Announcements: HW3 updated. Due next Thursday HW3 updated. Due next Thursday Written quiz today Written quiz today Computer quiz next Friday on breaking codes from chapter 2 Computer quiz next Friday on breaking codes from chapter 2Today: Three-pass protocol Three-pass protocol Quiz QuizQuestions? DTTF/NB479: DszquphsbqizDay 12

Wrapping up Fermat and Euler We skipped the proof of Fermat’s Little Theorem in the text. Be sure to read it Be sure to read it You are also prepared to read the rest of chapter 3 at your own pace.

Three-pass protocol How can Alice get a secret message to Bob without an established key? Can do it with locks. First 2 volunteers get to do the live demo

Three-pass protocol Situation: Alice wants to get a short message to Bob, but they don’t have an established key to transmit it. Can do with locks:

Three-pass protocol Situation: Alice wants to get a short message to Bob, but they don’t have an established key to transmit it. Can do with locks:

Three-pass protocol Situation: Alice wants to get a short message to Bob, but they don’t have an established key to transmit it. Can do with locks:

Three-pass protocol Situation: Alice wants to get a short message to Bob, but they don’t have an established key to transmit it. Can do with locks: Note: it’s always secured by one of their locks

Now with “Fermat’s locks” K: the secret message p: a public prime number > K The two locks: a: Alice’s random #, gcd(a,p-1)=1 a: Alice’s random #, gcd(a,p-1)=1 b: Bob’s random #, gcd(b,p-1)=1 b: Bob’s random #, gcd(b,p-1)=1 To unlock their locks: a -1 mod (p-1) a -1 mod (p-1) b -1 mod (p-1) b -1 mod (p-1)

Now with “Fermat’s locks” K: the secret message p: a public prime number > K The two locks: a: Alice’s random #, gcd(a,p-1)=1 a: Alice’s random #, gcd(a,p-1)=1 b: Bob’s random #, gcd(b,p-1)=1 b: Bob’s random #, gcd(b,p-1)=1 To unlock their locks: a -1 mod (p-1) a -1 mod (p-1) b -1 mod (p-1) b -1 mod (p-1) Three-pass protocol: Alice computes K a (mod p) and sends to Bob Bob computes (K a ) b (mod p) and sends it back Alice computes ((K a ) b ) inv(a) (mod p) and sends it back Bob computes (((K a ) b ) inv(a) ) inv(b) (mod p) and reads K

Now with “Fermat’s locks” K: the secret message p: a public prime number > K The two locks: a: Alice’s random #, gcd(a,p-1)=1 a: Alice’s random #, gcd(a,p-1)=1 b: Bob’s random #, gcd(b,p-1)=1 b: Bob’s random #, gcd(b,p-1)=1 To unlock their locks: a -1 mod (p-1) a -1 mod (p-1) b -1 mod (p-1) b -1 mod (p-1) Three-pass protocol: Alice computes K a (mod p) and sends to Bob Bob computes (K a ) b (mod p) and sends it back Alice computes ((K a ) b ) inv(a) (mod p) and sends it back Bob computes (((K a ) b ) inv(a) ) inv(b) (mod p) and reads K Toy example: (mod 59) = (mod 59) = (mod 59) = (mod 59) = Why’s it work?

Recall the basic principle When dealing with numbers mod n, we can deal with their exponents mod _____ Only look at once you’ve thought about this… Given integers a and b, Given integers a and b, Since aa -1 =bb -1 =1(mod p-1) Since aa -1 =bb -1 =1(mod p-1) What’s K^(aba -1 b -1 ) (mod p)? What’s K^(aba -1 b -1 ) (mod p)?

Final thought Trappe and Washington say that it’s vulnerable to an “intruder-in-the-middle” attack. Think about this…

Some levity before the exam Thanks to Nathan for the link!

Quiz Closed book and computer Get out note sheet: 1 handwritten sheet of 8.5 x 11 paper, one side only. 1 handwritten sheet of 8.5 x 11 paper, one side only.