The Mathematics Behind Bitcoin

The Mathematics Behind
Bitcoin
Double Spend Race
Cyril Grunspan
de Vinci Finance Group
ESILV
and Ricardo Perez-Marco
CNRS
University Paris 7
Thursday, January 12th 2017

1 Historical roots
ca 1800 BCE. Sîn-ka²id, tablets, centralized
government and harmonisation of tax. Silver
shekel, units of weight or currency.
[ca 680 BCE ¡ ca 547 BCE]. Mermnad
dynasty: Gyges, Ardys, Sadyattes II, Alyattes,
Croesus, Lydia, Pactolus,
Histories, Herodotus of Halicarnasse, (ca. 450).
Gold coins, Artemision, Ephesus. Manipulation
of the state.
1494. Venice, Tractatus XI Particularis de
Computibus et Scripturis, Luca Pacioli,
invention of Double-Entry Bookkeeping
System.
1923. Hyperination in the Weimar Republic.
Solved by Hjalmar Schacht. Must read : Le
Banquier du Diable, J.-F. Bouchard, Max Milo
(2015)
Europe. Individualism.
Italy: 10 cents Botticelli, 1€ Da Vinci, 2€ Dante,
Spain : Cervantès (10; 20; 50 cents)
Importance of middlemen in business, clearing
houses...
Daniel Pinto :
To sell a loan is a very cumbersome,
time-consuming process; settlement can
take weeks.
2

2 Political roots
Cypherpunks electronic mailing list, Tim May
The Crypto Anarchist Manifesto, T. May (1992)
Openbazar, decentralized e-commerce open source project,
Sam Patterson
PGP Zimmerman
Satoshi Nakamoto satoshi at vistomail.com
Thu Nov 6 15:15:40 EST 2008
You will not find a solution to
political problems in cryptography.
Yes, but we can win a major battle in
the arms race and gain a new territory
of freedom for several years.
Governments are good at cutting off
the heads of a centrally controlled
networks like Napster, but pure P2P
networks like Gnutella and Tor seem to
be holding their own.
Satoshi
The Cryptography Mailing List
www.metzdowd.com/pipermail/cryptography/2008-
November/014823.html
3

3 Advances in computer science
TCP/IP
Transmission Control Protocol / Internet Protocol. V.
G. Cerf, B. Kahn, 1972
Open, shared public network without any central
authority
Secure and scalable
Arpanet
Internet
E-mail
World Wide Web, R. Cailliau, T. Berners-Lee, 1987
Chat online, Instant messaging
P2P computer networks
= client-server model
Music-sharing application Napster (1999 ¡ 2001)
BitTorrent, Peer-to-peer le sharing
Distributed systems
Byzantine fault tolerance
NASA late 70s.
The Byzantine Generals Problem, L. Lamport,
R. Shostak, M. Pease, ACM Transactions on
Programming Languages and Systems (1982)
Another Advantage of Free Choice: completely
asynchronous agreement protocols, M. Ben-Or (1983)
Randomized Consensus
Paxos, Lamport (1989)
King Algorithm
ZYZZYVA
4

4 Bitcoin References
[1]. W.Dai,b-money,
http://www.weidai.com/bmoney.txt, 1998.
[2]. H. Massias, X.S. Avila, and J.-J. Quisquater,
Design of a secure timestamping service with
minimal trust requirements, In 20th Symposium on
Information Theory in the Benelux, May 1999
[3]. S. Haber, W.S. Stornetta, How to time-stamp a
digital document, In Journal of Cryptology, vol 3,
no 2, pages 99-111, 1991.
[4]. D. Bayer, S. Haber, W.S. Stornetta, Improving
the eciency and reliability of digital time-
stamping, In Sequences II: Methods in
Communication, Security and Computer Science,
pages 329-334, 1993.
[5]. S. Haber, W.S. Stornetta, Secure names for bit-
strings, In Proceedings of the 4th ACM Conference
on Computer and Communications Security, pages
28-35, April 1997.
[6]. A. Back, Hashcash - a denial of service counter-
measure,
http://www.hashcash.org/papers/hashcash.pdf,
2002.
[7]. R.C. Merkle, Protocols for public key
cryptosystems, In Proc. 1980 Symposium on
Security and Privacy, IEEE Computer Society,
pages 122-133, April 1980.
[8]. W. Feller, An introduction to probability theory
and its applications, 1957.
5

5 Advances in Cryptography
5.1 Public key cryptography
UK, seventies, J.H. Ellis, C. Cocks, M.J. Williamson
Research declassied by the British government in
1997.
DieHellman key exchange
New Directions in Cryptography, W. Die, M.
Hellman, IEE Transactions on Information theory
(1976).
1977, R. Rivest, A. Shamir, L. Adleman
Alice and Bob
Secret key, public key
RSA
ECDSA
Elliptic curve on Galois eld Fp secp256k1,
y2
= x3
+ 7
with
p = 2256
¡ 232
¡ 29
¡ 28
¡ 27
¡ 26
¡ 24
¡ 1
= 11579208923731619542357098500868790785326998466
5640564039457584007908834671663
Only used for Bitcoin? Gaining in popularity.
Elliptic curve useful for generating a nite group
Discrete logarithmic problem hard to solve
Base point G
Secret integer n
Public key =n G
6

5.2 Hash functions
Rabin, Yuval, Merkle, late 70.
Swiss army knife of cryptography
input of any size
output of xed-size
easy to calculate (in O(n) if input is n-bit
string)
i. collision resistance
ii. preimage resistance
iii. second preimage resistance
One way function
Random Oracles are Practical: A Paradigm for
Designing Ecient Protocols, M. Bellare, P. Rogaway,
ACM Conference on Computer and Communications
Security (1993).
Based on block ciphers
Compression function
Initialization Vector (IV)
MerkleDamgård construction
Birthday paradox
Integrity of transfered data
Message digest
Commitments
Puzzle
Digital signature
SHA-1, MD5 broken
SHA-2
7

5.3 Proof of Work
Use of hash function to create a puzzle
Time consuming
Cost function. A string, D integer, x integer
F : C [0; Dmax] [0; N] ¡! fTrue; Falseg
(A; D; x) 7¡! F(A; D; x)
Example: F(A; D; x) = True if Hash(AjDjx) starts
with D zeros and false else.
Problem. Given A; D, nd x such that
F(A; D; x) = True (1)
Solution x (not necessarily unique) called nonce
Very hard to solve
Use of computational power
Pricing via Processing or Combatting Junk Mail, C.
Dwork and M. Naor, (1993).
Denial-of-service counter measure technique in a
number of systems
Anti-spam tool
Hashcash, A Denial of Service Counter-Measure, A.
Back, preprint (2002)
Hashcash: a proof-of-work algorithm
Create a stamp to attach to mail
Cost functions proposed are dierent
Solution of (1) by brute-force.
Calculus of plenty of hash
8

5.4 Merkle root
Patent in 1979...
A Digital Signature Based on a Conventional
Encryption Function, R. C. Merkle (1988).
Merkle tree = Tree of hashes
Oriented Acyclic Rooted tree
Binary Tree
Leaf = Hash (block)
Top Hash = Merkle root
Used to check integrity of a list of blocks
How to prove that an element x belongs to a set S ?
Screen all S ? Solution in O(n).
Solution proportional to the logarithm of the number
of nodes of the tree O(ln(n))
Any permutation of leaves gives a new Merkle root...
9

5.5 Timestamping
Works of S. Haber, W.S. Stornetta, J.-J.
Quisquater,...
How should a Patent Oce timestamp a digital
document?
Let D be a document. How to dene Certicate(D)?
Certicate(D) : = Hash(D) ?
If D came just before D0
how to prove it using
certicates?
Idea: Certicate of D reused to dene
Certicate of D0
Certicate(D0
) := Hash(D0
jCerticate(D))
Proof that D came before...
Improving the Eciency and Reliability of Digital
Time-Stamping, D. Bayer, S. Haber, W.S. Stornetta
(1993)
To establish that a document was
created after a given moment in time, it
is necessary to report events that could
not have been predicted before they
happened. To establish that a document
was created before a given moment in
time, it is necessary to cause an event
based on the document, which can be
observed by others.
What if many documents came at the same period of
time? Solution with a Merkle tree...
10

6 Blockchain
Block of certicates
Each block contains a reference to previous block
Ledger of certicates.
Block = Merkle tree of certicates + header(block)
Header(Block) = Merkle root + Hash(previous block
header) + (possibly a date?)
Easy to check if a given certicate belongs to the ledger
Any modication of the ledger is automatically
detected
Linked list of hash pointers
Huge dierence between blockchain and proof of work
Concepts are dierent!
Blockchain popularized by bitcoin
Blockchain = Ledger
Hal Finney on the cypherpunk mailing list
11

7 The creation of Bitcoin
Bitcoin: A Peer-to-Peer Electronic Cash System,
October 31; 2008.
Probably more cryptography in payment cards than
in Bitcoin!
ECDSA
Secret/Public Key with secp256k1
Hash Functions RIPEMD160 SHA-256.
Bitcoin Adress = SHA-256 RIPEMD160(PublicKey)
Each block contains a Merkle tree of transactions
Blockchain, Ledger
Proof of work for mining blocks
Two natural questions
1. How to avoid double spending?
2. How can it work in a decentralized network?
Two clever answers
1. Make each transaction a patent certicate
2. Use of proof of work to get decentralization
Tour de force
12

More related questions.
1. How to avoid Sybill attacks?
2. How to solve the byzantine problem?
Answers
1. Naturally thanks to proof of work
2. Thanks to public key cryptography
asumption that The requirement is that the
good guys collectively have more CPU power
than any single attacker.
Commitment schemes in Lightning Network
Payment Channels
Many failed attempts
SET (Visa Mastercard)
Cybercash (bug 2000)
Bitgold (Nick Szabo, 1998)
Digicash (David Chaum 1990, banqueroute
1998)
B-money (Wei Dai 1998)
Paypal (1998)
Blind signatures for untraceable payments, David
Chaum (1983)
Satoshi Nakamoto (forum bitcointalk 2010) :
13

Bitcoin is an implementation of Wei
Dai's b-money proposal on Cypher-
punks in 1998 and Nick Szabo's Bitgold
proposal.
Satoshi's white paper
Double-spending is prevented with
a peer-to-peer network. No mint or
other trusted parties. Participants can
be anonymous. New coins are made
from Hashcash style proof-of-work. The
proof-of-work for new coin generation
also powers the network to prevent
double-spending.
Everything is public
Ledger of transactions
Page = block
Everybody can maintain the ledger
Writer = miner
Money transfer = smart contract
Satoshi
The only way to prevent double
spending is to have a ledger accounting
for all transactions, so that the recipient
can check that the transaction is
legitimate. If we don't want this ledger
to be centralized under the control of a
third party, then it must be public.
14

What is a (bit)coin?
We dene an electronic coin as a chain of
digital signatures. Each owner transfers
the coin to the next by digitally signing a
hash of the previous transaction and the
public key of the next owner and adding
these to the end of the coin. A payee can
verify the signatures to verify the chain
of ownership.
Transaction = 2 scripts = scriptsig + scriptpubkey
What does a miner do?
Verify transactions, gather valid transactions,
constitute a block, he tries to win the mining race
with other miners. The rst one to mine a new block
wins 12.5 bitcoins.
How to recognize the ocial blockchain?
It is (Bi)06i6N such that
P
i=0
N
Di is maximum with
Di = diculty associated with block Bi.
Diculty adjusted every 2016 blocks
Ocial blockchain longest chain
15

8 Why should we trust Bitcoin?
8.1 First results
Satoshi was wrong !
Underestimation of double spend success probability
Existence of closed form formulas
Mathematical foundation of Bitcoin
Bitcoin and Gamma functions
Notation 1. Let 0 q
1
2
(resp. p = 1 ¡ q), the
relative hash power of the group of attackers (resp. of
honest miners).
Theorem 2. After z blocks have been validated by
the honest miners, the probability of success of the
attackers is
P(z) = I4pq

z;
1
2

where Ix(a; b) is the regularized incomplete beta
function
Ix(a; b) :=
¡(a + b)
¡(a) ¡(b)
Z
0
x
ta¡1
(1 ¡ t)b¡1
dt
Corollary 3. Let s=4 p q 1. When z !1, we have
P(z)
sz
p (1 ¡ z) s
p
16

8.2 Other results
Given z 2 N, block generation time t for mining z
block(s) is publicly known.
Denition 4. We denote by P(z; t) the probability of
success of a double spend attack when z blocks have
been validated within a period of time of t.
What we'll obtain also:
Closed form formula for P(z; t).
Satoshi's formula PSN(z) is actually a P(z; t)
Asymptotics formulas for PSN(z) and P(z; t)
Explicit rank z0 such that P(z) PSN(z).
In particular,
PSN(z)
e
¡z

q
p
¡1¡ln
q
p

2
9 Other considerations
Economic cost of a double spend attack (with and
without eclipse attack):
An Analysis of Attacks on Blockchain Consensus
G. Bissias, B. Levine, A. Pinar-Pzisik, G. Andresen,
preprint, (2016/11/20).
Is it necessary to wait for conrmations?
17

Have a Snack, Pay With Bitcoins
O. Bamert, C. Decker, L. Elsen, R. Wattenhofer, S.
Welten, 13-th IEEE International Conference on Peer-
to-Peer Computing, Trento, Italy (2013).
18

10 Mathematics of mining
10.1 Introduction
Looking for a nonce y (number used once)
= Waiting for a bus!
Hashcash proof-of-work (Adam Back).
F = hash function=SHA2562
Looking for y such that F(xjy) Target
x = x1jx2jx3jx4jx5
x1 = Version
x2 = hash Previous Block
x3 = hash Merkle Root
x4 = Timestamp
x5 = Target
Block Header =xjy.
Reference: Bitcoin Wiki
https://en.bitcoin.it/wiki/Block_hashing_algorithm
Example 5. Block Hash 0
000000000019d6689c085ae165831e934763ae46a2a6c172b3f1b60a8ce26f
Example 6. Block Hash 447384
0000000000000000027175e4c9a3216c1331650e45eafdb94803ab59ef1778
Notation 7. (Random variable) Interblock Time is
T. Time used for mining k-th block Tk.
Bitcoin Protocol : E[T] = 0 := 600 (seconds)
Adjustment of target every 2 weeks (2016 = 2 7
24 6 blocs)
19

See https://bitcoinwisdom.com/bitcoin/diculty
Targetnew = Targetold
t
2016 0
where t = time spent for mining the last 2016 blocks.
10.2 Mining one block
The time it takes to mine a block is memoryless
P[T t1 + t2jT t2] = P[T t1]
Proposition 8. The random variable T has the
exponential distribution with parameter =
1
600
i.e.,
fT (t) = e¡t
Parameter seen as a mining speed, E[T ] =
1

.
Conrmation by studying timestamps sequence
10.3 Mining more blocks
Interblock times T1;:::;Tn are independent identically
distributed exponential random variables. The sum
Sn = T1 + :::Tn
is the time spent to get n blocks
Proposition 9. The random variable Sn has a
Gamma distribution with parameter (n; ):
fSn(t) =
n
(n ¡ 1)!
tn¡1
e¡t
20

Denition 10. Let N(t) be the number of blocks
already mined at t-time. Start is at t = 0.
Proposition 11. The random process N is a Poisson
process with parameter i.e.,
P[N(t) = k] =
( t)k
k!
e¡t
Notation 12. The letters T ; ; Sn; N (resp. T 0
; 0
;
Sn
0
;N) are reserved for honest miners (resp. attacker).
10.4 Interpretation of speed mining
Same notations as above. Mining speed (honest) and
0
(attacker). Probability p (honest) and q (attacker).
We note also 0 = 600 seconds =10 minutes.
Proposition 13. We have:
p = P[T T 0
] (2)
p =

+ 0
(3)
q =
0
+ 0
(4)
+ 0
=
1
0
(5)
=
p
0
(6)
0
=
q
0
(7)
Proof. The random variable Inf(T ; T 0
) has the
exponential distribution with parameter + 0
.
21

Proof. (Another proof). Denote by h (resp. h0
) the
hashrate of the honest miners (resp. attacker) and t0
(resp. t0
0
) the average time it takes for mining a block.
Total hashrate of the network =h + h0
.
Proof-of-work: search for a nonce in Block Header such
that
Hash(Block Header) Target
Set m =
2256
Target
We have
p =
h
h + h0
(8)
q =
h0
h + h0
(9)
(h + h0
) 0 = m (10)
h t0 = m (11)
h0
t0
0
= m (12)

So, ; h; p are proportionnal.
22

11 Classical Double Spend
Attack
No eclips attack
11.1 What is a double spend?
A single output may not be used as an input to
multiple transactions.
T =0. A merchant M receives a transaction tx
from A (= attacker). Transaction tx is issued
from an UTXO tx0
Honest Miners start mining openly,
transparently
Attacker A starts mining secretly
One block of honest miners include tx
No block of attacker include tx
On the contrary, one blocks of the attacker
includes another transaction tx' conicting
with tx from same UTXO tx0
As soon as the z-th block has been mined, M
sends his good to A
A keeps on mining secretly
As soon as A has mined a blockchain with a
lenght greater than the ocial one, A releases
his blockchain to the network
Transaction tx has disappeared from the
ocial blockchain.
Free Lunch!
23

12 Interlude: Gambler's ruin
problem
12.1 Original gambler's ruin problem
Gambler has probability p of winning one unit and
q = 1 ¡ p of loosing one unit.
What is the probability Pi that starting with i units,
gambler's fortune will reach N before reaching 0 ?
We denote by Xn gambler's fortune at time n.
Possible states: f0g; f1g; :::; fN g.
Process (Xn) = Markov chain
Transition probability Pk;l:
P0;0 = 1
PN ;N = 1
8k 2 f1; :::; N ¡ 1g Pk;k+1 = p
Pk;k¡1 = q
Conditionning on the outcome of the initial play
8i 2 f1; :::; N ¡ 1g Pi = p Pi+1 + q Pi¡1
P0 = 0
PN = 1
Pi+1 ¡ Pi =
q
p
(Pi ¡ Pi¡1)
So,
Pi =
8

:
1 ¡ (q/ p)i
1 ¡ (q/p)N ; if p=
1
2
i
N
; if p =
1
2
24

When N ! 1;
Pi !
8

:
1 ¡ (q/p)i
; if p=
1
2
0; if p =
1
2
12.2 Another gambler's ruin problem
Competition
Gambler against Banker.
At the beginning, gambler's fortune = banker's
fortune minus n units
Gambler's fortune can be negative
Game takes end if gambler's fortune = banker's
fortune at a certain time t.
What is the probability of success?
Note qn this probability. We have: q0 = 1 and qn ! 0
when n! 1. Also by Markov's property,
qn = q qn¡1 + p qn+1 (13)
Proposition 14. We have qn =

q
p
n
when n 0 and
qn = 1 when n 6 0.
An Introduction to Probability Theory and Its
Applications, W. Feller (1957)
25

13 Nakamoto's Analysis
13.1 Some denitions
Denition 15. Let n 2 Z. We denote by qn the
probability of the attacker A to catch up honest miners
whereas A's blockchain is n blocks behind.
Same problem as gambler's ruin problem!
qn =

q
p
n
(14)
Denition 16. For, z 2 N, the probability of success
of a double-spending attack is denoted by P(z).
Problem: P(z) = ?
Note 17. The probability P(z) is evaluated at t =
0. The double-spending attack cannot be successful
before t = Sz.
13.2 Formula for P (z)
When t = Sz, the attacker has mined N 0
(Sz) blocks.
By conditionning on N 0
(Sz), we get:
P(z) =
X
k=0
1
P[N 0
(Sz) = k] qz¡k
= P[N 0
(Sz) z] +
X
k=0
z¡1
P[N 0
(Sz) = k] qz¡k
= 1 ¡
X
k=0
z¡1
P[N 0
(Sz) = k]
26

+
X
k=0
z¡1
P[N 0
(Sz) = k] qz¡k
= 1 ¡
X
k=0
z¡1
P[N 0
(Sz) = k] (1 ¡ qz¡k)
13.3 Satoshi's approximation
White paper, Section 11 Calculations
According to Satoshi,
Sz E[Sz]
and
N 0
(Sz) N 0
(E[Sz])
N 0
(z E[T ])
N 0

z
0
p

So, N 0
(Sz) Poisson process with parameter given
by
= 0
z
0
p
= z
q
p
27

The recipient waits until the transaction
has been added to a block and z blocks
have been linked after it. He doesn't
know the exact amount of progress the
attacker has made, but assuming the
honest blocks took the average expected
time per block, the attacker's potential
progress will be a Poisson distribution
with expected value:
= z
q
p
Denition 18. We denote by PSN(z) the (false)
formula obtained by Satoshi in Bitcoin's white paper.
Then,
PSN(z) = 1 ¡
X
k=0
z¡1
k
e¡
k!

1 ¡

q
p
z¡k

(15)
Converting to C code...
#include math.h
double AttackerSuccessProbability(double q, int z)
{
double p = 1.0 - q;
double lambda = z * (q / p);
double sum = 1.0;
int i,k;
for (k=0; k=z; k++)
{
double poisson = exp(-lambda);
28

for (i=1; i=k; i++)
poisson *= lambda/i;
sum -= poisson * (1 - pow(q / p, z - k));
}
return sum;
}
However,
P(z) = PSN(z)
since
N 0
(Sz) = N 0
(E[Sz])
29

14 A correct analysis of double-
spending attack
14.1 Meni Rosenfeld's correction
Set Xn := N0
(Sn).
Proposition 19. The random variable Xn has a
negative binomial distribution with parameters (n; p),
i.e., for k 0
P[Xn = k] = pn
qk

k + n ¡ 1
k

Proof. We have Sn ¡(; n) i.e.,
fSn(t) =
n
(n ¡ 1)!
tn¡1
e¡t
with fSn(t) = density of Sn. So,
P[Xn = k] =
Z
0
+1
P[N0
(Sn) = kjSn = t] fSn(t) dt
=
Z
0
+1
(0
t)k
k!
e¡0
t n
(n ¡ 1)!
tn¡1
e¡t
dt
=
pn
qk
(n ¡ 1)! k!
Z
0
+1
tk+n¡1
dt
=
pn
qk
(n ¡ 1)! k!
(k + n ¡ 1)!

The attacker's potential progress is not a Poisson
distribution with expected value = z
q
p
...
Already remarked in 2012 (probably seen by
Satoshi...)
30

Analysis of Hashrate-Based Double-Spending, Meni
Rosenfeld, preprint, First Version December 11, 2012,
p.7.
Proposition 20. (Probability of success of the
attacker) The probabilitu of success of a double-
spending attack is
P(z) = 1 ¡
X
k=0
z¡1
(pz
qk
¡ qz
pk
)

k + z ¡ 1
k

Proof. Direct application of Section 13.2 and
Proposition 19.
14.2 Numerical Applications
For q = 0.1,
z P(z) PSN(z)
0 1 1
1 0.2 0.2045873
2 0.0560000 0.0509779
3 0.0171200 0.0131722
4 0.0054560 0.0034552
5 0.0017818 0.0009137
6 0.0005914 0.0002428
7 0.0001986 0.0000647
8 0.0000673 0.0000173
9 0.0000229 0.0000046
10 0.0000079 0.0000012
31

For q = 0.3,
z P(z) PSN(z)
0 1 1
5 0.1976173 0.1773523
10 0.0651067 0.0416605
15 0.0233077 0.0101008
20 0.0086739 0.0024804
25 0.0033027 0.0006132
30 0.0012769 0.0001522
35 0.0004991 0.0000379
40 0.0001967 0.0000095
45 0.0000780 0.0000024
50 0.0000311 0.0000006
Solving for P less than 0.1%:
q z zSN
0.1 6 5
0.15 9 8
0.20 18 11
0.25 20 15
0.3 32 24
0.35 58 41
0.40 133 89
Satoshi underestimates P(z)...
32

15 A closed form formula
References.
Hanbook of Mathematical Functions, M.
Abramovitch, I.A. Stegun, Dover NY (1970).
Digital Library of Mathematical Functions,
http://dlmf.nist.gov
Denition 21. The Gamma function is dened for
x 0 by
¡(x) :=
Z
0
+1
tx¡1
e¡t
dt
The incomplete Bêta function is dened for a;b0
and x 2 [0; 1] by
Bx(a; b) :=
Z
0
x
ta¡1
(1 ¡ t)b¡1
dt
The (classical) Bêta function is dened for a; b 0 by
B(a; b) := B1(a; b)
The regularized Bêta function is dened by
Ix(a; b) :=
Bx(a; b)
B(a; b)
Classical result: for a; b 0,
B(a; b) =
¡(a) ¡(b)
¡(a + b)
33

Theorem 22. We have:
P(z) = Is(z; 1/2)
with s = 4 p q 1.
Proof. It turns out that the cumulative distribution
function of a negative binomial random variable X
(same notation as above) is
FX(k) = P[X 6 k]
= 1 ¡ Ip(k + 1; z)
By parts,
Ip(k; z) ¡ Ip(k + 1; z) =
pk
qz
k B(k; z)
So,
P(z) = 1 ¡ Ip(z; z) + Iq(z; z)
Classical symmetry relation for Bêta function:
Ip(a; b) + Iq(b; a) = 1
(change of variable t 7! 1 ¡ t in the denition). So,
Ip(z; z) + Iq(z; z) = 1
We also use:
Iq(z; z) =
1
2
Is(z; 1/2)
with s = 4 p q.
Classical function pbeta implemented in R gives the
true double-spending attack success probability.
34

16 Asymptotic analysis
According to Satoshi,
Given our assumption that p q, the
probability drops exponentially as the
number of blocks the attacker has to
catch up with increases.
A result which has never been proven...
Lemma 23. Let f 2 C1
(R+) with f(0)=0 and absolut
convergent integral. Then,
Z
0
+1
f(u) e¡zu
du
f(0)
z
Lemma 24. For b 0 and s 2 [0; 1], we have when
z 1,
Bs(z; b)
sz
z
(1 ¡ s)b¡1
Proof. By the change of variable u = ln(s/t) in the
denition of Bs(z; b) =
R
0
s
tz¡1
(1 ¡ t)b¡1
dt,
Bs(z; b) = sz
Z
0
+1
(1 ¡ s e¡u
)b¡1
e¡zu
du
Then, we apply Lemma 23 with f(u):=(1¡s e¡u
)b¡1
.

35

Proposition 25. When z ! 1, we have:
P(z)
sz
p(1 ¡ s) z
p
with s = 4 p q 1.
Proof. By Stirling formula,
B(z; 1/2) =
¡(z) ¡(1/2)
¡(z + 1/2)

p
z
q
So,
P(z) = Is(z; 1/2)

(1 ¡ s)
¡
1
2
sz
z
p
z
q

sz
p(1 ¡ s) z
p

36

17 A more accurate risk analysis
The merchant waits for z blocks. Once it has been
done, he knows how long it took... Denote this number
by 1. In average, it should take E[z T ] =
z 0
p
.
Denition 26. Set :=
p 1
z 0
Dimensionless parameter.
Satoshi's approximation: = 1...
Instead of computing P(z), let us compute P(z; ).
Probability for a successful double-spending attack
knowing that z blocks have been mined by the honest
miners at Sz = 1.
Note 27. We have PSN(z) = P(z; 1).
Note 28. Two dierent probabilities.
Theoretical probability P(z) calculated at T =
0 by the attacker or the merchant.
concrete probability P(z; ) calculated at T =
1 by the merchant .
37

Number of bocks mined by the attacker at T = 1
unknown to the merchant = Poisson distribution
parameter (z; ):
(z; ) = 0
1
=
q
0

z 0
p
=
z q
p

i.e.,
P[N 0
(1) = k] =

z q
p

k
k!
e
¡
zq
p

Denition 29. The regularized Gamma function is
dened by:
¡(s; x) : =
Z
x
+1
ts¡1
e¡t
dt
The regularized incomplete Gamma function is:
Q(s; x) :=
¡(s; x)
¡(s)
It turns out that
Q(z; ) =
X
k=0
z¡1
k
k!
e¡
So,
38

Theorem 30. We have:
P(z; ) = 1 ¡ Q

z;
z q
p

+

q
p
z
e
z
p¡q
p
Q(z; z)
Proof. We have:
P(z; ) = P[N 0
(1) z] +
X
k=0
z¡1
P[N 0
(1) = k] qz¡k
= 1 ¡
X
k=0
z¡1
(z; )k
k!
e¡(z;)
+
X
k=0
z¡1
q
p
z¡k

(z; )k
k!
e¡(z;)
= 1 ¡ Q

z;
z q
p

+

q
p
z
e
z
p¡q
p
Q(z; z)

18 Asymptotics Analysis
Lemma 31. We have:
i. For 2 ]0; 1[; Q(z; z) ! 1 and
1 ¡ Q(z; z)
1
1 ¡
1
2 p z
p e¡z(¡1¡ln)
ii. For = 1; Q(z; z) !
1
2
and
1
2
¡ Q(z; z)
1
3 2 p z
p
39

iii. For 2 ]1; +1[,
Q(z; z)
1
¡ 1
1
2 p z
p e¡z(¡1¡ln)
Proposition 32. We have PSN (z)
e
¡zc

q
p

2
with
c() := ¡ 1 ¡ ln
Proof. It follows that
1 ¡ Q

z;
q
p
z

1
1 ¡
q
p
1
2 p z
p e
¡zc

q
p

q
p
z
e
z
p¡q
p
Q(z; z)
1
2
e
¡zc

q
p

More generally, we have 5 dierent regimes.
Proposition 33. When z ! +1, we have:
For 0 1; P(z; )
1
1 ¡
q
p
1
2 p z
p e
¡zc

q
p

For = 1; P(z; 1) = PSN(z)
e
¡zc

q
p

2
For 1
p
q
,
P(z; )

1 ¡
q
p

( ¡ 1)

1 ¡
q
p

1
2 p z
p e
¡zc

q
p

40

For =
p
q
, P

z;
p
q

!
1
2
and
P

z;
p
q

¡
1
2

1
2 p z
p

1
3
+
q
p ¡ q

For
p
q
, P(z; ) ! 1 and
1 ¡ P(z; )

1 ¡
q
p

q
p
¡ 1

( ¡ 1)
1
2 p z
p e
¡zc

q
p

Proof. Repetitive application of Lemma 31.
19 Comparison between P (z)
and PSN(z)
19.1 Asymptotic behaviours
The asymptotic behaviours of P(z) and PSN(z) are
quite dierent
Proposition 34. We have PSN(z) P(z)
Proposition 35. We have:
q
p
¡ 1 ¡ ln
q
p
¡ ln

1
4 p q

= ln 4 ¡ 2 + x ¡ 2 ln x
with x =
1
p
2 [1; 2].
41

19.2 Bounds for P (z) and PSN(z)
Goal: compute an explicit rank z0 such that
PSN(z) P(z)
for all z z0.
19.2.1 Upper and lower bounds for P (z)
Remember that s = 4 p q.
We'll use Gautschi's inequalities.
Proposition 36. For any z 1,
z
z + 1
r
sz
p z
p 6 P(z) 6
sz
p (1 ¡ s) z
p
Proof. The function x 7!(1¡ x)
¡
1
2 is non-decreasing.
So, by denition of Is,
P(z) = Is

z;
1
2

=
¡
¡
z +
1
2

¡
¡ 1
2

¡(z)
Z
0
s
tz¡1
(1 ¡ t)
¡
1
2 dt
6
¡
¡
z +
1
2

¡
¡ 1
2

¡(z)
Z
0
s
tz¡1
(1 ¡ s)
¡
1
2 dt
6
¡
¡
z +
1
2

z
p
¡(z)
!
sz
p (1 ¡ s) z
p
Then, we use Gautschi's inequality
¡
¡
z +
1
2

z
p
¡(z)
1
42

On the same way, using other side of Gautschi's
inequality,
P(z) = Is

z;
1
2

¡
¡
z +
1
2

¡
¡ 1
2

¡(z)
Z
0
s
tz¡1
dt

¡
¡
z +
1
2

¡
¡ 1
2

¡(z)
sz
p z
p

z
z + 1
r
sz
p z
p

19.2.2 An upper bound for PSN(z)
Lemma 37. Let z 2 N
and 2 R+

. We have:
i. If 2 ]0; 1[, then
1 ¡ Q(z; z)
1
1 ¡
1
2 p z
p e¡z(¡1¡ln)
ii. If = 1; Q(z; z)
1
2
.
Proof. Let us prove i. rst.
We use dlmf.nist.gov/8.7.1,

(a; x) = e¡x
xa
X
n=0
+1
¡(a)
¡(a + n + 1)
xn
valid for a; x 2 R and

(a; x) :=
Z
x
+1
ta¡1
e¡t
dt
= ¡(a) ¡ ¡(a; x)
43

Let 2 ]0; 1[. Using recursively ¡(z + 1) = z ¡(z),

(z; z ) = e¡z
(z )z
X
n=0
+1
¡(z)
¡(z + n + 1)
(z )n
=
z
zz¡1
e¡z
1 ¡
z (1 ¡ )

1
z
+
1
z (z + 1)
(z ) + :::

6
z
zz¡1
e¡z
1 ¡
z (1 ¡ )

1
z
+
1
z2
(z ) +
1
z3
(z )2
+ :::

6
z
zz¡1
e¡z
1 ¡
z (1 ¡ )
1
z
(1 + + :::)
6
z
zz¡1
e¡z
1 ¡
z (1 ¡ )
1
z

1
1 ¡
6
z
zz¡1
e¡z
1 ¡
By dlmf.nist.gov/5.6.1,
1
¡(z)

ez
2 p z
p
zz¡1
So, for 0 1,
1 ¡ Q(z; z) =

(z; z )
¡(z)

1
1 ¡
1
2 p z
p e¡z(¡1¡ln)
The second inequality (ii) comes directly from
dlmf.nist.gov/8.10.13
44

Proposition 38. We have
PSN(z)
1
1 ¡
q
p
1
2 p z
p e
¡zc

q
p

+
1
2
e
¡zc

q
p

with c() := ¡ 1 ¡ ln .
19.3 An explicit rank z0
Lemma 39. For ; ; x 0, the inequality
e¡ x

x + 1
p
is satised if x
1
2 2
p ¡
1 + 2
p
2 2
p
ln (2 2
)
.
Theorem 40. Let z 2 N
. A sucient condition to
get P(z) PSN(z) is z z0 with
z0 := Max
0
B
@
2
p

1 ¡
q
p
2;
1
2 2
p ¡
1 +
1
2
p
2
ln

2 0
p

0
1
C
A
with
0 :=
q
p
¡ 1 ¡ ln

q
p

¡ ln

1
4 p q

0
45

20 Securing Fast Payments
On the Scalability and Security of Bitcoin, C. Decker,
2016. Chapter 8.
Group Thesis T. Bamert, L. Elsen, S. Welten, R.
Wattenhofer, ETH Zurich.
Have a snack, pay with Bitcoins, 2014 ?
Tradeo between transaction speed and
confirmation reliability in the Bitcoin
network.
C. Decker and R. Wattenhofer, Information
propagation in the bitcoin network, 2013.
Two transactions from the same output TA and TV .
The attacker attempts to convince the
merchant about the validity of TV while
broadcasting TA to the network at the
same time.
Goal: Hide TA to the merchant but TA must be
included in a block of the blockchain
Inuence of node sample size. Double spending-attack
46

20.1 Risk of information eclipsing
If the merchant forwards TV to its
neighboring nodes, they will verify
and tentatively commit it to the local
ledger. Should they later receive TA,
it will not be considered valid as it
conicts with TV , and it will not
be forwarded to the merchant. The
merchant inadvertently shields itself
against conicting transactions like TA,
and will be unaware of the double-
spending attempt.
20.2 Countermeasures
The merchant should connect
to a suciently large random
sample of nodes in the Bitcoin
network.
The merchant should not accept
incoming connections.
The merchant can eectively
avoid isolation by not relaying
transaction TV
As soon as a single node is uninuenced
by the attacker, it will forward TA to the
merchant, thus informing the merchant
of the attempted double-spend
Many simulations: 192 200 000
47

At 100 nodes the merchant will not learn
of a double-spending attempt in only
0.77% of all attempted double-spends.
Time before detection.
The time until the merchant detects
the double-spending attack quickly
decreases for larger sample sizes. The
99 percentile is at 6; 29 seconds for 100
peers.
Transaction TV should be seen rst but not conrmed
by the blockchain
Conclusion of the study
Bitcoin can be used as a reliable
alternative for fast cashless payments.
But not scalable...
21 What is the cost of a double-
spending attack?
Economic evaluation
21.1 Cost of mining
Mining during t with hashing power h has a cost C
(for honest miners) which is proportionnal to t and h:
9 0 such that
C(h; t) = h t
Let B be the block reward. Today, B = 12; 5 BTC
48

Parameter is adjusted so that
C(h + h0
; 0) = B
Therefore,
(h + h0
) 0 = B
and
C(h; t) =
h t
(h + h0) 0
B
=
p t
0
B
Simirarly, for an attacker,
C(h; t) =
q t
0
B
21.2 Classical double spending attack
Competition attacker/honest miners
Cost is a random variable
Cost function at T = 0
C =
q
0
B
where is the stopping time:
:= Inf ft Sz /N 0
(t) N(t)g
Economic evaluation:
C = E

q
0
B

=
q B
0
E[]
= +1
49

Other possible stopping time:
T := Inf ft Sz /N 0
(t) N(t)g ^ T
Andresen al:
We assume that the attacker will stop
mining when he reaches z + 1 blocks
on the fraudulent branch or when the
honest miners reach z + 1 blocks on the
main branch, whichever happens rst.
~ := Sz+1 ^ Sz+1
0
Economic evaluation:
C = E

q ~
0
B

=
q B
0
E[Sz+1 ^ Sz+1
0
]
and Sz+1 and Sz+1
0
are two independent random
variables that has a Gamma distribution
21.3 Double spending attack and eclips
attack
It is simply the cost for mining z blocks
C =
q
0
B
with
: = Sz
0
With a deadline T:
C =
q
0
B
50

with
= Sz ^ T
See Andresen al...
The security of a transaction increases
roughly logarithmically with the number
of conrmations that it receives, where
an attacker benets from the increasing
goods at risk but is also throttled by
the increasing proof of work required.
Additionally, we have demonstrated
that, if merchants impose a conservative
conrmation deadline, the eclipse attack
does not increase an attacker's prot
when his share of the mining power
is less than 35% or more than 10
conrmations are required.
51

The Mathematics Behind Bitcoin

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