RSA

• I enjoyed playfair's method last class and wanted to build on it.
• This is from 12.3 starting on page 795.
• There is some math here I don't really expect you to get.
• RSA is named after the creators Rivest, Shamir, Adelman in 1977
  • Apparently a version was created by British Intelligence in 1973.
  • And not declassified until 1997
• This is a public-key private key encryption system.
• The security of the system is based on the difficulty of factoring large numbers.
  • There are no known methods to defeat the system when a large enough key is used.
• Because of the method RSA is slow, and not normally used for basic exchanges
  • But it is great for exchanges of a symmetric key.
• The basic of the system.
  • Given three appropriate integers, e, d and n
  • $(m^e)^d = m \pmod n$
  • $(m^d)^e = m \pmod n$
• IN the above example
  • the public key is (n, e)
  • The private key is (n,d)
• Key Generation
  • This relies on number theory.
  • The Euler Totient Function $\varphi(n)$
    • Also Euler's Phi function.
    • Returns the number of posotive integers up to a given integer n that are relatively prime to n.
    • For a prime number n, it is n-1, since all numbers less than n are prime to n.
    • If two numbers, p and q are relatively prime (gcd(p,q)=1), then $\varphi(nm) = \varphi(n)\varphi(m)$
    • I don't really know this, I asked Doug.
  • Select two relatively prime numbers p and q.
  • $n = pq$
  • Compute the Carmichael function
    • $\lambda(n)$.
    • is the smallest positive integer m such that $a^m = 1 \pmod n$ for every integer $a$ between 1 and n that is coprime to n.
    • We want $\lambda(n) = lcm(p-1, q-1)$.
  • Select an ecryption key $e$ such that $ 1 \le e \le \lambda(n)$ and e coprime to $\lambda(n)$
    • e is the public key.
  • Find $d$ such that $de = 1 \pmod n$
    • d is the private key.
• This algorithm is strong because factoring is an open problem.
• Typically
  • p and q are selected to be nearly 100 digits each.
  • Thus n is 200 digits, or about 512 bits.
• Practical
  • ssh-keygen is part of openssh.
  • A discussion of the file format.