A mathematical introduction to cryptographic primitives in blockchain technology.

crypto graphic and block chains updated-01-01.png

cryptographic.tar.gz

ellipticCurves

Algorithms

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

 Given point $P(4,2)$ on the elliptic curve $E_{11}(1,2)$, find 2P.


Given point <math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><mi>P</mi><mo stretchy="false">(</mo><mn>4</mn><mo separator="true">,</mo><mn>2</mn><mo stretchy="false">)</mo></mrow><annotation encoding="application/x-tex">P(4,2)</annotation></semantics></math>P(4,2) on the elliptic curve <math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><msub><mi>E</mi><mn>11</mn></msub><mo stretchy="false">(</mo><mn>1</mn><mo separator="true">,</mo><mn>2</mn><mo stretchy="false">)</mo></mrow><annotation encoding="application/x-tex">E_{11}(1,2)</annotation></semantics></math>E11​(1,2), find 2P.

 Let $E: y^{2}=x^{3}+x+1$ be an elliptic curve over $\mathbb{F}_{5}$ .
Then:
$$ 
\left|E\left(\mathbb{F}_{5}\right)\right|=?
$$


 

Let <math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><mi>E</mi><mo>:</mo><msup><mi>y</mi><mn>2</mn></msup><mo>=</mo><msup><mi>x</mi><mn>3</mn></msup><mo>+</mo><mi>x</mi><mo>+</mo><mn>1</mn></mrow><annotation encoding="application/x-tex">E: y^{2}=x^{3}+x+1</annotation></semantics></math>E:y2=x3+x+1 be an elliptic curve over <math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><msub><mi mathvariant="double-struck">F</mi><mn>5</mn></msub></mrow><annotation encoding="application/x-tex">\mathbb{F}_{5}</annotation></semantics></math>F5​ .
Then:
<math xmlns="http://www.w3.org/1998/Math/MathML" display="block"><semantics><mrow><mrow><mo fence="true">∣</mo><mi>E</mi><mrow><mo fence="true">(</mo><msub><mi mathvariant="double-struck">F</mi><mn>5</mn></msub><mo fence="true">)</mo></mrow><mo fence="true">∣</mo></mrow><mo>=</mo><mo stretchy="false">?</mo></mrow><annotation encoding="application/x-tex">\left|E\left(\mathbb{F}_{5}\right)\right|=?
</annotation></semantics></math>∣E(F5​)∣=?

$
\# E\left(\mathbb{F}_{p}\right)=\mathbb{0}
$

An elliptic curve is said to be anomalous if:


An elliptic curve is said to be anomalous if:

Which of the following is not a generic attack on the ECDLP?

Which of the following is not a generic attack on the ECDLP?

The curve must not be $\mathbb{F}_{p}$ anomalous, i.e., $\# E\left(\mathbb{F}_{p}\right) \neq p$.

The order $n$ of the base point $P$ should be prime or at least contain a very large prime factor.

The order $n$ of the base point $P$ should be chosen as large as possible.

The curve must be $\mathbb{F}_{p}$ anomalous, i.e., $\# E\left(\mathbb{F}_{p}\right) = p$.

How can we prevent an anomalous attack on ECDLP?

How can we prevent an anomalous attack on ECDLP?

Which of the following elliptic curves is the most secure against all known attacks?

Which of the following elliptic curves is the most secure against all known attacks?

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: Elliptic Curves Cryptography