1. A Method for Obtaining Digital Signatures and Public-key Cryptosystems
R.L Rivest, A. Shamir, L. Adleman
Presented by Wesley Varela

2. Ronald L. Rivest, Adi Shamir, and Leonard Adleman
Inventers of RSA

3. OUTLINE Introduction The Basic Idea of RSA A Small Example (Simple)
The Big Hairy Math Stuff
A Better Example
Known Weaknesses

4. Introduction
“The era of “electronic mail” [10] may soon be upon us”

5. Why Encryption ??
What uses ??

6. So Bad People Don’t Do Bad Things

7. Public vs. Private Key Cryptosystems
Encryption and decryption keys can be derived from each other
The key must be securely delivered to the person(s) who need to decrypt or encrypt messages
Person(s) receiving the key must be trustworthy
More useable in single-user enviroments
Public Key
More secure and easier to use
Encryption keys are publicly available
Digital signatures

8. RSA – The Basic Idea Properties of RSA
Encrypting a plaintext message M gives the ciphertext. Deciphering the ciphertext returns M.
D(E(M)) = M
The inverse is also true (digital signatures).
E(D(M)) = M
The encryption key (E) and decryption key (D) are easily computed
E is publicly available but does not compromise the cryptosystem.

9. Basic Definitions you can’t live without
Trap-door one-way functions: easily computed in one direction but not the other, unless the decryption key is known
Trap-door one-way permutations: Every message is ciphertext from some message and every ciphertext is a permissible message.

10. A Small Example Assume we have two people Oliver Katie
Encryption Key Eo
Decryption Key Do
Katie
Encryption Key Ek
Decryption Key Dk

11. Oliver sends a message to Katie
Ek(M) = ciphertext
Sends the ciphertext to Katie
Katie receives the message and response
Dk(C) = plaintext
Eo(M2) = ciphertext
Sends the ciphertext to Oliver
Oliver receives the return ciphertext
Do(C2) = plaintext

12. Signatures Oliver sends a message to Katie Katie receives the message
Do(M) = Signature
Ek(Signature) = Ciphertext
Sends the ciphertext to Katie
Katie receives the message
Dk(C) = Signature
Eo(Signature) = plaintext
Has the pair (Signature, Plaintext)

13. Big Hairy Math Stuff Eo = (E(e, m) mod n ) + some algorithm
Do = (D(d, m) mod n ) + some algorithm
d and e are exponents.
m is the message
n is the product of 2 very large primes

14. Why Use Primes for p & q? They have no non-trivial factors
It makes it even harder to find Ф(n)
No algorithm to easily factor large numbers
Everything that’s cool has the word prime attached to it.
Prime rib
Prime time
Optimus Prime
Prime Directive

15. Eulers totient function Ф
Ф(n) - gives the number of positive integers less then n which are relatively prime to n.

16. Computing Ф(n) Ф(n) = Ф(p*q) = Ф(p)* Ф(q) = (p-1)*(q-1)

17. Selecting “d” d should be relatively prime to Ф(n)
Select d to be a prime larger than max(p,q)
“since d is relatively prime to Ф(n), it has a multiplicative inverse e in the ring of integers modulo Ф(n)”

18. Multiplicative Inverse Example
Two relatively prime numbers 5 and 7
1 * 5 = 5 ≡ 5 (Mod 7)
2 * 5 = 10 ≡ 3 (Mod 7)
3 * 5 = 15 ≡ 1 (Mod 7)
4 * 5 = 20 ≡ 6 (Mod 7)
5 * 5 = 25 ≡ 4 (Mod 7)
6 * 5 = 30 ≡ 2 (Mod 7)
7 * 5 = 35 ≡ 0 (Mod 7)
Z7 is a cyclic group

19. Computing e from d Using the Extended Euclidian Algorithm
We find e such that e*d ≡ 1 (modФ(n))

20. The following theorem from Euler and Fermat will be useful
MФ(n) ≡ 1 (mod n)

21. “since d is relatively prime to Ф(n), it has a multiplicative inverse e in the ring of integers modulo Ф(n)”
e*d ≡ 1 (mod Ф(n))
MФ(n) ≡ 1 (mod n)
(Me)d = Med = MkФ(n) +1
= MkФ(n) * M1
= (MФ(n))k * M
≡ 1K * M
= M

22. ALL about M
Choosing how to represent your message M is very important.
One method (ASCII)
hello world
m < n

23. Computing Me mod n Represent e as a binary number
ekek-1…e1e0 = 5 = 101
C = 1
For(i=k, i>=0; i--){
C = (C*C) mod n
if(ei = 1)
C = (C * M) mod n }
At most 2*log2(e) + 2*log2(e)

24. Primarily Testing How to find a really big prime
Randomly generate a large odd number b of the size you want
Use Solovay and Strassen’s probabilistic algorithm
Select some number a from {0, …, b-1}
gcd(a,b) = 1 and J(a,b) = a(b-1)/2
If false b is composite.
If true b is prime with a probability of at least ½

25. Factoring
Factoring (n)
Computing (Ф(n))
Computing d

26. “512-bit number RSA-155 was factored in seven months during 1999 ”
Faster Computers?
Pick a bigger prime.
“RSA Laboratories currently recommends key sizes of 1024 bits for corporate use and 2048 bits for extremely valuable keys like the root key pair used by a certifying authority. Several recent standards specify a 1024-bit minimum for corporate use. Less valuable information may well be encrypted using a 768-bit key”
“512-bit number RSA-155 was factored in seven months during 1999 ”

27. Known Weaknesses d<N.5 Lattice Attack
Low public exponent (Coppersmith)
Broadcast Attack (Hastad)
Related message Attack (Franklin-Reiter)

28. Provable
Is RSA provably secure?

29. Physical Security
Don’t let anyone copy your key or your primes