Public Key Cryptography From Scratch In Python

Diffie hellman key exchange protocol enables to generate shared keys between parties in a secure way. We'll mention the background of the algorithm.

We'll adapt Diffie Hellman Key Exchange algorithm in python. Also, we'll use shared key in symmetric encryption.

In this section, we will focus on RSA. It depends on Euler's generalization of Fermat's Little Theorem. In this video, we will prove Fermat's Little Theorem by Induction. If you are not interested in the math behind algorithm, you can jump to "RSA Overview" video directly.

We have proven Fermat's Little Theorem by induction in the recent video. In this video, we will prove it with necklace method and then prove Euler's generalization of Fermat's Little Theorem with number theory. If you are not interested in the math behind this algorithm, you can jump to "RSA Overview" video directly.

RSA is the most common public key cryptography algorithm. Today, we'll mention its key generation and encryption - decryption procedures.

We have shown encryption and decryption with RSA algorithm in the recent video. This algorithm is based on Euler's generalization of Fermat's Little Theorem. In this video, we are going to show how RSA is working. If you are not interested in the math behind algorithm, you can jump to next video.

In RSA Overview, we've mentioned that randomly generated e and totient function must be coprime (or relatively prime). The question is that how can this tested? Euclidean algorithm finds the greatest common divisors. If GCD is equal to 1, then these two numbers are coprime. In this lecture, we'll mention the euclidean algorithm.

In RSA Overview, we've mentioned that randomly generated e and its multiplicative inverse d are public and private key pairs. So, how can we calculate the multiplicative inverse? Extended euclidean algorithm finds the multiplicative inverse of an integer faster. We'll mention the extended euclidean algorithm.

Until now, we have mentioned the key generation and encryption - decryption procedures. Now, we'll adapt all of these steps into a python code. Also, we'll work on really large prime numbers.

RSA can be used for signing messages digitally and verifying them. In this video, we'll mention the overview of digital signatures.

We are going to use RSA algorithm to exchange key between parties. 

We are going to adapt key exchange schema to python in this lecture

Decryption is handled by m = c^privatekey mod n formula. One can restore the message for known ciphertext, private key and n. However, extracting private key is not easy task for knwon message, ciphertext and n even though an attacker has plaintext, ciphertext pair. This refers to discrete logarithm problem.

In this video, I will provide a comprehensive overview of the various forms in which elliptic curves can be represented. The video begins by introducing the concept of elliptic curves and their significance in mathematics and cryptography. Through clear explanations and visual aids, the presenter goes on to summarize and compare the most commonly used forms of elliptic curves, including Weierstrass form, Koblitz form, and Edwards form. Each form is discussed in terms of its defining equations, advantages, and specific applications. By the end of the video, viewers gain a solid understanding of the distinct characteristics of each form and are equipped with valuable knowledge to explore further aspects of elliptic curves in their preferred representation.

In this video, you are guided through a comprehensive demonstration of the addition law proof within the context of Weierstrass form in elliptic curves. The video begins by introducing the foundational concepts of elliptic curves and their Weierstrass form representation. Through clear explanations and visual aids, the presenter illustrates how the addition of two points on an elliptic curve can be geometrically and algebraically computed using the Weierstrass equations. By the end of the video, viewers gain a solid understanding of the rigorous proof behind the addition law in Weierstrass form for elliptic curves, empowering them to explore further applications and insights in this intriguing field of mathematics.

In this video, you are taken on a comprehensive journey through the proof of the addition law within the context of Koblitz form in elliptic curves. The video commences by providing a concise introduction to elliptic curves and their representation in Koblitz form. With the aid of clear explanations and visual aids, the presenter elucidates how the addition of two points on an elliptic curve can be algebraically and geometrically computed using the Koblitz equations. By the conclusion of the video, viewers are equipped with a solid grasp of the rigorous proof behind the addition law in Koblitz form for elliptic curves, enabling them to delve deeper into the applications and intricacies of this fascinating mathematical realm.

In this video, you are presented with a comprehensive demonstration of the addition law proof within the framework of Edwards form, also known as Edwards curves, in elliptic curve theory. The video begins by introducing the fundamental concepts of elliptic curves and their representation in Edwards form. Through clear explanations and visual aids, the presenter illustrates how the addition of two points on an Edwards curve can be algebraically and geometrically computed using the Edwards equations. By the end of the video, viewers gain a thorough understanding of the rigorous proof behind the addition law in Edwards form for elliptic curves, empowering them to explore further applications and insights in this captivating field of mathematics.

In the video titled "Elliptic Curve Scalar Multiplication and Double-and-Add Method," viewers are guided through a detailed explanation of scalar multiplication on elliptic curves and the popular double-and-add method. The video begins by introducing the concept of scalar multiplication and its significance in elliptic curve cryptography. The presenter explains how scalar multiplication is computed using the double-and-add method, which involves doubling points on the curve and adding them together in a sequential manner. Through clear explanations and visual demonstrations, viewers gain a comprehensive understanding of how the double-and-add method efficiently calculates scalar multiplication on elliptic curves. By the end of the video, viewers are equipped with valuable knowledge of this essential operation, enabling them to explore further applications and cryptographic protocols in the fascinating realm of elliptic curve scalar multiplication.

In the video titled "Explaining the Order of Elliptic Curve Groups," viewers are provided with a comprehensive explanation of the order of elliptic curve groups. The video begins by introducing the concept of elliptic curves and their associated group structures. The presenter delves into the definition of the order of an elliptic curve group, which represents the total number of points on the curve, including the point at infinity. Through clear explanations and visual aids, viewers are guided through the process of determining the order of an elliptic curve group using mathematical techniques such as counting points. By the end of the video, viewers gain a solid understanding of how to calculate and interpret the order of elliptic curve groups, empowering them to explore further applications and cryptographic protocols that rely on this fundamental property.

In the video titled "Explaining the Neutral Element (Identity Element) in Elliptic Curves," viewers are provided with a clear and detailed explanation of the neutral element, also known as the identity element, in elliptic curves. The video begins by introducing the concept of elliptic curves and their group structure. The presenter focuses on the neutral element, which is a special point on the curve that serves as the identity for the group operation. Through visual aids and mathematical explanations, viewers gain a deep understanding of the properties and characteristics of the neutral element in elliptic curve groups. The video explores how the neutral element interacts with other points on the curve, emphasizing its significance in operations such as addition and scalar multiplication. By the end of the video, viewers have a solid grasp of the role and importance of the neutral element in elliptic curves, enabling them to apply this knowledge in further studies or cryptographic applications.

In the video titled "Explaining the Elliptic Curve Discrete Logarithm Problem," viewers are presented with a comprehensive explanation of the Elliptic Curve Discrete Logarithm Problem (ECDLP). The video begins by introducing the ECDLP as a fundamental challenge in elliptic curve cryptography. The presenter explains how the problem involves finding the exponent or scalar value that, when multiplied by a given base point, results in a specific point on the elliptic curve. Through clear explanations and illustrative examples, viewers gain a deep understanding of the difficulty and significance of solving the ECDLP, which forms the foundation of many cryptographic schemes and protocols. The video further explores the security implications of the ECDLP and discusses various algorithms and techniques employed to mitigate its vulnerability. By the end of the video, viewers are equipped with a comprehensive understanding of the ECDLP, enabling them to appreciate its role in secure elliptic curve cryptography and engage in further studies in this field.

In the video titled "Explaining Elliptic Curve Diffie-Hellman Key Exchange," viewers are presented with a detailed explanation of the Elliptic Curve Diffie-Hellman (ECDH) key exchange protocol. The video begins by introducing the concept of key exchange and the need for secure communication in modern cryptography. The presenter then dives into the specifics of the ECDH protocol, which utilizes the mathematical properties of elliptic curves to establish a shared secret key between two parties. Through clear explanations and visual aids, viewers gain a solid understanding of how the ECDH protocol operates, including the generation of private and public keys, the exchange of public keys, and the derivation of the shared secret. The video also highlights the advantages of ECDH, such as its efficiency and resistance to attacks. By the end of the video, viewers are equipped with a comprehensive understanding of the ECDH key exchange protocol, enabling them to appreciate its significance and apply it in secure communication scenarios.

In the video titled "Implementing Elliptic Curve Diffie-Hellman Key Exchange from Scratch in Python," viewers are provided with a comprehensive demonstration of how to implement the Elliptic Curve Diffie-Hellman (ECDH) key exchange protocol without relying on external libraries or modules, using only Python. The video begins by explaining the key concepts and mathematics behind elliptic curve cryptography and the ECDH protocol. The presenter then guides viewers through the step-by-step process of generating elliptic curve parameters, performing point operations, generating private and public keys, exchanging keys, and deriving the shared secret key, all through their own custom code implementation. By following along with the code explanations and examples, viewers gain a deep understanding of the inner workings of the ECDH protocol and develop the skills to implement it from scratch using Python. This empowers viewers to not only grasp the theoretical aspects of elliptic curve cryptography but also acquire the practical knowledge needed to build secure communication systems and applications.

In the video titled "Explaining the Elliptic Curve Digital Signature Algorithm (ECDSA)," viewers are provided with a comprehensive explanation of the Elliptic Curve Digital Signature Algorithm. The video begins by introducing the concept of digital signatures and their importance in ensuring the authenticity and integrity of digital messages. The presenter then delves into the specifics of ECDSA, which is a widely used digital signature scheme based on elliptic curve cryptography. Through clear explanations and visual aids, viewers gain a deep understanding of how ECDSA works, including key generation, signature creation, and signature verification. The video also explores the mathematical principles and security properties behind ECDSA, highlighting its advantages such as shorter key lengths and faster computations compared to traditional signature algorithms. By the end of the video, viewers are equipped with a comprehensive understanding of ECDSA, enabling them to appreciate its significance in secure digital communications and cryptographic applications.

In the video titled "Implementing Elliptic Curve Digital Signature Algorithm from Scratch in Python," viewers are taken on a practical journey of implementing the Elliptic Curve Digital Signature Algorithm (ECDSA) entirely using Python, without relying on external libraries or modules. The video begins by providing a thorough explanation of the ECDSA algorithm, its mathematical foundations, and its significance in digital signature schemes. The presenter then guides viewers through the step-by-step process of generating elliptic curve parameters, creating private and public keys, signing messages, and verifying signatures, all through their own custom code implementation in Python. By following along with the code examples and explanations, viewers gain hands-on experience in implementing the ECDSA algorithm from scratch, enhancing their understanding of the underlying mathematics and the practical aspects of digital signature generation and verification. This video empowers viewers to not only grasp the theoretical concepts of ECDSA but also develop the skills to build secure and efficient digital signature systems using Python.

In the video titled "Explaining and Implementing the Importance of Random Keys in Elliptic Curve Digital Signature Algorithm (ECDSA) in Python from Scratch," viewers are provided with a detailed understanding of the crucial role of random keys in the ECDSA algorithm. The video begins by highlighting the significance of random key generation in cryptographic systems, particularly in the context of digital signature algorithms. The presenter then explains the potential consequences of using static or predictable keys by citing the historical example of the Sony PlayStation security breach, where the use of a static key compromised the system's integrity. By emphasizing the importance of random key generation in ECDSA, the video showcases the potential vulnerabilities and pitfalls of using non-random or easily guessable keys. Through a step-by-step implementation in Python, viewers gain practical experience in generating random keys for ECDSA, ensuring the integrity and security of digital signatures. By the end of the video, viewers are equipped with a comprehensive understanding of the significance of random key generation in ECDSA, allowing them to build more robust and secure digital signature systems.

In the video titled "Explaining the Edwards Curve Digital Signature Algorithm," viewers are presented with a comprehensive explanation of the Edwards Curve Digital Signature Algorithm (EdDSA). The video begins by introducing the concept of digital signatures and their importance in ensuring the authenticity and integrity of digital messages. The presenter then focuses on EdDSA, which is a modern digital signature scheme based on the Edwards curve. Through clear explanations and visual aids, viewers gain a deep understanding of how EdDSA works, including key generation, signature creation, and signature verification. The video also highlights the advantages of EdDSA, such as its resistance to side-channel attacks and efficient implementation. By the end of the video, viewers have a comprehensive understanding of EdDSA, enabling them to appreciate its significance in secure digital communications and cryptographic applications using Edwards curves.

In the video titled "Implementing Edwards Curve Digital Signature Algorithm from Scratch in Python," viewers are taken on a practical journey of implementing the Edwards Curve Digital Signature Algorithm (EdDSA) entirely using Python, without relying on external libraries or modules. The video begins by providing a comprehensive explanation of the EdDSA algorithm, its mathematical foundations, and its significance in digital signature schemes using Edwards curves. The presenter then guides viewers through the step-by-step process of generating curve parameters, creating private and public keys, signing messages, and verifying signatures, all through their own custom code implementation in Python. By following along with the code examples and explanations, viewers gain hands-on experience in implementing the EdDSA algorithm from scratch, enhancing their understanding of the underlying mathematics and the practical aspects of digital signature generation and verification. This video empowers viewers to not only grasp the theoretical concepts of EdDSA but also develop the skills to build secure and efficient digital signature systems using Python with Edwards curves.

In the video titled "Comparing ECDSA and EdDSA: Choosing the Right Digital Signature Algorithm," viewers are presented with a detailed comparison between the two popular digital signature algorithms, ECDSA (Elliptic Curve Digital Signature Algorithm) and EdDSA (Edwards Curve Digital Signature Algorithm). The video begins by introducing the significance of digital signatures and the role they play in securing digital communication. The presenter then delves into the technical aspects of both algorithms, highlighting their similarities and differences in terms of mathematical foundations, security properties, efficiency, and implementation considerations. Through clear explanations and visual aids, viewers gain a comprehensive understanding of the strengths and weaknesses of each algorithm, allowing them to make informed decisions in choosing the appropriate digital signature scheme for their specific use cases. By the end of the video, viewers are equipped with valuable knowledge that enables them to evaluate and select between ECDSA and EdDSA based on their unique requirements and security considerations.

In the video titled "Explaining Elliptic Curve ElGamal Encryption," viewers are provided with a comprehensive explanation of the Elliptic Curve ElGamal encryption scheme. The video begins by introducing the concept of public-key encryption and the need for secure communication. The presenter then delves into the specifics of the ElGamal encryption scheme, which is based on elliptic curve cryptography. Through clear explanations and visual aids, viewers gain a deep understanding of how the Elliptic Curve ElGamal encryption works, including key generation, encryption, and decryption processes. The video also explores the mathematical principles behind the scheme, highlighting its security properties and advantages. By the end of the video, viewers are equipped with a comprehensive understanding of Elliptic Curve ElGamal encryption, enabling them to appreciate its significance in secure communication and cryptographic applications.

In the video titled "Implementing Elliptic Curve ElGamal Encryption from Scratch in Python," viewers are taken on a practical journey of implementing the Elliptic Curve ElGamal encryption scheme entirely using Python, without relying on external libraries or modules. The video begins by providing a thorough explanation of the ElGamal encryption scheme, its mathematical foundations, and its significance in secure communication. The presenter then guides viewers through the step-by-step process of generating elliptic curve parameters, creating encryption and decryption keys, encrypting messages, and decrypting ciphertexts, all through their own custom code implementation in Python. By following along with the code examples and explanations, viewers gain hands-on experience in implementing the Elliptic Curve ElGamal encryption scheme from scratch, enhancing their understanding of the underlying mathematics and the practical aspects of secure message encryption and decryption. This video empowers viewers to not only grasp the theoretical concepts of Elliptic Curve ElGamal encryption but also develop the skills to build secure communication systems using Python.

In the video titled "Explaining the Elliptic Curve Integrated Encryption Scheme (ECIES)," viewers are presented with a comprehensive explanation of the Elliptic Curve Integrated Encryption Scheme. The video begins by introducing the need for secure communication and the concept of public-key encryption. The presenter then delves into the specifics of ECIES, a widely used encryption scheme based on elliptic curve cryptography. Through clear explanations and visual aids, viewers gain a deep understanding of how ECIES combines symmetric-key encryption and public-key encryption techniques to achieve secure and efficient communication. The video explores the steps involved in ECIES, including key generation, encryption, and decryption processes. Additionally, the presenter discusses the advantages of ECIES, such as its resistance to attacks and the ability to provide confidentiality, integrity, and authenticity of encrypted data. By the end of the video, viewers are equipped with a comprehensive understanding of ECIES, enabling them to appreciate its significance and apply it in secure communication scenarios.

Learn how to implement Elliptic Curve Integrated Encryption Scheme (ECIES) in Python from scratch with authentication. This tutorial will take you through the steps of generating EC keys, encrypting and decrypting messages, and verifying the authenticity of the sender using ECDSA signatures. Perfect for anyone who wants to enhance their encryption skills with a secure and efficient encryption scheme.

Learn how to implement Elliptic Curve MQV (Menezes-Qu-Vanstone) key agreement from scratch using Python. This tutorial breaks down the cryptographic protocol, guides you through key generation, commitment, and shared key computation, and provides a hands-on Python implementation. Enhance your cryptography skills and secure key agreement with Elliptic Curve MQV in Python.

Homomorphic encryption is letting us to make calculations on encrypted data without private key. Algorithms could be homomorphic with respect to the multiplication or addition. These are called partially homomorphic encryption algorithms.

RSA is partially homomorphic with respect to the multiplication. In this video, we will implement and then show the math behind its homomorphic feature.

Homomorphic encryption is letting us to make calculations on encrypted data without private key. Algorithms could be homomorphic with respect to the multiplication or addition. These are called partially homomorphic encryption algorithms.

Standard ElGamal is partially homomorphic with respect to the multiplication. Its modified version - Exponential ElGamal - would be homomorphic with respect to the addition but it loses its multiplicative homomorphic feature then. In this video, we will implement and then show the math behind its homomorphic features.

Homomorphic encryption is letting us to make calculations on encrypted data without private key. Algorithms could be homomorphic with respect to the multiplication or addition. These are called partially homomorphic encryption algorithms.

Paillier is partially homomorphic with respect to the addition. Also, a plain power of its ciphertext would be multiplicative homomorphic as well . In this video, we will implement and then show the math behind its homomorphic features.

In the video titled "Explaining and Implementing the Partially Homomorphic Properties of Elliptic Curve ElGamal Encryption in Python from Scratch," viewers are provided with a comprehensive explanation and practical demonstration of the partially homomorphic properties of Elliptic Curve ElGamal encryption. The video begins by introducing the concept of homomorphic encryption and its significance in performing computations on encrypted data. The presenter then focuses on the partially homomorphic properties of Elliptic Curve ElGamal encryption, which allows for the homomorphic addition of ciphertexts. Through clear explanations and step-by-step code implementation in Python, viewers gain a deep understanding of how to perform addition operations on encrypted data using Elliptic Curve ElGamal encryption. By following along with the code examples and explanations, viewers gain hands-on experience in implementing and utilizing the partially homomorphic properties of Elliptic Curve ElGamal encryption, enabling them to perform computations on encrypted data while preserving confidentiality. This video equips viewers with valuable knowledge to explore and leverage the partially homomorphic properties of Elliptic Curve ElGamal encryption in various cryptographic applications.

We'll explore homomorphic encryption's power by demonstrating its key features: additive homomorphism, scalar multiplication, and ciphertext regeneration. In this video, we'll split the screen to show encryption done on-premises and homomorphic operations performed in the cloud. This setup highlights the practical use of homomorphic encryption in maintaining data security in untrusted environments. We'll compare the benefits of homomorphic encryption over traditional symmetric key encryption, particularly for cloud operations. Discover how you can enhance data security without compromising functionality. Join us to unlock the potential of homomorphic encryption with Paillier (or RSA, ElGamal and more) and learn how it can transform data security.