1. Ryan Henry I 538 /B 609 : Introduction to Cryptography

2. Ryan Henry 1

3. 1 Tuesday’s lecture: Basic number theory Today’s lecture: More number theory Introduction to groups

4. Ryan Henry Assignment 4 is due TODAY! Assignment 5 has been posted! 3

5. Ryan Henry ab∈G Groups Def n : Let G be a non-empty set and let ‘’ be a binary operation acting on ordered pairs of elements from G. The pair (G,) is called a group if 1. Closure: ∀a,b∈G, 2. Associativity: ∀a,b,c∈G, 3. Identity: ∃e∈G, 4. Inverses: ∀a∈G, The group (G,) is abelian (or commutative) if 5. Commutative: ∀a,b∈G, 4 ??? (ab)c=a(bc) ∀a∈G, ae=ea=a ∃a -1 ∈G such that aa -1 =a -1 a=e a b=b a ??? ??? ??? ??? In our case this is typically multiplication

6. Ryan Henry Examples of groups ▪(▪(ℤ,+), ( ℚ,+), and (ℝ,+) –I–Identity: –I–Inverse of a: ▪(▪(ℝ∖{0}, ) and (ℚ∖{0}, ) where is regular multiplication –I–Identity: –I–Inverse of a: ▪(▪(ℤ n,⊞) where ⊞ is addition modulo n –I–Identity: –I–Inverse of a: Q: Is (ℤ n,⊡) a group, where ⊡ is multiplication modulo n? A: No! Not all elements of ℤ n have a multiplicative inverse modulo n 5 0 -a 1 1⁄a 0 n-a ?? ?? ?? ?? ?? ??

7. Ryan Henry Examples of groups ▪(▪(℥ n,⊡) where ⊡ is multiplication modulo n –I–Identity: –I–Inverse of a: Q: Is (℥ n,⊞) where ⊞ is addition modulo n? A: NO! ℥ n is not closed under addition modulo n. Q: Is (ℕ,+) a group? A: NO! ℕ does not have additive inverses! 6 1 a -1 mod n ?? ??

8. Ryan Henry Examples of groups Q: Let ℤ odd denote the set of odd integers. Explain why (ℤ odd,-) is not a group. A1: ℤ odd has no identity: 0 is even –A–Also, no identity ⇒ no inverses A2: ℤ odd is not closed under subtraction: odd-odd=even A3: Subtraction is not associative: (a-b)-c≠a-(b-c) 7

9. Ryan Henry Elementary properties of groups Thm (uniqueness of identity): In a group (G,), there is only one identity element. 8 Thm (uniqueness of inverses): Let (G,) be a group. For each a∈G, there exists a unique inverse. - Proofs of these facts are very simple (you are asked to prove them on assignment 5!)

10. Ryan Henry Elementary properties of groups 9 Thm (cancellation): Let (G,) be a group. The left and right cancellation laws both hold; that is, for all a,b,c∈G, – Left cancellation: ab=ac⇒b=c – Right cancellation: ba=ca ⇒b=c Proof (for right cancellation):Suppose ba=ca. Multiplying on the right by a -1 yields (ba)a -1 =(ca)a -1 By associativity, (ba)a -1 =b(aa -1 )=b and (ca)a -1 =c(aa -1 )=c Hence b=c. A symmetric argument proves left cancellation holds.

11. Ryan Henry Exponentiation ▪F▪For n∈{1,2,3,…} we define a n =aaaa ▪F▪For n=0, we define a n =e ▪F▪For n∈{-1,-2,-3, …} we define a n =(a -1 ) -n Q: Is (ab) n =a n b n ? A: Sometimes! Specifically, (a b) n =a n b n if a b=b a 10 n times Thm (law of exponents): Let (G,) be a group and let m,n∈ℤ. For each a∈G, a m a n =a m+n and (a m ) n =a mn.

12. Ryan Henry Order Def n : The number of elements in a group (G,) is called its order. We write |G| to denote the order of (G,). 11 Def n : Let (G,) be a group and let a∈G. The smallest positive integer i such that a i =e is called the order of a in (G,). We write |a| to denote the order of a in G. If |a|=|G|, then we call a a generator of (G,).

13. Ryan Henry That’s all for today, folks! 12