The Mathematics of Public-Key Cryptography, Lecture 3

September 19, 2000

Reference

• read section 4.4 of the textbook for background

Summary of material covered in lecture 3

• Parameter generation algorithm for RSA
  • generate p,q
  • compute n and phi(n)
  • choose b such that gcd(b,phi(n)) = 1
  • compute a = b^{-1} mod n
  • publish (n,b) as the public key and store (p,q,a) as the secret key
• encryption / decryption operations for RSA
• security of RSA
  • factorization of RSA-155 in August 1999
  • recommend p,q to be 512-bit integers
• complexity of arithmetic operations
  • integer addition and multiplication
  • modular addition and multiplication
  • modular exponentiation
    • square-and-multiply algorithm
    • number of multiplications
    • complexity analysis
    • proof of correctness
• complexity of RSA operations for k-bit moduli
  • computing n, phi(n): complexity O(k^2)
  • computing a: complexity O(k^2) using extended Euclidean algorithm
  • encryption and decryption: complexity O(k^3) for "random" exponents
  • effect of choice of encryption exponent on efficiency of RSA encryption
  • examples: b=3 and b=2^{16}+1
• danger of choosing small decryption exponent for RSA