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Lecture Six: KEX and Asymmetric Operations 1. Public Key Cryptography and Digital Signature: • Public Key Cryptography: Public key cryptography is a form of asymmetric encryption where one key (public) is used to encrypt the data, and a different key (private) is used to decrypt it. This eliminates the need for both parties to share a single key in advance. This means, encrypt with public key and decrypt with private key. Example: When you visit a website, your browser uses public key cryptography to encrypt data before sending it. The website uses its private key to decrypt it, ensuring secure communication. • Digital Signature: is a cryptographic method used to verify the authenticity and integrity of a message, document, or piece of data. It ensures that the message was sent by the legitimate sender (authentication) and that the content hasn’t been altered (integrity). Example: When Alice sends Bob an important document, she signs it with her private key. Bob can then use Alice’s public key to confirm that the document really came from her and hasn’t been tampered with during transmission. 2. Trapdoor Function: • Trapdoor function: A trapdoor function is a type of mathematical function that is easy to compute in one direction but extremely difficult to reverse without special information (the "trapdoor"). • Example: Multiplying two large prime numbers is easy, but factoring their product back into those primes (without knowing the original numbers) is incredibly difficult. This principle is used in encryption algorithms like RSA to make decryption nearly impossible without the private key. 3. Finite-Field Diffie-Hellman (DH) and Canonical Vulnerabilities: • Finite-Field DH: In this version of Diffie-Hellman, calculations are performed in a finite field, meaning they use modular arithmetic, which allows for secure key exchange even in an insecure environment. • Canonical Vulnerabilities: Finite-Field DH is vulnerable to man-in-the-middle attacks if the exchange isn’t properly authenticated. An attacker can intercept the key exchange and generate their own key with each party, effectively decrypting the communication. • Example: If Alice and Bob don’t authenticate each other during the key exchange, Mallory can trick both parties into thinking they’re communicating securely, while she’s decrypting and re-encrypting the messages between them. 4. Elliptic Curve Cryptography (ECC): • Why ECC is Used: ECC is a more efficient form of asymmetric cryptography that uses elliptic curves over finite fields. It offers the same level of security as traditional methods like RSA but with much smaller key sizes, making it faster and requiring less computational power. • How It Works: ECC relies on the difficulty of solving the elliptic curve discrete logarithm problem. This problem is similar to the trapdoor function in that it’s easy to compute in one direction but nearly impossible to reverse without the private key. • Example: ECC is widely used in secure messaging apps and mobile devices because it provides strong security without draining battery life or requiring heavy computational resources.