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Unformatted text preview: Great Theoretical Ideas In Computer Science COMPSCI 102 Fall 2010 Lecture 16 October 27, 2010 Duke University Modular Arithmetic and the RSA Cryptosystem p1 ≡ 1 Starring Rivest Shamir Adleman Euler Fermat The RSA Cryptosystem R ivest, S hamir, and A delman (1978) RSA is one of the most used cryptographic protocols on the net. Your browser uses it to establish a secure session with a site. Pick secret, random large primes: p,q “Publish”: n = p*q φ (n) = φ (p) φ (q) = (p1)*(q1) Pick random e ∈ Z * φ (n) “Publish”: e Compute d = inverse of e in Z * φ (n) Hence, e*d = 1 [ mod φ (n) ] “Private Key”: d Mumbo jumbo… More Mumbo jumbo… n,e is my public key. Use it to send me a message. p,q random primes, e random ∈ Z * φ (n) n = p*q e*d = 1 [ mod φ (n) ] n, e p,q prime, e random ∈ Z * φ (n) n = p*q e*d = 1 [ mod φ (n) ] message m m e [mod n] (m e ) d ≡ n m But how does it all work? What is φ (n)? What is Z φ (n) * ? … Why do all the steps work? To understand this, we need a little number theory... MAX(a,b) + MIN(a,b) = a+b nm means that m is an integer multiple of n. We say that “ n divides m ”. Greatest Common Divisor: GCD(x,y) = greatest k ≥ 1 s.t. kx and ky. Least Common Multiple: LCM(x,y) = smallest k ≥ 1 s.t. xk and yk. Fact: GCD(x,y) × LCM(x,y) = x × y GCD(x,y) × LCM(x,y) = xy MAX(a,b) + MIN(a,b) = a+b (a mod n) means the remainder when a is divided by n. If a = dn + r with 0 ≤ r < n Then r = (a mod n) and d = (a div n) Defn: Modular equivalence of integers a and b a ≡ b [mod n] (a mod n) = (b mod n) ⇔ n(ab) Written as a ≡ n b, and spoken “a and b are equivalent modulo n” 31 ≡ 81 [mod 2] 31 ≡ 2 81 ≡ n is an equivalence relation In other words, Reflexive: a ≡ n a Symmetric: (a ≡ n b) ⇒ (b ≡ n a) Transitive: (a ≡ n b and b ≡ n c) ⇒ (a ≡ n c) a ≡ n b ⇔ n(ab) “a and b are equivalent modulo n” ≡ n induces a natural partition of the integers into n classes. a and b are said to be in the same “residue class” or “congruence class” exactly when a ≡ n b. a ≡ n b ⇔ n(ab) “a and b are equivalent modulo n” Define Residue class [i] = the set of all integers that are congruent to i modulo n . Residue Classes Mod 3: [0] = { …, 6, 3, 0, 3, 6, ..} [1] = { …, 5, 2, 1, 4, 7, ..} [2] = { …, 4, 1, 2, 5, 8, ..} [6] = { …, 6, 3, 0, 3, 6, ..} [7] = { …, 5, 2, 1, 4, 7, ..} [1] = { …, 4, 1, 2, 5, 8, ..} Fact : equivalence mod n implies equivalence...
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 Fall '09

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