The document provides an overview of bitcoin, focusing on its network properties, address generation, and transaction mechanisms. It explains how bitcoin addresses are formed, the cryptographic principles behind transactions, and the role of the blockchain in maintaining security and preventing double spending. Key concepts such as elliptic curve cryptography, the transaction process, and the structure of blocks in the blockchain are also discussed.

1. Introduction
Bitcoin addresses
Transactions
Block chain
Bitcoins
Mathematical Background
Akram El-Korashy1
1 Computer
Science and Engineering
German University in Cairo
CSEN 1033 Seminar in Topics and Applications in
Cryptography, 2013
Akram El-Korashy Bitcoins Math

2. Introduction
Bitcoin addresses
Transactions
Block chain
Outline
1 Introduction
Bitcoins network properties
2 Bitcoin addresses
BTC address generation overview
BTC address cryptography
3 Transactions
4 Block chain
Akram El-Korashy Bitcoins Math

3. Introduction
Bitcoin addresses
Bitcoins network properties
Transactions
Block chain
Outline
1 Introduction
Bitcoins network properties
2 Bitcoin addresses
BTC address generation overview
BTC address cryptography
3 Transactions
4 Block chain
Akram El-Korashy Bitcoins Math

4. Introduction
Bitcoin addresses
Bitcoins network properties
Transactions
Block chain
Main features
Bitcoins can be transferred between arbitrary nodes.
Transactions can be received at any time.
Transactions are broadcast within seconds.
Akram El-Korashy Bitcoins Math

5. Introduction
Bitcoin addresses
Bitcoins network properties
Transactions
Block chain
Main features
Bitcoins can be transferred between arbitrary nodes.
Transactions can be received at any time.
Transactions are broadcast within seconds.
Transactions are verified within 10 to 60 minutes.
Transactions are irreversible.
Double spending is prevented by using a block chain.
Akram El-Korashy Bitcoins Math

6. Introduction
Bitcoin addresses
Bitcoins network properties
Transactions
Block chain
Main features
Bitcoin addresses
Bitcoins can be transferred between arbitrary nodes.
Transactions can be received at any time.
Transactions are broadcast within seconds.
Transactions are verified within 10 to 60 minutes.
Transactions are irreversible.
Double spending is prevented by using a block chain.
Akram El-Korashy Bitcoins Math

7. Introduction
Bitcoin addresses
Bitcoins network properties
Transactions
Block chain
Main features
Bitcoin addresses
Bitcoins can be transferred between arbitrary nodes.
Transactions can be received at any time.
Transactions are broadcast within seconds.
Bitcoin blockchains
Transactions are verified within 10 to 60 minutes.
Transactions are irreversible.
Double spending is prevented by using a block chain.
Akram El-Korashy Bitcoins Math

8. Introduction
Bitcoin addresses BTC address generation overview
Transactions BTC address cryptography
Block chain
Outline
1 Introduction
Bitcoins network properties
2 Bitcoin addresses
BTC address generation overview
BTC address cryptography
3 Transactions
4 Block chain
Akram El-Korashy Bitcoins Math

9. Introduction
Bitcoin addresses BTC address generation overview
Transactions BTC address cryptography
Block chain
Bitcoin Address
Definition
Definition
Bicoin address is an identifier of 27-34 alphanumeric
characters, that represents a possible destination for a Bitcoin
payment.
Akram El-Korashy Bitcoins Math

10. Introduction
Bitcoin addresses BTC address generation overview
Transactions BTC address cryptography
Block chain
Bitcoin Address
Definition
Definition
Bicoin address is an identifier of 27-34 alphanumeric
characters, that represents a possible destination for a Bitcoin
payment.
Address Generation
A Bitcoin address is a 160-bit hash of the public portion of a
public/private ECDSA keypair.
Akram El-Korashy Bitcoins Math

11. Introduction
Bitcoin addresses BTC address generation overview
Transactions BTC address cryptography
Block chain
Bitcoin Address
Address Generation
Choose a random Elliptic-Curve key-pair. The public part is
converted into a BTC address
Akram El-Korashy Bitcoins Math

12. Introduction
Bitcoin addresses BTC address generation overview
Transactions BTC address cryptography
Block chain
Bitcoin Address
Address Generation
The 20 bytes Message Digest from RIPEMD-160..
Akram El-Korashy Bitcoins Math

13. Introduction
Bitcoin addresses BTC address generation overview
Transactions BTC address cryptography
Block chain
Outline
1 Introduction
Bitcoins network properties
2 Bitcoin addresses
BTC address generation overview
BTC address cryptography
3 Transactions
4 Block chain
Akram El-Korashy Bitcoins Math

14. Introduction
Bitcoin addresses BTC address generation overview
Transactions BTC address cryptography
Block chain
Bitcoin Address
Definitions and some Math
Definition
Base58
Akram El-Korashy Bitcoins Math

15. Introduction
Bitcoin addresses BTC address generation overview
Transactions BTC address cryptography
Block chain
Bitcoin Address
Definitions and some Math
Definition
Base58 encoding is a binary-to-text encoding, that uses only
the alphanumeric characters
Akram El-Korashy Bitcoins Math

16. Introduction
Bitcoin addresses BTC address generation overview
Transactions BTC address cryptography
Block chain
Bitcoin Address
Definitions and some Math
Definition
Base58 encoding is a binary-to-text encoding, that uses only
the alphanumeric characters (except 0,O,I, and l.)
Akram El-Korashy Bitcoins Math

17. Introduction
Bitcoin addresses BTC address generation overview
Transactions BTC address cryptography
Block chain
Bitcoin Address
Definitions and some Math
Definition
Base58 encoding is a binary-to-text encoding, that uses only
the alphanumeric characters (except 0,O,I, and l.)
Definition
RIPEMD-160 is a cryptographic hash function that generates a
160-bit digest of an arbitrary size block of data.
Example
RIPEMD-160("The quick brown fox jumps over the lazy
dog") = 37f332f68db77bd9d7edd4969571ad671cf9dd3b
RIPEMD-160("The quick brown fox jumps over the lazy
cog") = 132072df690933835eb8b6ad0b77e7b6f14acad7
Akram El-Korashy Bitcoins Math

18. Introduction
Bitcoin addresses BTC address generation overview
Transactions BTC address cryptography
Block chain
Bitcoin Address
Secure Hash Algorithm-256
Input: arbitrary length data, Output: 256-bit digest
Merkle-Damgard Construction (one-way compression
function, block ciphers)
Akram El-Korashy Bitcoins Math

19. Introduction
Bitcoin addresses BTC address generation overview
Transactions BTC address cryptography
Block chain
Definition
Elliptic curve over a field Fp can be defined as the set of points (x,y)
satisfying the equation
y 2 mod p = (x 3 + ax + b) mod p
Akram El-Korashy Bitcoins Math

20. Introduction
Bitcoin addresses BTC address generation overview
Transactions BTC address cryptography
Block chain
Definition
Elliptic curve over a field Fp can be defined as the set of points (x,y)
satisfying the equation
y 2 mod p = (x 3 + ax + b) mod p
Example
The set of affine points of the elliptic curve y 2 = x 3 − x over F61 .
Akram El-Korashy Bitcoins Math

21. Introduction
Bitcoin addresses BTC address generation overview
Transactions BTC address cryptography
Block chain
Bitcoin Address
Elliptic Curve Digital Signature Algorithm
Any signature algorithm has a Sign phase and a Verify
phase.
Akram El-Korashy Bitcoins Math

22. Introduction
Bitcoin addresses BTC address generation overview
Transactions BTC address cryptography
Block chain
Bitcoin Address
Elliptic Curve Digital Signature Algorithm
Any signature algorithm has a Sign phase and a Verify
phase. Some parameters agreed upon between sender and
receiver:
1 q field size
Akram El-Korashy Bitcoins Math

23. Introduction
Bitcoin addresses BTC address generation overview
Transactions BTC address cryptography
Block chain
Bitcoin Address
Elliptic Curve Digital Signature Algorithm
Any signature algorithm has a Sign phase and a Verify
phase. Some parameters agreed upon between sender and
receiver:
1 q field size
2 FR the basis used - can assume a fixed basis
Akram El-Korashy Bitcoins Math

24. Introduction
Bitcoin addresses BTC address generation overview
Transactions BTC address cryptography
Block chain
Bitcoin Address
Elliptic Curve Digital Signature Algorithm
Any signature algorithm has a Sign phase and a Verify
phase. Some parameters agreed upon between sender and
receiver:
1 q field size
2 FR the basis used - can assume a fixed basis
3 a, b parameters of the equation
Akram El-Korashy Bitcoins Math

25. Introduction
Bitcoin addresses BTC address generation overview
Transactions BTC address cryptography
Block chain
Bitcoin Address
Elliptic Curve Digital Signature Algorithm
Any signature algorithm has a Sign phase and a Verify
phase. Some parameters agreed upon between sender and
receiver:
1 q field size
2 FR the basis used - can assume a fixed basis
3 a, b parameters of the equation (field elements)
Akram El-Korashy Bitcoins Math

26. Introduction
Bitcoin addresses BTC address generation overview
Transactions BTC address cryptography
Block chain
Bitcoin Address
Elliptic Curve Digital Signature Algorithm
Any signature algorithm has a Sign phase and a Verify
phase. Some parameters agreed upon between sender and
receiver:
1 q field size
2 FR the basis used - can assume a fixed basis
3 a, b parameters of the equation (field elements)
4 G base point, and
5 n its order - group element order, point addition, point
doubling
Akram El-Korashy Bitcoins Math

27. Introduction
Bitcoin addresses BTC address generation overview
Transactions BTC address cryptography
Block chain
Bitcoin Address
Elliptic Curve Digital Signature Algorithm
Any signature algorithm has a Sign phase and a Verify
phase. Some parameters agreed upon between sender and
receiver:
1 q field size
2 FR the basis used - can assume a fixed basis
3 a, b parameters of the equation (field elements)
4 G base point, and
5 n its order - group element order, point addition, point
doubling (can be calculated from G!)
Akram El-Korashy Bitcoins Math

28. Introduction
Bitcoin addresses BTC address generation overview
Transactions BTC address cryptography
Block chain
Bitcoin Address
Elliptic Curve Digital Signature Algorithm
Any signature algorithm has a Sign phase and a Verify
phase. Some parameters agreed upon between sender and
receiver:
1 q field size
2 FR the basis used - can assume a fixed basis
3 a, b parameters of the equation (field elements)
4 G base point, and
5 n its order - group element order, point addition, point
doubling (can be calculated from G!)
6 h cofactor - order of the curve divided by n
Akram El-Korashy Bitcoins Math

29. Introduction
Bitcoin addresses
Transactions
Block chain
Transactions
Definitions
Definition
A transaction is a signed section of data that is broadcast to the
network and collected into blocks.
It typically references previous transaction(s) and dedicates a
certain number of bitcoins from it to one or more new public
key(s) (Bitcoin address).
It is not encrypted (nothing in Bitcoin is encrypted).
Akram El-Korashy Bitcoins Math

30. Introduction
Bitcoin addresses
Transactions
Block chain
Transactions
Definitions
Definition
A transaction is a signed section of data that is broadcast to the
network and collected into blocks.
It typically references previous transaction(s) and dedicates a
certain number of bitcoins from it to one or more new public
key(s) (Bitcoin address).
It is not encrypted (nothing in Bitcoin is encrypted).
Transactions and Blocks are the main data structures.
A Block is composed of a header and a group of transactions.
Transactions are identified by their hash (SHA-256).
Blocks are identified by a hash of the header.
Akram El-Korashy Bitcoins Math

31. Introduction
Bitcoin addresses
Transactions
Block chain
Transactions
Example
Example
Principle example of a transaction
Script is a set of instructions for the receiver to follow in order to
be able to spend the Bitcoins.
An input is a reference to the output of a different transaction.
Previous tx is the pointer to (hash value of) the transaction
from which the bitcoin Value will be transferred.
Akram El-Korashy Bitcoins Math

32. Introduction
Bitcoin addresses
Transactions
Block chain
Transactions
Example
The main script instructions are:
The hash of the public key should match the bitcoin
address specified in the transaction.
The signature on the sent data should be verifiable by the
public key.
Instructions are executed from the Output field of the
referenced transaction Previous tx
scriptSig is an ECDSA signature on small data (part of the
transaction). It serves as a proof that the transaction in which it
exists was created by the real owner of the public address which
the Previous tx was intended to.
Akram El-Korashy Bitcoins Math

33. Introduction
Bitcoin addresses
Transactions
Block chain
Block Chain
Definition
Definition
Block chain is a transaction database shared by all nodes
participating in a system based on the Bitcoin protocol.
Every block contains a hash of the previous block
BitCoin uses the SHA-256 hash algorithm to generate verifiably
"random" numbers in a way that requires a predictable amount
of CPU effort.
Akram El-Korashy Bitcoins Math

34. Introduction
Bitcoin addresses
Transactions
Block chain
Block Chain
Concept
Block chain: Because a block can only reference one previous
block, it is impossible for two forked chains to merge.
(Tree goes bottom-up)
Akram El-Korashy Bitcoins Math

35. Appendix
References I
https://en.bitcoin.it/wiki/Address
Accessed March, 2013
Simple Tutorial on Elliptic Curve Cryptography
Accessed March, 2013
https://en.bitcoin.it/w/images/en/9/9b/PubKeyToAddr.png
Accessed March, 2013
Bitcoin wiki: Technical background of bitcoin addresses
Accessed March, 2013
rnks.informatik.tu-cottbus.de - A tutorial on elliptic curve cryptography
Accessed March, 2013
https://en.bitcoin.it/wiki/Category:Technical
Accessed March, 2013
Akram El-Korashy Bitcoins Math