Announcements: HW3 updated. Due next Thursday Written quiz tomorrow on chapters 1-2 (next slide)

1. DTTF/NB479: Dszquphsbqiz Day 11 • Announcements: • HW3 updated. Due next Thursday • Written quiz tomorrow on chapters 1-2 (next slide) • Computer quiz next week on breaking codes from chapter 2 • Questions? • Today: • Finish Modular Exponents example • Fermat’s little theorem • Euler’s theorem

2. Tomorrow’s Quiz • Rules: • Written problems • Closed book and computer • You may bring a note sheet: 1 handwritten sheet of 8.5 x 11 paper, one side only. • Content: • Concepts of the algorithms we discussed, how they work, how you can break them using various attacks • Inverses of integers and matrices (mod n) • Working out some examples by hand, like 5-1 mod (7) • Anything else from ch 1-2, but nothing that will require a computer.

3. Modular Exponentiation (All congruences are mod 152) • Compute 3^2000 (mod 152) • Technique: • Repeatedly square 3, but take mod at each step. • Then multiply the terms you need to get the desired power. • Matlab’s powermod()

4. Is there an easier way to compute 32000 (mod 152)? • Consider first a similar example, 32000 (mod 17) (chosen so p is prime). • Today’s theorems will be really important when dealing with RSA encryption – pay careful attention!

5. Fermat’s Little Theorem • if p is prime and doesn’t divide a. • Examples: 22(mod 3), 44 (mod ???) • So what’s (32002)(mod 17)?

6. Converse when a=2 • If p is prime and doesn’t divide a, • Converse: • If , p is prime and doesn’t divide a. • This is almost always true when a = 2. Rare counterexamples: • n = 561 =3*11*17, but • n = 1729 = 7*13*19 • Can do first one by hand if use Fermat and combine results with Chinese Remainder Theorem

7. Using Fermat within a primality testing scheme n Even? no div by other small primes? no Prime by Factoring/advanced techn.? yes prime

8. Using Fermat within a primality testing scheme (ch 6) n Use Fermat as a filter since it’s faster than factoring (if calculated using the powermod method). Odd? no div by other small primes? no yes Fermat: p prime 2p-1 = 1 (mod p) Contrapositive? Prime by Factoring/advanced techn.? Why can’t we just compute 2n-1(mod n) Using Fermat if it’s so much faster? yes prime

9. Euler’s Theorem • Analog to Fermat for composite modulus • What’s f(n)? • f(n) = the number of integers a, s.t., 1<= a <= n where gcd(a,n) = 1. • Ex: f(10) = 4. • What’s f(p) for p prime? • What about f(n), where n =pq (a product of 2 primes)

10. Euler’s f-function • Notes: The p are taken from the set of distinct primes that divide n • Eg, for n=60, use p = 2 (only once), 3, and 5. • Answer: f(60)=16 • When we compute the ratios, what about mutual exclusion? Consider the intuition: • Crossing out every number divisible by 3 leaves 2/3 of them. • If I crossed out the even numbers first, then crossing out every odd number divisible by 3 still leaves 2/3 of those left! *Thanks to Bill Waite for this good insight

11. Back to Euler’s Theorem • As long as gcd(a,n,) = 1 • Examples: • Find last 3 digits of 7803 • Find 32007 (mod 12) • Find 243210 (mod 101) Basic Principle: when working mod n, view the exponents mod f(n).