Cryptography: Number Theory in Practice
Cryptography: Number Theory in Practice
💡 Key Takeaways
- How primes and modular arithmetic underpin secure encryption and key exchange (RSA, DH).
- Why public-key systems use easy computations with large numbers but hard reverse problems (factoring, discrete log).
- Real-world protocols based on number theory: RSA, Diffie-Hellman, and digital signatures.
- Practical security choices: key sizes, RSA vs ECC, and common attack considerations.
❓ Frequently Asked Questions
What is modular arithmetic and why is it fundamental in cryptography?
Modular arithmetic deals with integers wrapped around by a modulus n (remainders after division). It underpins many cryptographic schemes (e.g., RSA, Diffie-Hellman, elliptic-curve cryptography) because operations are easy to perform in one direction and hard to reverse without the modulus.
What is public-key cryptography and how does number theory enable it?
Public-key cryptography uses a public key for encryption and a private key for decryption. Its security relies on number-theoretic problems that are easy to perform in one direction but hard to reverse, such as factoring large numbers or computing discrete logarithms.
What is the discrete logarithm problem and why is it hard?
In a group, given a base g and y = g^x, find the exponent x. For appropriately chosen groups and large sizes, computing x from g and y is computationally hard, which secures many protocols like Diffie-Hellman and ECC.
What is RSA and what number theory concepts does it use?
RSA uses two large primes p and q to form n = pq. The public key is (n, e) and the private key is d, where d is the modular inverse of e modulo φ(n) = (p−1)(q−1). Security rests on the difficulty of factoring n and the properties of modular arithmetic.