> For the complete documentation index, see [llms.txt](https://www.notbank.com/learn/llms.txt). Markdown versions of documentation pages are available by appending `.md` to page URLs; this page is available as [Markdown](https://www.notbank.com/learn/academy/en/bitcoin/mining.md).

# Mining

Mining, Proof of Work and Difficulty Adjustment

***

> “Proof of Work is the soul of Bitcoin: it turns energy into security.”\
> — *Cypherpunk principle*

***

## 1. Introduction

The **mining** is the mechanism that secures the Bitcoin network by enabling:

* Issuance of new bitcoins
* Protection against attacks
* Temporal ordering of transactions
* Probabilistic consensus
* Cumulative immutability
* Defense against tampering

The tool that makes this possible is **Proof of Work (PoW)**.

This chapter explains:

* How PoW works
* Why it makes the network secure
* The critical role of difficulty adjustment
* Economic incentives of miners
* Mathematical security models
* Potential attacks (and why they are impractical)

***

## 2. What is mining in Bitcoin?

Mining is the process by which specialized nodes:

1. group valid transactions,
2. compete to find a valid hash for the header,
3. propagate the winning block to the network,
4. receive reward (subsidy + fees).

***

### 2.1. Formal definition of PoW

A miner must find:

$$H(\text{header}) < \text{Target}$$

where:

* $$H = SHA256(SHA256(x))$$
* *Target* is a number that defines the difficulty
* the probability of success is inversely proportional to the `hashrate` global

***

## 3. Structure of the header used in PoW

The **80-byte header** is the only data subjected to PoW:

| Field             | Bytes | Purpose                   |
| ----------------- | ----- | ------------------------- |
| version           | 4     | Consensus rules           |
| prev\_block\_hash | 32    | Chaining                  |
| merkle\_root      | 32    | Transactions              |
| timestamp         | 4     | Timestamp                 |
| bits              | 4     | Target compression        |
| nonce             | 4     | Value that is incremented |

***

## 4. The “nonce” and the search space

The miner tests billions of nonces per second.

The nonce is 32 bits → 4 billion possible values.\
But if it is exhausted, the miner modifies:

* extraNonce (in coinbase)
* merkle\_root
* timestamp

which creates a practically infinite space for PoW.

***

## 5. Difficulty: heart of economic security

Bitcoin adjusts the difficulty every **2016 blocks** (\~2 weeks) to maintain an average interval of:

$$10 \text{ minutos por bloque}$$

If blocks are mined faster:

* hashrate ↑
* difficulty ↑

If they are mined slower:

* hashrate ↓
* difficulty ↓

***

### 5.1. Difficulty adjustment formula

The new difficulty is calculated as:

$$D\_{new} = D\_{old} \cdot \frac{t\_{real}}{t\_{esperado}}$$

where:

* $$t\_{esperado} = 2016 \cdot 600 = 1,209,600 s$$ (2 weeks)
* $$t\_{real}$$ = actual time it took for the last 2016 blocks

Limits:

* maximum adjustment ±4×
* prevents exaggerated manipulation

***

## 6. The target: numerical threshold

Difficulty is an abstract value.\
The target is the concrete number the hash must be less than.

$$\text{Target} = \frac{2^{224} - 1}{Dificultad}$$

Simplified example:

HASH (in hex): 0000000af3c1... TARGET: 0000000fffff...

Since HASH < TARGET → valid block

***

## 7. Pseudocode example of PoW

```python
while True:
    header = build_header(block, nonce)
    hash_val = sha256(sha256(header))
    if hash_val < target:
        return header
    nonce += 1
```

## 8. Why is PoW secure?

Because it physically costs energy to generate hashes, while verifying them is trivial:

<p align="center"><span class="math">Costo \ de \ minar \gg Costo \ de \ verificar</span></p>

Costs for the attacker:

* specialized hardware (ASICs)
* electricity
* infrastructure
* lost economic opportunity
* financial risk
* social coordination

Costs for the verifier:

* a single hash per node
* less than 1 ms per block

### 8.1. Accumulated security

PoW creates a “cryptographic wall”:

<p align="center"><span class="math">Seguridad \ de \ la \ cadena \propto PoW \ acumulado</span></p>

No matter how many nodes exist: **security comes from the total work buried under each block**.

## 9. Economic incentives: game theory applied

We could define mining as a strategic interaction between economic agents.

### 9.1. Block reward

It is composed of:

<p align="center"><span class="math">Recompensa = Subsidio + Fees</span></p>

Subsidy:

<p align="center"><span class="math">50 \times 2^{-n}</span></p>

where $$n = número \ de \ halvings$$.

### 9.2. Aligned incentives

* Miners → maximize profits
* Nodes → verify rules
* Users → want security
* Developers → improve efficiency

Incentives are set up to favor:

* Honesty > Attack

### 9.3. Why attacking the network is irrational

To rewrite old blocks you must redo all the accumulated PoW competing against all honest miners at a cost higher than the benefit.

Even an attacker with 51%:

* cannot steal funds
* cannot modify rules
* cannot create inflation
* can only reverse their own payments

And at a ruinous cost.

## 10. Global hashrate: security metric

<p align="center"><span class="math">Hashrate \ global \thickapprox TH/s</span></p>

<p align="center">(terahashes/second)</p>

The higher it is, the more expensive it is to attack the network.

### 10.1. Conceptual example

If the network has:

<p align="center"><span class="math">300 EH/s</span></p>

<p align="center">(300 quintillion hashes/s)</p>

An actor would have to match that power **just to compete**.

## 11. Mining pools

Miners usually group into pools.

Reasons:

* reduce variance
* stable income
* share work

The pool:

Miners → send shares → Pool aggregates PoW → Pool produces block → Distributes reward

### 11.1. Does this increase centralization risk?

Yes, but not as much as it seems:

* changing pools is trivial
* pools do not control private keys
* they only coordinate work
* if they abuse, miners migrate

## 12. Types of hardware in the history of mining

| Era          | Hardware | Efficiency | Duration |
| ------------ | -------- | ---------- | -------- |
| 2009         | CPU      | Very low   | Months   |
| 2010         | GPU      | 50× CPU    | \~1 year |
| 2012         | FPGA     | 10× GPU    | Short    |
| 2013-present | ASIC     | 1,000× GPU | Dominant |

ASIC = `Application-Specific Integrated Circuit`

## 13. Mathematical models of block time

Mining follows a **Poisson process**.

Expected time per block:

<p align="center"><span class="math">E[T]= \dfrac{1}{λ}</span></p>

where:

<p align="center"><span class="math">λ =\dfrac{dificultad}{hashrate}</span></p>

Variability: some blocks take seconds, others hours. But the long-run mean converges to **\~10 minutes**.

## 14. Attacks on the PoW system

### 14.1. 51% attack

Allows:

* reversing own payments
* censoring transactions (temporarily)

Does not allow:

* creating bitcoins
* stealing funds
* breaking signatures
* changing consensus rules

### 14.2. Selfish mining

Theory: miner withholds blocks to gain advantage.

Reality:

* requires threshold > 33%
* not very profitable
* easy to detect
* partially solved with `BIP32`, P2P improvements and topologies

### 14.3. Timestamp attacks

Manipulate time to adjust difficulty. Mitigation:

* strict limits
* allowed window ± 2h
* nodes reject impossible timestamps

## 15. Why PoW is superior to PoS (according to original design)

Cypherpunk / engineering perspective:

* **Based on physics, not on wealth**
* **Objective cost of attack**
* **History protected with irreversible work**
* **Australian Fairness → everyone can compete**
* **Proven dynamics since 2009 without failures**

PoW is a mechanism **neutral, resilient, verifiable and politically independent**.

## 16. Chapter conclusion

Mining and Proof of Work are:

* The physical security layer of Bitcoin
* The source of its immutability
* The engine of its issuance
* The foundation of its consensus
* The economic barrier against attackers

PoW turns electricity into digital security. Difficulty adjustment turns fluctuations into stability. Incentives turn competition into collective protection.

> Mining is not “wasting energy”.\
> It is **spending energy to prevent fraud**, a military function without violence.


---

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