A mathematical introduction to cryptographic primitives in blockchain technology.

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Blockchain-based distributed public ledgers can securely store our data without the need for a centralized controlling authority. Blockchain technology uses intriguing mathematical and cryptographical concepts, which are usually misunderstood. 

This course presents a comprehensive overview of the basic concepts of blockchain while providing deep insights into its background in mathematics and cryptography. Next, you’ll learn about the critical cryptographical vulnerabilities and attacks against blockchain technology and how to encounter them. Finally, the course describes possible attacks against blockchains by a fully-capable quantum computer and the possible remedies.

By the end of this course, you’ll have a deep understanding of blockchain technology and its mechanisms, particularly the mathematics and cryptography involved in blockchain security.

Cryptographic Primitives in Blockchain Technology

What is the answer to <math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><mi>g</mi><mi>c</mi><mi>d</mi><mo stretchy="false">(</mo><mn>25</mn><mo separator="true">,</mo><mn>145</mn><mo stretchy="false">)</mo></mrow><annotation encoding="application/x-tex">gcd(25, 145)</annotation></semantics></math>gcd(25,145)?

Which of these pairs is relatively prime (or coprime)?

Which of these pairs is relatively prime (or coprime)?

How many numbers are relatively prime to 81?
$$\varphi(81) = ?$$


How many numbers are relatively prime to 81?
<math><semantics><mrow><mi>φ</mi><mo>(</mo><mn>8</mn><mn>1</mn><mo>)</mo><mo>=</mo><mo>?</mo></mrow><annotation encoding="application/x-tex">\varphi(81) = ?
</annotation></semantics></math>φ(81)=?

The addition modulo 
$n$ in $\mathbb{Z} / n \mathbb{Z}$ is defined by the following functions: $\oplus: \mathbb{Z} / n \mathbb{Z} \times \mathbb{Z} / n \mathbb{Z} \rightarrow \mathbb{Z} / n \mathbb{Z}$ 

if $n$ = 6,
what is the answer to the function below? 
$$\bar{3} \oplus \bar{4}=?$$

The addition modulo
<math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><mi>n</mi></mrow><annotation encoding="application/x-tex">n</annotation></semantics></math>n in <math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><mi mathvariant="double-struck">Z</mi><mi mathvariant="normal">/</mi><mi>n</mi><mi mathvariant="double-struck">Z</mi></mrow><annotation encoding="application/x-tex">\mathbb{Z} / n \mathbb{Z}</annotation></semantics></math>Z/nZ is defined by the following functions: <math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><mo>⊕</mo><mo>:</mo><mi mathvariant="double-struck">Z</mi><mi mathvariant="normal">/</mi><mi>n</mi><mi mathvariant="double-struck">Z</mi><mo>×</mo><mi mathvariant="double-struck">Z</mi><mi mathvariant="normal">/</mi><mi>n</mi><mi mathvariant="double-struck">Z</mi><mo>→</mo><mi mathvariant="double-struck">Z</mi><mi mathvariant="normal">/</mi><mi>n</mi><mi mathvariant="double-struck">Z</mi></mrow><annotation encoding="application/x-tex">\oplus: \mathbb{Z} / n \mathbb{Z} \times \mathbb{Z} / n \mathbb{Z} \rightarrow \mathbb{Z} / n \mathbb{Z}</annotation></semantics></math>⊕:Z/nZ×Z/nZ→Z/nZ
if <math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><mi>n</mi></mrow><annotation encoding="application/x-tex">n</annotation></semantics></math>n = 6,
what is the answer to the function below?
<math xmlns="http://www.w3.org/1998/Math/MathML" display="block"><semantics><mrow><mover accent="true"><mn>3</mn><mo>ˉ</mo></mover><mo>⊕</mo><mover accent="true"><mn>4</mn><mo>ˉ</mo></mover><mo>=</mo><mo stretchy="false">?</mo></mrow><annotation encoding="application/x-tex">\bar{3} \oplus \bar{4}=?
</annotation></semantics></math>3ˉ⊕4ˉ=?

**(True or False)**$(\mathbb{Z},+),(\mathbb{R},+)$, and $(\mathbb{C},+)$ form abelian groups with $e=0$ and $a^{-1}=$ $-a$.



(True or False)<math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><mo stretchy="false">(</mo><mi mathvariant="double-struck">Z</mi><mo separator="true">,</mo><mo>+</mo><mo stretchy="false">)</mo><mo separator="true">,</mo><mo stretchy="false">(</mo><mi mathvariant="double-struck">R</mi><mo separator="true">,</mo><mo>+</mo><mo stretchy="false">)</mo></mrow><annotation encoding="application/x-tex">(\mathbb{Z},+),(\mathbb{R},+)</annotation></semantics></math>(Z,+),(R,+), and <math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><mo stretchy="false">(</mo><mi mathvariant="double-struck">C</mi><mo separator="true">,</mo><mo>+</mo><mo stretchy="false">)</mo></mrow><annotation encoding="application/x-tex">(\mathbb{C},+)</annotation></semantics></math>(C,+) form abelian groups with <math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><mi>e</mi><mo>=</mo><mn>0</mn></mrow><annotation encoding="application/x-tex">e=0</annotation></semantics></math>e=0 and <math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><msup><mi>a</mi><mrow><mo>−</mo><mn>1</mn></mrow></msup><mo>=</mo></mrow><annotation encoding="application/x-tex">a^{-1}=</annotation></semantics></math>a−1= <math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><mo>−</mo><mi>a</mi></mrow><annotation encoding="application/x-tex">-a</annotation></semantics></math>−a.

**(Fill in the blank.)** For any group $G=(G, *)$ with identity $e_{G},\left\{e_{G}\right\}$ and $G$ itself are always subgroups of $G$, whereas $\left\{e_{G}\right\}$ is called a ________.

(Fill in the blank.) For any group <math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><mi>G</mi><mo>=</mo><mo stretchy="false">(</mo><mi>G</mi><mo separator="true">,</mo><mo>∗</mo><mo stretchy="false">)</mo></mrow><annotation encoding="application/x-tex">G=(G, *)</annotation></semantics></math>G=(G,∗) with identity <math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><msub><mi>e</mi><mi>G</mi></msub><mo separator="true">,</mo><mrow><mo fence="true">{</mo><msub><mi>e</mi><mi>G</mi></msub><mo fence="true">}</mo></mrow></mrow><annotation encoding="application/x-tex">e_{G},\left\{e_{G}\right\}</annotation></semantics></math>eG​,{eG​} and <math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><mi>G</mi></mrow><annotation encoding="application/x-tex">G</annotation></semantics></math>G itself are always subgroups of <math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><mi>G</mi></mrow><annotation encoding="application/x-tex">G</annotation></semantics></math>G, whereas <math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><mo fence="true">{</mo><msub><mi>e</mi><mi>G</mi></msub><mo fence="true">}</mo></mrow><annotation encoding="application/x-tex">\left\{e_{G}\right\}</annotation></semantics></math>{eG​} is called a ________.

Consider the additive group $\left(\mathbb{Z}_{6},+\right) .$ This group ($G$) has order $|G|=6$. Then, what is the order of its element $|2|$? 



Consider the additive group <math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><mrow><mo fence="true">(</mo><msub><mi mathvariant="double-struck">Z</mi><mn>6</mn></msub><mo separator="true">,</mo><mo>+</mo><mo fence="true">)</mo></mrow><mi mathvariant="normal">.</mi></mrow><annotation encoding="application/x-tex">\left(\mathbb{Z}_{6},+\right) .</annotation></semantics></math>(Z6​,+). This group (<math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><mi>G</mi></mrow><annotation encoding="application/x-tex">G</annotation></semantics></math>G) has order <math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><mi mathvariant="normal">∣</mi><mi>G</mi><mi mathvariant="normal">∣</mi><mo>=</mo><mn>6</mn></mrow><annotation encoding="application/x-tex">|G|=6</annotation></semantics></math>∣G∣=6. Then, what is the order of its element <math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><mi mathvariant="normal">∣</mi><mn>2</mn><mi mathvariant="normal">∣</mi></mrow><annotation encoding="application/x-tex">|2|</annotation></semantics></math>∣2∣?

Consider the following statement:
$$
U_{n} =  \mathbb{Z}_{n}^{*} =\{  a + n\mathbb{Z} | gcd(a,n) = 1 \}.
$$

According to the equation 
$$\mathbb{Z}_{4}^{*} = \{1,3\},$$

what will be the result of
$$\mathbb{Z}_{6}^{*} = ?$$

Consider the following statement:
<math xmlns="http://www.w3.org/1998/Math/MathML" display="block"><semantics><mrow><msub><mi>U</mi><mi>n</mi></msub><mo>=</mo><msubsup><mi mathvariant="double-struck">Z</mi><mi>n</mi><mo lspace="0em" rspace="0em">∗</mo></msubsup><mo>=</mo><mo stretchy="false">{</mo><mi>a</mi><mo>+</mo><mi>n</mi><mi mathvariant="double-struck">Z</mi><mi mathvariant="normal">∣</mi><mi>g</mi><mi>c</mi><mi>d</mi><mo stretchy="false">(</mo><mi>a</mi><mo separator="true">,</mo><mi>n</mi><mo stretchy="false">)</mo><mo>=</mo><mn>1</mn><mo stretchy="false">}</mo><mi mathvariant="normal">.</mi></mrow><annotation encoding="application/x-tex">U_{n} = \mathbb{Z}_{n}^{*} =\{ a + n\mathbb{Z} | gcd(a,n) = 1 \}.
</annotation></semantics></math>Un​=Zn∗​={a+nZ∣gcd(a,n)=1}.
According to the equation
<math xmlns="http://www.w3.org/1998/Math/MathML" display="block"><semantics><mrow><msubsup><mi mathvariant="double-struck">Z</mi><mn>4</mn><mo lspace="0em" rspace="0em">∗</mo></msubsup><mo>=</mo><mo stretchy="false">{</mo><mn>1</mn><mo separator="true">,</mo><mn>3</mn><mo stretchy="false">}</mo><mo separator="true">,</mo></mrow><annotation encoding="application/x-tex">\mathbb{Z}_{4}^{*} = \{1,3\},
</annotation></semantics></math>Z4∗​={1,3},
what will be the result of
<math xmlns="http://www.w3.org/1998/Math/MathML" display="block"><semantics><mrow><msubsup><mi mathvariant="double-struck">Z</mi><mn>6</mn><mo lspace="0em" rspace="0em">∗</mo></msubsup><mo>=</mo><mo stretchy="false">?</mo></mrow><annotation encoding="application/x-tex">\mathbb{Z}_{6}^{*} = ?
</annotation></semantics></math>Z6∗​=?

**(True or False)** The sets $\mathbb{Z}, \mathbb{Q}, \mathbb{R}, \mathbb{C}$ are commutative unitary rings under addition $(+)$ and multiplication $(\cdot)$.

(True or False) The sets <math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><mi mathvariant="double-struck">Z</mi><mo separator="true">,</mo><mi mathvariant="double-struck">Q</mi><mo separator="true">,</mo><mi mathvariant="double-struck">R</mi><mo separator="true">,</mo><mi mathvariant="double-struck">C</mi></mrow><annotation encoding="application/x-tex">\mathbb{Z}, \mathbb{Q}, \mathbb{R}, \mathbb{C}</annotation></semantics></math>Z,Q,R,C are commutative unitary rings under addition <math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><mo stretchy="false">(</mo><mo>+</mo><mo stretchy="false">)</mo></mrow><annotation encoding="application/x-tex">(+)</annotation></semantics></math>(+) and multiplication <math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><mo stretchy="false">(</mo><mo>⋅</mo><mo stretchy="false">)</mo></mrow><annotation encoding="application/x-tex">(\cdot)</annotation></semantics></math>(⋅).

Let $f(n)=5 n \log n+3 n(n+1) +n.$  

In this case, what is $f(n)=$ $\mathcal{O}(?)$?

Let <math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><mi>f</mi><mo stretchy="false">(</mo><mi>n</mi><mo stretchy="false">)</mo><mo>=</mo><mn>5</mn><mi>n</mi><mi>log</mi><mo>⁡</mo><mi>n</mi><mo>+</mo><mn>3</mn><mi>n</mi><mo stretchy="false">(</mo><mi>n</mi><mo>+</mo><mn>1</mn><mo stretchy="false">)</mo><mo>+</mo><mi>n</mi><mi mathvariant="normal">.</mi></mrow><annotation encoding="application/x-tex">f(n)=5 n \log n+3 n(n+1) +n.</annotation></semantics></math>f(n)=5nlogn+3n(n+1)+n.
In this case, what is <math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><mi>f</mi><mo stretchy="false">(</mo><mi>n</mi><mo stretchy="false">)</mo><mo>=</mo></mrow><annotation encoding="application/x-tex">f(n)=</annotation></semantics></math>f(n)= <math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><mi mathvariant="script">O</mi><mo stretchy="false">(</mo><mo stretchy="false">?</mo><mo stretchy="false">)</mo></mrow><annotation encoding="application/x-tex">\mathcal{O}(?)</annotation></semantics></math>O(?)?

Introduction

Preliminaries

Cryptographic Primitives

Elliptic Curve Cryptography

Information Security in Software Systems

Distributed Systems

Introduction to Blockchain Technology

Bitcoin

Introduction to Quantum Computing

Bitcoin Under Broken Crypto Primitives

Post-Quantum Blockchains

In Closing

Quiz: Preliminaries