish-Mine, the two are unrelated, as our work that deals with an attack by
the pool on the system.
Various systems build services on top of the Bitcoin global log, e.g., improved
coin anonymity [22], namespace maintenance [24] and virtual notaries [16,3].
These services that rely on Bitcoin are at risk in case of a Bitcoin collapse.
3

In alphabetical order

8Discussion

We briefly discuss below several points at the periphery of our scope.
System Collapse The Bitcoin protocol is designed explicitly to be decentralized.
We therefore refer to a state in which a single entity controls the entire currency
system as a collapse of Bitcoin.
Note that such a collapse does not immediately imply that the value of a
Bitcoin drops to 0. The controlling entity will have an incentive to accept mosttransactions, if only to reap their fees, and because if it mines all Bitcoins, it has
strong motivation that they maintain their value. It may also choose to remaincovert, and hide the fact that it can control the entire currency. An analysis of a
Bitcoin monopolist’s behavior is beyond the scope of this paper, but we believethat a currency that is de facto or potentially controlled by a single entity may
deter many of Bitcoin’s clients.
Detecting Selfish Mining There are two telltale network signatures of selfish
mining that can be used to detect when selfish mining is taking place, but neither
are easy to measure definitively.
The first and strongest sign is that of abandoned (orphaned) chains, where
the block race that takes place as part of selfish mining leaves behind blocks
that were not incorporated into the blockchain. Unfortunately, it is difficult todefinitively account for abandoned blocks, as the current protocol prunes anddiscards such blocks inside the network. A measurement tool that connects tothe network from a small number of vantage points may miss abandoned blocks.
The second indicator of selfish mining activity is the timing gap between
successive blocks. A selfish miner who squelches an honest chain of length N witha chain of length N + 1 will reveal a block very soon after its predecessor. Since
normal mining events should be independent, one would expect block discovery
times to be exponentially distributed. A deviation from this distribution would
be suggestive of mining activity. The problems with this approach are that it
detects only a subset of the selfish miner’s behavior (the transition from state 2 tostate 0 in the state machine), the signature behavior occurs relatively rarely, andsuch a statistical detector may take a long time to accrue statistically significant
data.
Measures and Countermeasures Although miners may choose to collude in aselfish mining effort, they may prefer to hide it in order to avoid public criticism
and countermeasures. It is easy to hide Selfish-Mine behavior, and difficult toban it. A selfish pool may never reveal its size by using different Bitcoin addresses
and IP addresses, and by faking block creation times. The rest of the network
would not even suspect that a pool is near a dangerous threshold.
Moreover, the honest protocol is public, so if a detection mechanism is set
up, a selfish pool would know its parameters and use them to avoid detection.
For instance, if the protocol was defined to reject blocks with creation time
below a certain threshold, the pool could publish its secret blocks just before
this threshold.

A possible line of defense against selfish mining pools is for counter-attackers
to infiltrate selfish pools and expose their secret blocks for the honest miners.
However, selfish pool managers can, in turn, selectively reveal blocks to subsetsof the members in the pool, identify spy nodes through intersection, and expel
nodes that leak information.
Thieves and Snowballs Selfish mining poses two kinds of danger to the Bitcoin
ecosystem: selfish miners reap disproportionate rewards, and the dynamics favor
the growth of selfish mining pools towards a majority, in a snowball effect. The
system would be immune to selfish mining if there were no pools above thethreshold size. Yet, since the current protocol has no guaranteed lower boundon this threshold, it cannot automatically protect against selfish miners.
Even with our proposed fix that raises the threshold to 25%, the systemremains vulnerable: there already exist pools whose mining power exceeds the
25% threshold [26], and at times, even the 33% theoretical hard limit. Responsible miners should therefore break off from large pools until no pool exceeds thethreshold size.
Responsible Disclosure Because of Bitcoin’s decentralized nature, selfish mining can only be thwarted by collective, concerted action. There is no central
repository, no push mechanism and no set of privileged developers; all protocol
modifications require public discussion prior to adoption. In order to promote a
swift solution and to avoid a scenario where some set of people had the benefitof selective access, we published a preliminary report [14] and explained boththe problem and our suggested solution in public forums [13].

9Conclusion

Bitcoin is the first widely popular cryptocurrency with a broad user base anda rich ecosystem, all hinging on the incentives in place to maintain the critical Bitcoin blockchain. Our results show that Bitcoin’s mining protocol is notincentive-compatible. We presented Selfish-Mine, a mining strategy that enables
pools of colluding miners that adopt it to earn revenues in excess of their mining power. Higher revenues can lead new miners to join a selfish miner pool,a dangerous dynamic that enables the selfish mining pool to grow towards amajority. The Bitcoin system would be much more robust if it were to adopt
an automated mechanism that can thwart selfish miners. We offer a backwardscompatible modification to Bitcoin that ensures that pools smaller than 1/4 ofthe total mining power cannot profitably engage selfish mining. We also showthat at least 2/3 of the network needs to be honest to thwart selfish mining; asimple majority is not enough.
Acknowledgements We are grateful to Raphael Rom, Fred B. Schneider, Eva
Tardos, and Dror Kronstein for their valuable advice on drafts of this paper, as
well as our shepherd Rainer Böhme for his guidance.

References
1. andes: Bitcoin’s kryptonite: The 51% attack. https://bitcointalk.org/index.php?topic=12435(June 2011)
2. Andresen, G.: March 2013 chain fork post-mortem. BIP 50, https://en.bitcoin.it/wiki/BIP_50,
retrieved Sep. 2013
3. Araoz, M.: Proof of existence. http://www.proofofexistence.com/, retrieved Sep. 2013
4. Babaioff, M., Dobzinski, S., Oren, S., Zohar, A.: On Bitcoin and red balloons. In: ACM Conference on Electronic Commerce. pp. 56–73 (2012)
5. Barber, S., Boyen, X., Shi, E., Uzun, E.: Bitter to better, how to make Bitcoin a better currency.
In: Financial Cryptography and Data Security, pp. 399–414. Springer (2012)
6. Bitcoin community: Bitcoin source. https://github.com/bitcoin/bitcoin, retrieved Sep. 2013
7. Bitcoin community: Protocol rules. https://en.bitcoin.it/wiki/Protocol_rules, retrieved
Sep. 2013
8. Bitcoin
community:Protocolspecification.
https://en.bitcoin.it/wiki/Protocol_
specification, retrieved Sep. 2013
9. bitcoincharts.com: Bitcoin network. http://bitcoincharts.com/bitcoin/, retrieved Nov. 2013
10. blockchain.info: Bitcoin market capitalization. http://blockchain.info/charts/market-cap, retrieved Jan. 2014
11. Chaum, D.: Blind signatures for untraceable payments. In: Crypto. vol. 82, pp. 199–203 (1982)
12. Decker, C., Wattenhofer, R.: Information propagation in the Bitcoin network. In: IEEE P2P(2013)
13. Eyal, I., Sirer, E.G.: Bitcoin is broken. http://hackingdistributed.com/2013/11/04/
bitcoin-is-broken/ (2013)
14. Eyal, I., Sirer, E.G.: Majority is not enough: Bitcoin mining is vulnerable. arXiv preprint
arXiv:1311.0243 (2013)
15. Felten, E.W.: Bitcoin research in Princeton CS. https://freedom-to-tinker.com/blog/felten/
bitcoin-research-in-princeton-cs/ (November 2013)
16. Kelkar, A., Bernard, J., Joshi, S., Premkumar, S., Sirer, E.G.: Virtual notary. http://
virtual-notary.org/, retrieved Sep. 2013
17. King, S.: Primecoin: Cryptocurrency with prime number proof-of-work. http://primecoin.org/
static/primecoin-paper.pdf (2013)
18. King, S., Nadal, S.: PPCoin: Peer-to-peer crypto-currency with proof-of-stake. https://archive.
org/details/PPCoinPaper (2012)
19. Kroll, J.A., Davey, I.C., Felten, E.W.: The economics of Bitcoin mining or, Bitcoin in thepresence of adversaries. In: Workshop on the Economics of Information Security (2013)
20. Lee, T.B.: Four reasons Bitcoin is worth studying. http://www.forbes.com/sites/timothylee/
2013/04/07/four-reasons-bitcoin-is-worth-studying/2/ (April 2013)
21. Litecoin Project: Litecoin, open source P2P digital currency. https://litecoin.org, retrieved
Sep. 2013
22. Miers, I., Garman, C., Green, M., Rubin, A.D.: Zerocoin: Anonymous distributed e-cash from
Bitcoin. In: IEEE Symposium on Security and Privacy (2013)
23. Nakamoto, S.: Bitcoin: A peer-to-peer electronic cash system (2008)
24. Namecoin Project: Namecoin DNS – DotBIT project. https://dot-bit.org, retrieved Sep. 2013
25. Narayanan, A., Miller, A.: Why the Cornell paper on Bitcoin mining is important. https://freedom-to-tinker.com/blog/randomwalker/why-the-cornell-paper-on-bitcoinmining-is-important/ (November 2013)
26. Neighborhood Pool Watch: October 27th 2013 weekly pool and network statistics. http://organofcorti.blogspot.com/2013/10/october-27th-2013-weekly-pool-and.html, retrieved Oct. 201327. Pacia, C.: Bitcoin mining explained like you’re five: Part 1 – incentives. http://chrispacia.
wordpress.com/2013/09/02/bitcoin-mining-explained-like-youre-five-part-1-incentives/(September 2013)
28. RHorning, mtgox, btchris, ByteCoin: Mining cartel attack. https://bitcointalk.org/index.php?
topic=2227 (December 2010)
29. Ron, D., Shamir, A.: Quantitative analysis of the full Bitcoin transaction graph. In: Financial
Cryptography. pp. 6–24 (2013)
30. Rosenfeld, M.: Analysis of Bitcoin pooled mining reward systems. arXiv preprint
arXiv:1112.4980 (2011)31. Swanson, E.: Bitcoin mining calculator. http://www.alloscomp.com/bitcoin/calculator, retrieved Sep. 2013
32. Vishnumurthy, V., Chandrakumar, S., Sirer, E.G.: Karma: A secure economic framework forpeer-to-peer resource sharing. In: Workshop on Economics of Peer-to-Peer Systems (2003)
33. Wikipedia: List of cryptocurrencies. https://en.wikipedia.org/wiki/List_of_cryptocurrencies,
retrieved Oct. 2013
34. Yang, B., Garcia-Molina, H.: PPay: Micropayments for peer-to-peer systems. In: Proceedingsof the 10th ACM conference on Computer and communications security. pp. 300–310. ACM(2003)