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Algorithmentheorie 03 – Randomized Algorithms (Public Key Cryptosystems)

Algorithmentheorie 03 – Randomized Algorithms (Public Key Cryptosystems). Prof. Dr. Th. Ottmann. Randomized algorithms. Classes of randomized algorithms Randomized Quicksort Randomized algorithm for Closest Pair Randomized primality test Cryptography. Classes of randomized algorithms.

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Algorithmentheorie 03 – Randomized Algorithms (Public Key Cryptosystems)

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  1. Algorithmentheorie03 – Randomized Algorithms(Public Key Cryptosystems) Prof. Dr. Th. Ottmann WS 2006-07

  2. Randomized algorithms • Classes of randomized algorithms • Randomized Quicksort • Randomized algorithm for Closest Pair • Randomized primality test • Cryptography WS 2006-07

  3. Classes of randomized algorithms • Las Vegas algorithms always correct; expected running time (“probably fast”) Example: randomized Quicksort • Monte Carlo algorithms (mostly correct): probably correct; guaranteed running time Example: randomized primality test WS 2006-07

  4. Application: cryptosystems Traditional encryption of messages with secret keys Disadvantages: • The key k has to be exchanged between A and B before the transmission of the message. • For messages between n parties n(n-1)/2 keys are required. Advantage: Encryption and decryption can be computed very efficiently. WS 2006-07

  5. Duties of security providers Guarantee… • confidential transmission • integrity of data • authenticity of the sender • reliable transmission WS 2006-07

  6. Public-key cryptosystems Diffie and Hellman (1976) Idea:Each participant A has two keys: • a public key PA accessible to every other participant • a private (or: secret) key SA only known to A. WS 2006-07

  7. Public-key cryptosystems D = set of all legal messages, e.g. the set of all bit strings of finite length Three conditions: 1. PA and SA can be computed efficiently 2. SA(PA(M)) = M and PA(SA(M)) = M (PA, SA are inverse functions) 3. SAcannot be computed from PA (without unreasonable effort) 1-1 WS 2006-07

  8. Encryption in a public-key system Asends a message M to B. Dear Bob,I just checkedthe new... #*k- + ;}?,@-) #$<9{o7::-&$3(-##!]?8... Dear Bob,I just checkedthe new... WS 2006-07

  9. Encryption in a public-key system 1. Aaccesses B’s public key PB(from a public directory or directly from B). 2. Acomputes the encrypted message C= PB(M) and sends C toB. 3. After Bhas received message C, Bdecrypts the message with his own private key SB: M = SB(C) WS 2006-07

  10. Generating a digital signature Asends a digitally signed message M´to B: 1. Acomputes the digital signature σ for M´ with her own private key: σ =SA(M´) 2. Asends the pair (M´,σ) to B. 3. After receiving (M´,σ), B verifies the digital signature: PA(σ) = M´ σ can by verified by anybody via the public PA. WS 2006-07

  11. RSA cryptosystems R. Rivest, A. Shamir, L. Adleman Generating the public and private keys: 1. Randomly select two primes pand qof similar size, each with l+1 bits (l ≥ 500). 2. Let n = p·q 3. Let ebe an integer that does not divide (p - 1)·(q - 1). 4. Calculate d = e-1 mod (p - 1)(q - 1) i.e.: d · e ≡ 1mod (p - 1)(q - 1) WS 2006-07

  12. RSA cryptosystems 5. Publish P = (e, n)as public key 6. Keep S = (d, n)as private key Divide message (represented in binary) in blocks of size 2·l. Interpret each block M as a binary number: 0 ≤ M < 22·l P(M) = Memod n S(C) = Cdmod n WS 2006-07

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