Related Ideas: DHM Key Mechanics
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1 Related Ideas: DHM Key Mechanics Example (DHM Key Mechanics) Two parties, Alice and Bob, calculate a key that a third person Carl will never know, even if Carl intercepts all communication between Alice and Bob. First o, Alice & Bob agree on numbers n and M (not secret). What Alice does: 1 Choose a secret value a. 2 Compute = M a (mod n). 3 Sends to Bob. 4 Receives from Bob. 5 Key is K = a (mod n). What Bob does: 1 Choose a secret value b. 2 Compute = M b (mod n). 3 Sends to Alice. 4 Receives from Alice. 5 Key is K = b (mod n).
2 DHM Practice 3, Part 1 Example (DHM Practice 3) Bob and Alice are trying to send a key over unsecured communication lines. They agree to use M = 10 and n = 41. Compute M 2 (mod n) = 10 2 (mod 41). Compute M 5 (mod n) = 10 5 (mod 41). Can you use your answers above to easily calculate = M 17 (mod 41)?
3 DHM Practice 3, Part 2 Example (DHM Practice 3) Bob and Alice are trying to send a key over unsecured communication lines. They agree to use M =6andn = 23. Alice receives = 18(mod 41) from Bob. Her secret value is a = 13. Calculate 4 (mod 41) = 18 4 (mod 41). Use your answer above to quickly calculate 12 (mod 41) = (18 4 (mod 41)) 3. What is the key for this exchange?
4 Master Spy 1 Example (Master Spy 1) You ve been promoted to the highest rank in the Spy Agency! It s now time to learn about a modern and sophisticated code. See if you can handle the following questions: How long does it take you to factor 2173 as a product of two primes 2173 = p q? How long does it take you to multiply the numbers 41 and 53? If n is a big number, is it easy to factor? If p and q are big numbers, is it easy to multiply them?
5 Encryption: RSA Cipher Definition (Encryption: RSA Cipher) The RSA Cipher is a public key cipher publicly discovered in the 1970s. The RSA cipher uses a form of multiplication for encryption and is secure because factoring large numbers is (currently) very di cult to do. RSA stands for Rivest, Shamir, and Adleman, the people responsible for first publicizing the RSA cipher. The British and US governments may have known about RSA prior to the 1970s, but did not announce their discovery. Even though this is the basis for most modern cryptography, there is current speculation that the US government (specifically the NSA) has the ability to break this code.
6 Related Idea: RSA Encryption Step 1 Example (RSA Encryption: Step 1) Here is how Alice and Bob can do to share a secret from Carl: What Alice Does 1. Alice chooses two (large) prime numbers p and q, which she keeps secret. 2. She then multiplies to find n = p q. This can be done quickly because multiplication is easy. 3. Alice also calculates a value m =(p 1)(q 1). 4. She selects a value e(mod m) thatisaunit. So any choice of e with gcd(e, m) =1willwork. Thevaluee is called the encryption exponent.
7 Related Idea: RSA Encryption Step 2 Example (RSA Encryption: Step 2) 5. Next, Alice tells Bob (and anyone else) the values for n and e. The fact that Alice can publicly state n and e is what makes RSA a public key cipher. What Bob Does To Send Alice a Message 6. Bob converts letters (or blocks of letters) into numbers. We can do this is the standard way, but in real-life this gets done by a computer. 7. For each letter,he uses the rule e (mod n) = to find the ciphertext. He sends this ciphertext to Alice.
8 Master Spy 2 Example (Master Spy 2) You ve been promoted to the highest rank in the Spy Agency! It s now time to learn about a modern and sophisticated code. See if you can handle the following questions: If p = 71 and q = 59, find n = p q. If p = 71 and q = 59, find m =(p 1) (q 1). If p = 101 and q = 103, find n = p q. If p = 101 and q = 103, find m =(p 1) (q 1).
9 Master Spy 3 Example (Master Spy 3) You ve been promoted to the highest rank in the Spy Agency! It s now time to learn about a modern and sophisticated code. See if you can handle the following questions: If p = 41 and q = 53, find n and m. If p = 101 and q = 107, find n and m. If p = 521 and q = 641, find n and m.
10 Master Spy 4 Example (Master Spy 4) You ve been promoted to the highest rank in the Spy Agency! It s now time to learn about a modern and sophisticated code. See if you can handle the following questions: If p = 17 and q = 19, find n and m. If p =7andm = 132, find q and n. If p =3andq = 5, find all units (mod m). If p =5andq = 11, what is 27 3(mod m)?
11 Master Spy 5 Example (Master Spy 5) A fellow agent wants you to send her a message. She broadcasts the numbers n = 33 and e = 3, expecting that these will be intercepted. Use this RSA cipher to encrypt the letter H as a number. Use this RSA cipher to encrypt the letter I as a number. Use this RSA cipher to encrypt the letter J as a number. The letters H, I and J are consecutive. Does RSA encrypt these letters as consecutive numbers?
12 Master Spy 6 Example (Master Spy 6) You want a fellow agent to send you a secret message. You decide on the numbers n = 77 and e = 7 and publish these to an open webpage. What number will the letter B be encrypted as? What number will the letter C be encrypted as? Encrypt the number 0203? Is this connected to the answers above in any way?
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