🔐 Cryptography Unit 2

🎯 Unit 2 Overview

Unit 2 covers advanced encryption and public key cryptography. In this unit, AES, Diffie-Hellman, RSA, digital signatures, primality testing, elliptic curve cryptography and Chinese Remainder Theorem are studied.

🔒 Advanced Encryption Standard (AES)

AES is a symmetric key block cipher used for secure data encryption. It replaced DES because DES became weak due to its small key size.

Features of AES

• Symmetric key encryption algorithm
• Block size: 128 bits
• Key sizes: 128-bit, 192-bit and 256-bit
• More secure than DES
• Fast and widely used

⚙️ AES Round Operations

AES performs multiple rounds of transformation on plaintext to generate ciphertext.

1. SubBytes: Byte substitution using S-box.
2. ShiftRows: Rows of state matrix are shifted.
3. MixColumns: Columns are mixed for diffusion.
4. AddRoundKey: Round key is XORed with state matrix.

🔑 Public Key Cryptosystem

Public key cryptosystem uses two keys: public key and private key. Public key is shared with everyone, while private key is kept secret by the owner.

Advantages

• Solves key distribution problem
• Supports digital signatures
• Provides authentication
• Useful for secure communication over internet

📌 Discrete Logarithmic Problem

The discrete logarithmic problem is a mathematical problem used in many public key cryptosystems. It is easy to calculate powers modulo a number, but difficult to find the exponent.

🤝 Diffie-Hellman Key Exchange

Diffie-Hellman is a key exchange algorithm used to securely share a secret key over an insecure channel.

Steps

1. Alice and Bob agree on public values p and g.
2. Alice chooses private key a and sends A = g^a mod p.
3. Bob chooses private key b and sends B = g^b mod p.
4. Alice calculates shared key K = B^a mod p.
5. Bob calculates shared key K = A^b mod p.
6. Both get the same secret key.

Advantages

• Secure key exchange
• No need to send secret key directly
• Used in secure communication protocols

⚠️ Computational & Decisional Diffie-Hellman Problem

🔐 RSA Cryptosystem

RSA is a public key cryptosystem based on the difficulty of factorizing large prime numbers. It is used for encryption, decryption and digital signatures.

RSA Key Generation Steps

1. Choose two large prime numbers p and q.
2. Calculate n = p × q.
3. Calculate φ(n) = (p − 1)(q − 1).
4. Choose public key e such that gcd(e, φ(n)) = 1.
5. Calculate private key d such that d × e ≡ 1 mod φ(n).
6. Public key = (e, n), Private key = (d, n).

Encryption and Decryption

📄 RSA Assumptions

• Large prime factorization is computationally difficult.
• It is difficult to find private key from public key.
• Security depends on the size of prime numbers.
• Breaking RSA requires factoring n into p and q.

✍️ RSA Signature

RSA digital signature is used to verify authenticity and integrity of a message.

Process

1. Sender creates hash of message.
2. Sender encrypts hash using private key.
3. This encrypted hash is the digital signature.
4. Receiver decrypts signature using sender public key.
5. Receiver compares received hash with calculated hash.

🆔 Schnorr Identification Scheme

Schnorr identification is a cryptographic protocol used to prove identity without revealing the secret key.

Features

• Based on discrete logarithm problem
• Used for authentication
• Efficient and secure
• Does not reveal private key

🔎 Primality Testing

Primality testing is used to check whether a number is prime or not. Prime numbers are very important in cryptography, especially RSA.

Common Methods

• Trial Division
• Fermat Primality Test
• Miller-Rabin Test

📈 Elliptic Curve Cryptography (ECC)

Elliptic Curve Cryptography is a public key cryptographic technique based on elliptic curves. It provides strong security with smaller key sizes.

Elliptic Curve Equation

Advantages of ECC

• Smaller key size
• High security
• Faster computation
• Less memory requirement
• Useful in mobile and IoT devices

🧮 Elliptic Curve over Reals and Modulo Prime

🧩 Chinese Remainder Theorem

Chinese Remainder Theorem is used to solve simultaneous congruence equations. It is also used in RSA to speed up computations.

Example Form

Applications

• RSA optimization
• Modular arithmetic
• Cryptographic computations
• Number theory problems

⚖️ AES vs RSA

⭐ Important Questions

1. Explain AES algorithm and its round operations.
2. Explain public key cryptosystem.
3. Explain Diffie-Hellman Key Exchange algorithm.
4. Explain Computational and Decisional Diffie-Hellman problem.
5. Explain RSA algorithm with key generation steps.
6. Explain RSA encryption and decryption.
7. Explain RSA digital signature.
8. Write short note on Schnorr Identification Scheme.
9. Explain Elliptic Curve Cryptography.
10. Explain Chinese Remainder Theorem and its applications.

🔥 Last Minute Revision

• AES = symmetric key block cipher.
• AES block size = 128 bits.
• AES key sizes = 128, 192, 256 bits.
• Public key cryptosystem uses public and private key.
• Diffie-Hellman is used for secure key exchange.
• RSA security depends on factorization difficulty.
• RSA public key = (e,n), private key = (d,n).
• ECC gives high security with smaller key size.
• CRT is used for solving congruences and RSA optimization.

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