Bitcoin Exposed: Today's Complete Guide to Tomorrow's Currency (13 page)

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Authors: Daniel Forrester,Mark Solomon

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Appendix D - Bitcoins Mathematical Basis

Bitcoin Mathematical Details:

This information comes from Akram El-korashy, Computer Science and Engineering Department, German University in Cairo (
http://www.slideshare.net/akram-elkorashy/bitcoins-math
)

Bitcoin address 27-34 alphanumerical address that is a potential destination or recipient for Bitcoin payments. A Bitcoin Address is 160-digit hash of /files/03/63/69/f036369/public/private portion of the ECDSA key pair.

Uses a random Elliptical Curve keypair with a public key portion turned into the BTC address.

Here is what this looks like from the results of the math problems solved to create the Bitcoin Address:

A Bitcoin Address looks like this:

1Lez1wcwPLWNmPtcfGXtSy8F34ZF45fwYh

Base58 encoding from binary to text /alphanumeric encoding except 0, O, I, l.

RIPEMD-160 is a hashing function that generates a 160 bit digest of an arbitrary block size.

Essentially this function turns random words letters, numbers, etc..into a block of alphanumerics of an arbitrary number of characters of 256 bit data.

This is a critical part of the Bitcoin code that creates uniformity from the random inputs:

Uses Merkle-Damguard Construction for compressing blocks. Creating cyphers.

The basis of the math is the elliptical curve function over a field Fp is a set of points (x,y) satisfying the equation:

y2 mod p = (x3 + ax + b) mod p

There is a Signature and Verify phase of the signature algorithm.

q is field size

a, b are field elements

fr is field size - and is fixed

G is base point

n is the order (of multiplication, how many times an element is multiplied by itself point element, group element, point doubling, etc. This is calculated from G!)

h is a cofactor (order of the curve divided by n)

Here is what this looks like for key generation for the ECRSA form of the function as taken from the Elliptical Curve Solution Primer:

This is advanced mathematics for most readers. For a complete explanation of the math behind elliptical curve cryptography, one of the best papers is here:

http://www.eis.mdx.ac.uk/staffpages/m_cheng/link/ecc_simple.pdf

A transaction section of data that is broadcast to the network and collected into blocks.

- It usually references previous transactions and describes sending bitcoins from one address to another. - It is not encrypted.

- Transactions and blocks are the main data structures.

- Blocks have a header and a group of transactions.

- Transactions are identified by their hash SHA-256

- Blocks are identified by a hash of the header.

- Blocks are transaction database shared by all nodes in the system using the bitcoin protocol.

Script is a set of instructions for the receiver to follow to spend the bitcoins.

Script instructions include:

- The hash of the public key should match the bitcoin address for the transaction.

- The signature of the sent date should be verifiable by the public key.

- The script includes a ECDSA signature to verify the identity of the real owner of the bitcoin address.

- Every block contains a hash of the previous block.

- Bitcoin uses the SHA-256 algorithm to generate verifiable “random” numbers that requires a predictable amount of CPU effort.

Block Chain: A block can only reference one previous block so it is impossible for two forked chains to merge.

SHA-256 is fully explained for higher-level mathematicians here:

http://www.quadibloc.com/crypto/mi060501.htm

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