Bitcoin
Wednesday, February 11, 2026
10:11 AM
Bitcoin blockchain is a decentralized, (peer-to-peer digital monetary system) cryptocurrency 
that uses  cryptography to control its creation and management. 
Bitcoin allows value to be transferred over the internet without intermediaries such as banks. 
In 2008 Satoshi Nakamoto published the Bitcoin whitepaper: "Bitcoin: A Peer-to-Peer Electronic Cash System"
Bitcoin consists of three tightly connected layers:
The currency (BTC) – the unit of value, the smales unit of BTC is Satoshi: 1 BTC = 1 000 000 000 Sat.
The network – thousands-millions of computers (nodes) running Bitcoin software
The protocol – the rules that govern how Bitcoin works
Unlike traditional payment systems, Bitcoin removes trusted intermediaries such as banks and relies on:
Blockchain
Cryptographic verification
Distributed consensus mechanism
Incentives for block mining (validation)
Fundamental Problem Bitcoin Solves
Ownership verification
Prevention of double spending
Trustless operation
Before Bitcoin, the payment systems relied on:
User → Bank → Ledger → Payment Confirmation
This creates problems:
central point of failure
censorship
trust dependency
Bitcoin replaces the central authority with cryptographic proof and distributed consensus mechanism.
Core Problem: Double Spending
Digital information can be copied two or many times. This creates the double spending problem.
Example:
Alice owns 1 BTC
Alice could make a copy of this bit string attempt to send the same coin to:
Bob and Charlie
Both transactions may appear valid unless a system ensures only one can succeed. 
Bitcoin solves this problem using:
Cryptographic signatures and hash functions
Public transactions ledger
Timestamped blocks of transactions
Proof-of-Work consensus mechanism
Mathematical Foundations of Bitcoin
Bitcoin relies heavily on three mathematical domains:

Field
Role
Number Theory
modular arithmetic in finite fields
Cryptography
signatures and hashing
Probability Theory
mining security and attacks
Bitcoin ownership is controlled through public-key cryptography.

When a user spends coins, they must prove ownership of the private key corresponding to a public key 
embedded in the transaction output: performed by digital signature.
Bitcoin originally uses: Elliptic Curve Digital Signature Algorithm - ECDSA of secp256k1 kindf of signatures 
and Hashing basedd on SHA-256

A cryptographic hash function H(x) must satisfy:
Deterministic
Preimage resistance
Second preimage resistance
Collision resistance
SHA-256 Function
h = sha256(m)
Output:
256-bit digest
Bitcoin frequently uses double hashing:
h = sha256(sha256(m))
>> sha256('Hello Bob')
ans = 2D6A12FFC0A952FBD09F8909DE4E0E4B20BA2B906CD12AC22BBEF4EE5BD9003E
>> h=sha256(ans)
h = 476D03360CCD1CF6F25B2B4F8038E9B526BF6F0F7386F2781CA92D31DC362873
Hash function is used for:
Address computation
Transaction IDs
Transactions signing
Block hashing
Proof-of-Work (PoW)
How does Bitcoin Works:
Bitcoin relies on blockchain and works through a sequence of processes combining cryptography, 
networking, distributed consensus mechanism, and incentives for mining. 
Every time a transaction happens, it gets written onto a "block." That block is linked to the previous one, 
forming a chain. This record is copied onto thousands-millions of computers around the world (called nodes).
Also, anyone can participate in the ecosystem by downloading Bitcoin's open-source software 
and create bitcoin based applications.
Among the nodes there are full node in the Bitcoin network and thes full nodes independently stores 
a block chain containing only blocks validated by that node. When several nodes all have the same blocks 
in their block chain, they are considered to be in consensus. The validation rules of these nodes follow 
to maintain consensus are called consensus mechanism. 
Bitcoin transactions are operating using so called Unspent Transactions Output - UTxO mechanism:
The sum of transaction Incomes (In) must be equal to the sum of transaction Expenses (Ex) or
the sum of transaction Inputs must be equal to the sum of transaction Outputs.
In1=10 BTC
AliceTransaction - Tx
Unspent Transaction Output
UTxO
Ex1=17 BTC
In2=20 BTC
Ex2=23 BTC
In3=30 BTC
In4=40 BTC
Ex3=60 BTC
Transactions flow
Transactions are included in blocks
Bitcoin block structure
1 rectangular represents 1 Byte = 8 bits
32 Bytes == 64 hex numbers == 256 bits
Nonce is a number represented by 4 Bytes
Antraštė
Transactions list is represented by Merkle Tree
which is represented by Root Hash.
This Root Hash is placed into the block’s header along with the hash of the previous block  and a random number called a nonce to be explained later.  
Merkle tree: is a graph consisting of leafs representing transactions data, graph vertices and edges.
The block’s header is then hashed with sha256 producing an output that will serve as the block’s identifier. 
Structure of a Bitcoin Transaction Blockheader.
Private Key
Random 256-bit number
Must remain secret
Loss = loss of funds
Bitcoin block mining ir based on PoW consensus mechanism referencing to Difficulty Target - DT.
Symbolically it can represented in the following way:
>> sha256('Block header data hashed using the arbitrary Nonce ...')
ans = 41F1801FC7E4B36D44DA242013CE7C6E8C111D19B5FB8795EF718A1030AD6980
Difficulty Target - DT defines the number of most significant hexadecimal digits having zero values required to mine a block.
For example DT=18.
This means that block is mined if the Nonce value is found such that h-value of block header has 
18 leading hexadecimal zeroes.
Let such a Nonce is found, then the block is mind if e.g. miner obtains the following h-value:
>> sha256('Block header data hashed using the concrete Nonce of 4 bytes')
ans = 000000000000000000E7B8F4C1AAAC8D2D85A326BEE0B9E242E0E360B09C1347
Key generation: is based on Elliptic Curve Digital Signature Algorithm - ECDSA
ECDSA Public Parameters are the following PP=(EC secp256k1; BasePoint-Generator G; prime p; param. a, b);
Prime p is of 256 bit length and defines the finite fielf Fp = {0, 1, 2, 3, …, p-1} where operations are performed mod p.
Parameters a, b defines the equation of Elliptic Curve (EC) over the finite fielf Fp:
 y2=x3+ax+b  mod p
Private Key PrK = z generation.
        Is generated at random as a hexadecimal number having 64 hexadecimal digits z=randi(p-1)
        Where p - is a large prime having 256 bit length.
Public Key PuK = A generation.
Derived mathematically from private key z.
Bitcoin uses the elliptic curve: secp256k1 it means that the number of Elliptic Curve (EC) points is equal to p.
The public key is generated using EC generator G multiplication by number (scalar) z we denote by *:
PuK = A = z*G
Public key A is an EC point and hence is represented by coordinates (x, y)
In secp256k1  every coordinate ocuupies 256 bits.
Then The representation of A requires 512 bits.
Address creation
Private Key z → Public Key A (EC multiplication operation *) → sha256(A) → Bitcoin Address by taking 
34 least significant hexadecimal digits
Let user's PuK=(x, y) is represented by the following hexadecimal number
x = 52A6B8D367854D62EABF9F415A5E68505506727009472936417C747610270624
y = 3EF4698C496E1475B7ECB6A2E4C85B6C7C1C57AA124DFCA859AE3B6CBB3F4A75
To compute Address PuK coordinates x and y must be hashed
>> h=sha256('52A6B8D367854D62EABF9F415A5E68505506727009472936417C7476102706243EF4698
C496E1475B7ECB6A2E4C85B6C7C1C57AA124DFCA859AE3B6CBB3F4A75')
h = 16A134BC52822E45372C9FE6411939B8B60747EAEA5FAF9AB40137FC97BD05F0
Bitcoin Address  Example: B8B60747EAEA5FAF9AB40137FC97BD05F0
Diagrams & Visual References (With Links)
You can paste the following diagrams directly into OneNote using the links.
Bitcoin Transaction Flow Diagram
Shows how a transaction moves from wallet → network → block → blockchain.
UTXO Model Diagram
Explains how inputs consume previous outputs and create new ones.
🔗 [link]
Block Structure Diagram
Shows block header, transactions, nonce, and hashes.
🔗 [link]
Blockchain Linked Blocks Diagram
Visual representation of how blocks are chained together.
🔗 [link]
Merkle Tree Diagram
Explains how transaction hashes are combined efficiently.
🔗 [link]
Mining & Proof of Work Diagram
Shows hashing, difficulty target, and nonce search.
🔗 [link]
34. Authoritative Learning Links (Highly Recommended)
Official & Core Resources
Bitcoin Whitepaper (original): [link]
Bitcoin.org Developer Guide: [link]
Bitcoin Core GitHub: [link]
Technical Learning (Beginner → Advanced)
Learn Me A Bitcoin (excellent technical explanations): [link]
Bitcoin Wiki: [link]
Mastering Bitcoin (Andreas M. Antonopoulos – free online): [link]
Blockchain Explorers (Live Data)
Blockchain.com Explorer: [link]
Mempool Visualizer: [link]