Foundational cryptography concepts built on discrete math, such as modular exponentiation and public key basics.
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- What is modular arithmetic?
- Arithmetic where computations are performed with respect to a modulus, working only with remainders after division by a fixed number.
- What does the notation a ≡ b (mod n) mean?
- a and b are congruent modulo n, meaning they have the same remainder when divided by n.
- Define modular exponentiation.
- Computing the result of a^b mod n efficiently without first calculating a^b.
- Why is modular exponentiation important for cryptography?
- It allows efficient computation of large powers while keeping intermediate results bounded, enabling practical implementation of cryptographic algorithms.
- State Fermat's Little Theorem.
- If p is prime and gcd(a, p) = 1, then a^(p-1) ≡ 1 (mod p).
- State Euler's Theorem.
- If gcd(a, n) = 1, then a^φ(n) ≡ 1 (mod n), where φ(n) is Euler's totient function.
- What is Euler's totient function φ(n)?
- The count of positive integers less than or equal to n that are coprime with n.
- What is symmetric key cryptography?
- An encryption method using a single shared secret key for both encryption and decryption.
- What is public key cryptography?
- An encryption method using a pair of mathematically related keys: a public key for encryption and a private key for decryption.
- What two numbers form an RSA public key?
- The modulus n and the public exponent e.
- What two numbers form an RSA private key?
- The modulus n and the private exponent d.
- How is the RSA modulus n constructed?
- By multiplying two distinct large prime numbers: n = p * q.
- What is the relationship between the RSA exponents e and d?
- e and d are multiplicative inverses modulo φ(n), meaning e * d ≡ 1 (mod φ(n)).
- Define the discrete logarithm problem.
- Given g, h, and prime p, find the exponent x such that g^x ≡ h (mod p), believed to be computationally hard.
- Why is the discrete logarithm problem important to cryptography?
- Many public key cryptosystems derive their security from the computational difficulty of solving the discrete logarithm problem.
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