[bitcoin-dev] Floating-Point Nakamoto Consensus

Mike Brooks f at in.st.capital
Thu Sep 24 19:40:46 UTC 2020

  Hey Everyone,

 A lot of work has gone into this paper, and the current revision has been
well received and there is a lot of excitement on this side to be sharing
it with you today. There are so few people that truly understand this
topic, but we are all pulling in the same direction to make Bitcoin better
and it shows.  It is wildly underrated that future proofing was never
really a consideration in the initial design - but here we are a decade
later with amazing solutions like SegWit which gives us a real
future-proofing framework.  The fact that future-proofing was added to
Bitcoin with a softfork gives me goosebumps. I'd just like to take the time
to thank the people who worked on SegWit and it is an appreciation that
comes up in conversation of how difficult and necessary that process
was, and this appreciation may not be vocalized to the great people who
worked on it. The fact that Bitcoin keeps improving and is able to respond
to new threats is nothing short of amazing - thank you everyone for a great

This current proposal really has nothing to do with SegWit - but it is an
update that will make the network a little better for the future, and we
hope you enjoy the paper.


Pull Request:


Floating-Point Nakamoto Consensus

Abstract — It has been shown that Nakamoto Consensus is very useful in the
formation of long-term global agreement — and has issues with short-term
disagreement which can lead to re-organization (“or-org”) of the
blockchain.  A malicious miner with knowledge of a specific kind of
denial-of-service (DoS) vulnerability can gain an unfair advantage in the
current Bitcoin network, and can be used to undermine the security
guarantees that developers rely upon.  Floating-Point Nakamoto consensu
makes it more expensive to replace an already mined block vs. creation of a
new block, and by resolving ambiguity of competition solutions it helps
achieve global consumers more quickly.  A floating-point fitness test
strongly incentivises the correct network behavior, and prevents
disagreement from ever forming in the first place.

The Bitcoin protocol was created to provide a decentralized consensus on a
fully distributed p2p network.  A problem arises when more than one
proof-of-work is presented as the next solution block in the blockchain.
Two solutions of the same height are seen as authoritative equals which is
the basis of a growing disagreement. A node will adopt the first solution
seen, as both solutions propagate across the network a race condition of
disagreement is formed. This race condition can be controlled by byzentiene
fault injection commonly referred to as an “eclipsing” attack.  When two
segments of the network disagree it creates a moment of weakness in which
less than 51% of the network’s computational resources are required to keep
the network balanced against itself.
Nakamoto Consensus

Nakamoto Consensus is the process of proving computational resources in
order to determine eligibility to participate in the decision making
process.  If the outcome of an election were based on one node (or
one-IP-address-one-vote), then representation could be subverted by anyone
able to allocate many IPs. A consensus is only formed when the prevailing
decision has the greatest proof-of-work effort invested in it. In order for
a Nakamoto Consensus to operate, the network must ensure that incentives
are aligned such that the resources needed to subvert a proof-of-work based
consensus outweigh the resources gained through its exploitation. In this
consensus model, the proof-of-work requirements for the creation of the
next valid solution has the exact same cost as replacing the current
solution. There is no penalty for dishonesty, and this has worked well in
practice because the majority of the nodes on the network are honest and
transparent, which is a substantial barrier for a single dishonest node to

A minimal network peer-to-peer structure is required to support Nakamoto
Conesus, and for our purposes this is entirely decentralized. Messages are
broadcast on a best-effort basis, and nodes can leave and rejoin the
network at will, accepting the longest proof-of-work chain as proof of what
happened while they were gone.  This design makes no guarantees that the
peers connected do not misrepresent the network or so called “dishonest
nodes.” Without a central authority or central view - all peers depend on
the data provided by neighboring peers - therefore a dishonest node can
continue until a peer is able to make contact an honest node.

In this threat model let us assume a malicious miner possesses knowledge of
an unpatched DoS vulnerability (“0-day”) which will strictly prevent honest
nodes from communicating to new members of the network - a so-called “total
eclipse.”  The kind of DoS vulnerability needed to conduct an eclipse does
not need to consume all CPU or computaitly ability of target nodes - but
rather prevent target nodes from forming new connections that would
undermine the eclipsing effect. These kinds of DoS vulnerabilities are
somewhat less substional than actually knocking a powerful-mining node
offline.  This class of attacks are valuable to an adversary because in
order for an honest node to prove that a dishonest node is lying - they
would need to form a connection to a segment of the network that isn’t
entirely suppressed. Let us assume a defense-in-depth strategy and plan on
this kind of failure.

Let us now consider that the C++ Bitcoind has a finite number of worker
threads and a finite number of connections that can be serviced by these
workers.  When a rude client occupies all connections - then a pidgin-hole
principle comes into play. If a network's maximum capacity for connection
handlers ‘k’, is the sum of all available worker threads for all nodes in
the network, establishing ‘k+1’ connections by the pidgin-hole principle
will prevent any new connections from being formed by honest nodes -
thereby creating a perfect eclipse for any new miners joining the network
would only be able to form connections with dishonest nodes.

Now let’s assume a dishonest node is modified in two ways - it increases
the maximum connection handles to hundreds of thousands instead of the
current value which is about 10. Then this node is modified to ignore any
solution blocks found by honest nodes - thus forcing the dishonest side of
the network to keep searching for a competitive-solution to split the
network in two sides that disagree about which tip of the chain to use.
Any new solution propagates through nodes one hop at a time. This
propagation can be predicted and shaped by dishonest non-voting nodes that
are being used to pass messages for honest nodes.

At this point an attacker can expedite the transmission of one solution,
while slowing another. If ever a competing proof-of-work is broadcasted to
the network, the adversary will use their network influence to split
knowledge of the proof-of-work as close to ½ as possible. If the network
eclipse is perfect then an adversary can leverage an eigen-vector of
computational effort to keep the disagreement in balance for as long as it
is needed. No mechanism is stopping the attacker from adding additional
computation resources or adjusting the eclipsing effect to make sure the
system is in balance.   As long as two sides of the network are perfectly
in disagreement and generating new blocks - the attacker has intentionally
created a hard-fork against the will of the network architects and
operators. The disagreement needs to be kept open until the adversary’s
transactions have been validated on the honest chain - at which point the
attacker will add more nodes to the dishonest chain to make sure it is the
ultimate winner - thus replacing out the honest chain with the one
generated by dishonest miners.

This attack is convenient from the adversary’s perspective,  Bitcoin being
a broadcast network advertises the IP addresses of all active nodes - and
Shodan and the internet scanning project can find all passive nodes
responding on TCP 8333.  This should illuminate all honest nodes on the
network, and even honest nodes that are trying to obscure themselves by not
announcing their presence.  This means that the attacker doesn’t need to
know exactly which node is used by a targeted exchange - if the attacker
has subdued all nodes then the targeted exchange must be operating a node
within this set of targeted honest nodes.

During a split in the blockchain, each side of the network will honor a
separate merkel-tree formation and therefore a separate ledger of
transactions. An adversary will then broadcast currency deposits to public
exchanges, but only on the weaker side, leaving the stronger side with no
transaction from the adversary. Any exchange that confirms one of these
deposits is relying upon nodes that have been entirely eclipsed so that
they cannot see the competing chain - at this point anyone looking to
confirm a transaction is vulnerable to a double-spend. With this currency
deposited on a chain that will become ephemeral, the attacker can wire out
the account balance on a different blockchain - such as Tether which is an
erc20 token on the Ethereum network which would be unaffected by this
attack.  When the weaker chain collapses, the transaction that the exchange
acted upon is no longer codified in Bitcoin blockchain's global ledger, and
will be replaced with a version of the that did not contain these deposits.

Nakamoto Consensus holds no guarantees that it’s process is deterministic.
In the short term, we can observe that the Nakamoto Consensus is
empirically non-deterministic which is evident by re-organizations (re-org)
as a method of resolving disagreements within the network.   During a
reorganization a blockchain network is at its weakest point, and a 51%
attack to take the network becomes unnecessary. An adversary who can
eclipse honest hosts on the network can use this as a means of byzantine
fault-injection to disrupt the normal flow of messages on the network which
creates disagreement between miners.

DeFi (Decentralized Finance) and smart-contract obligations depend on
network stability and determinism.  Failure to pay contracts, such as what
happened on “black thursday” resulted in secured loans accidentally falling
into redemption.  The transactions used by a smart contract are intended to
be completed quickly and the outcome is irreversible.  However, if the
blockchain network has split then a contract may fire and have it’s
side-effects execute only to have the transaction on the ledger to be
replaced.  Another example is that a hard-fork might cause the payer of a
smart contract to default - as the transaction that they broadcasted ended
up being on the weaker chain that lost. Some smart contracts, such as
collateral backed loans have a redemption clause which would force the
borrower on the loan to lose their deposit entirely.

With two sides of the network balanced against each other - an attacker has
split the blockchain and this hard-fork can last for as long as the
attacker is able to exert the computational power to ensure that
proof-of-work blocks are regularly found on both sides of the network.  The
amount of resources needed to balance the network against itself is far
less than a 51% attack - thereby undermining the security guarantees needed
for a decentralized untrusted payment network to function.  An adversary
with a sufficiently large network of dishonest bots could use this to take
a tally of which miners are participating in which side of the network
split. This will create an attacker-controlled hard fork of the network
with two mutually exclusive merkle trees. Whereby the duration of this
split is arbitrary, and the decision in which chain to collapse is up to
the individual with the most IP address, not the most computation.

In Satoshi Nakamoto’s original paper it was stated that the electorate
should be represented by computational effort in the form of a
proof-of-work, and only these nodes can participate in the consues
process.  However, the electorate can be misled by non-voting nodes which
can reshape the network to benefit an individual adversary.
Chain Fitness

Any solution to byzantine fault-injection or the intentional formation of
disagreements must be fully decentralized. A blockchain is allowed to split
because there is ambiguity in the Nakamoto proof-of-work, which creates the
environment for a race-condition to form. To resolve this, Floating-Point
Nakamoto Consensus makes it increasingly more expensive to replace the
current winning block. This added cost comes from a method of disagreement
resolution where not every solution block is the same value, and a more-fit
solution is always chosen over a weaker solution. Any adversary attempting
to have a weaker chain to win out would have to overcome a kind of
relay-race, whereby the winning team’s strength is carried forward and the
loser will have to work harder and harder to maintain the disagreement.  In
most cases Floating-Point Nakamoto Consensus will prevent a re-org
blockchain from ever going past a single block thereby expediting the
formation of a global consensus.  Floating-Point Nakamoto Consensus cements
the lead of the winner and to greatly incentivize the network to adopt the
dominant chain no matter how many valid solutions are advertised, or what
order they arrive.

The first step in Floating-Point Nakamoto Consensus is that all nodes in
the network should continue to conduct traditional Nakamoto Consensus and
the formation of new blocks is dictated by the same zero-prefix
proof-of-work requirements.  If at any point there are two solution blocks
advertised for the same height - then a floating-point fitness value is
calculated and the solution with the higher fitness value is the winner
which is then propagated to all neighbors. Any time two solutions are
advertised then a re-org is inevitable and it is in the best interest of
all miners to adopt the most-fit block, failing to do so risks wasting
resources on a mining of a block that would be discarded.  To make sure
that incentives are aligned, any zero-prefix proof of work could be the
next solution, but now in order to replace the current winning solution an
adversary would need a zero-prefix block that is also more fit that the
current solution - which is much more computationally expensive to produce.

Any changes to the current tip of the blockchain must be avoided as much as
possible. To avoid thrashing between two or more competitive solutions,
each replacement can only be done if it is more fit, thereby proving that
it has an increased expense.  If at any point two solutions of the same
height are found it means that eventually some node will have to replace
their tip - and it is better to have it done as quickly as possible so that
consensus is maintained.

In order to have a purely decentralized solution, this kind of agreement
must be empirically derived from the existing proof-of-work so that it is
universally and identically verifiable by all nodes on the network.
Additionally, this fitness-test evaluation needs to ensure that no two
competing solutions can be numerically equivalent.

Let us suppose that two or more valid solutions will be proposed for the
same block.  To weigh the value of a given solution, let's consider a
solution for block 639254, in which the following hash was proposed:


There are 19 zeros, and the remaining hash in base 16 starts with 9e3 and
ends with f8.  This can value can be represented in floating point as:


To simplify further lets give this block a single whole number to represent
one complete solution, and use a rounded floating-point value to represent
some fraction of additional work exerted by the miner.


Now let us suppose that a few minutes later another solution is advertised
to the network shown in base16 below:


The solution above also has 19 prefixed zeros, and is being broadcast for
the same blockheight value of 639254 - and a fitness score of 1.282.  With
Nakamoto Consensus both of these solutions would be equivalent and a given
node would adopt the one that it received first.  In Floating-Post Nakamoto
Consensus, we compare the fitness scores and keep the highest.  In this
case no matter what happens - some nodes will have to change their tip and
a fitness test makes sure this happens immediately.

With both solutions circulating in the network - any node who has received
both proof-of-works should know 1.847 is the current highest value, and
shouldn’t need to validate any lower-valued solution.  In fact this fitness
value has a high degree of confidence that it won’t be unseated by a larger
value - being able to produce a proof-of-work with 19 0’s and a decimal
component greater than 0.847 is non-trivial.  As time passes any nodes that
received a proof-of-work with a value 1.204 - their view of the network
should erode as these nodes adopt the 1.847 version of the blockchain.

All nodes are incentivized to support the solution with the highest fitness
value - irregardless of which order these proof-of-work were validated.
Miners are incentivized to support the dominant chain which helps preserve
the global consensus.

Let us assume that the underlying cryptographic hash-function used to
generate a proof-of-work is an ideal primitive, and therefore a node cannot
force the outcome of the non-zero component of their proof-of-work.
Additionally if we assume an ideal cipher then the fitness of all possible
solutions is gaussian-random. With these assumptions then on average a new
solution would split the keyspace of remaining solutions in half.  Given
that the work needed to form a  new block remains a constant at 19 blocks
for this period - it is cheaper to produce a N+1 block that has any
floating point value as this is guaranteed to be adopted by all nodes if it
is the first solution.  To leverage a chain replacement on nodes conducting
Floating-Point Nakamoto Consensus a malicious miner would have to expend
significantly more resources.

Each successive n+1 solution variant of the same block-height must
therefore on average consume half of the remaining finite keyspace.
Resulting in a the n+1 value not only needed to overcome the 19 zero
prefix, but also the non-zero fitness test.   It is possible for an
adversary to waste their time making a 19 where n+1 was not greater, at
which point the entire network will have had a chance to move on with the
next solution.  With inductive reasoning, we can see that a demissiniong
keyspace increases the amount of work needed to find a solution that also
meets this new criteria.

Now let us assume a heavily-fragmented network where some nodes have gotten
one or both of the solutions.  In the case of nodes that received the
proof-of-work solution with a fitness of 1.847, they will be happily mining
on this version of the blockchain. The nodes that have gotten both 1.847
and .240 will still be mining for the 1.847 domainite version, ensuring a
dominant chain.  However, we must assume some parts of the network never
got the message about 1.847 proof of work, and instead continued to mine
using a value of 1.240 as the previous block.   Now, let’s say this group
of isolated miners manages to present a new conflicting proof-of-work
solution for 639255:


The above base16 block has a fitness score of 1.532  The fitness value for
the previous block 639254 is added together:

     2.772 = 1.240 + 1.532

In this specific case, no other solution has been broadcast for block
height 639255 - putting the weaker branch in the lead.  If the weaker
branch is sufficiently lucky, and finds a solution before the dominant
branch then this solution will have a higher overall fitness score, and
this solution will propagate as it has the higher value.  This is also
important for transactions on the network as they benefit from using the
most recently formed block - which will have the highest local fitness
score at the time of its discovery.  At this junction, the weaker branch
has an opportunity to prevail enterally thus ending the split.

Now let us return to the DoS threat model and explore the worst-case
scenario created by byzantine fault injection. Let us assume that both the
weaker group and the dominant group have produced competing proof-of-work
solutions for blocks 639254 and 639255 respectively.  Let’s assume that the
dominant group that went with the 1.847 fitness score - also produces a
solution with a similar fitness value and advertises the following solution
to the network:




3.262 = 1.847 + 1.415

A total of 3.262 is still dominant over the lesser 2.772 - in order to
overcome this - the 2nd winning block needs to make up for all of the
losses in the previous block.  In this scenario, in order for the weaker
chain to supplant the dominant chain it must overcome a -0.49 point
deficit. In traditional Nakamoto Consensus the nodes would see both forks
as authoritative equals which creates a divide in mining capacity while two
groups of miners search for the next block.  In Floating-Point Nakamoto
Consensus any nodes receiving both forks, would prefer to mine on the chain
with an overall fitness score of +3.262 - making it even harder for the
weaker chain to find miners to compete in any future disagreement, thereby
eroding support for the weaker chain. This kind of comparison requires an
empirical method for determining fitness by miners following the same same
system of rules will insure a self-fulfilled outcome.  After all nodes
adopt the dominant chain normal Nakamoto Consuess can resume without having
to take into consideration block fitness. This example shows how
disagreement can be resolved more quickly if the network has a mechanism to
resolve ambiguity and de-incentivise dissent.
Soft Fork

Blockchain networks that would like to improve the consensus generation
method by adding a fitness test should be able to do so using a “Soft Fork”
otherwise known as a compatible software update.  By contrast a “Hard-Fork”
is a separate incompatible network that does not form the same consensus.
Floating-Point Nakamoto Consensus can be implemented as a soft-fork because
both patched, and non-patched nodes can co-exist and non-patched nodes will
benefit from a kind of herd immunity in overall network stability.  This is
because once a small number of nodes start following the same rules then
they will become the deciding factor in which chain is chosen.  Clients
that are using only traditional Nakamoto Consensus will still agree with
new clients over the total chain length. Miners that adopt the new strategy
early, will be less likely to lose out on mining invalid solutions.

Floating-Point Nakamoto consensus allows the network to form a consensus
more quickly by avoiding ambiguity allowing for determinism to take hold.
Bitcoin has become an essential utility, and attacks against our networks
must be avoided and adapting, patching and protecting the network is a
constant effort. An organized attack against a cryptocurrency network will
undermine the guarantees that blockchain developers are depending on.

Any blockchain using Nakamoto Consensus can be modified to use a fitness
constraint such as the one used by a Floating-Point Nakamoto Consensus.  An
example implementation has been written and submitted as a PR to the
bitcoin core which is free to be adapted by other networks.

A complete implementation of Floating-Point Nakamoto consensus is in the
following pull request:




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