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Public Key Cryptography INFSCI 1075: Network Security – Spring 2013 Amir Masoumzadeh

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What we have looked at so far 2 CRYPTOLOGY CRYPTOGRAPHYCRYPTANALYSIS Private Key (Secret Key) Public Key Block CipherStream CipherInteger Factorization Discrete Logarithm

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Outline 3 Problems with secret key schemes Public key cryptography Integer factorization Discrete logarithms How to achieve confidentiality, authentication, or both

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Conventional Encryption Model 4 EncryptDecrypt Key Source Insecure channel AliceBob y xx y = e k (x) : Ciphertext x = d k (y) : Plaintext k k Oscar Secure Channel

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Secret Key Cryptosystems 5 Block ciphers and stream ciphers Use the same secret key on both sides for encryption and decryption Operations for e k and d k are identical A separate key for each communication Alice Bob Carol K bob&Carol K Alice&Bob K Alice&Carol

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Problems with Secret Key Schemes 6 Key distribution and management is a problem If the key is disclosed, communications are compromised How many secret keys do we need? How to provide non-repudiation? What if a receiver forges a message and claims that is sent by a sender! Both have access to the secret key! Authentication, which secret key cryptosystems do not provide

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Problems with Secret Key Schemes (cont.) 7 A secret key algorithm implies every pair of communicating entities share a secret key Total number of keys is O(n 2 ) For n users, we need n(n – 1)/2 pairs of keys It is like having a mailbox for EACH pair of communicating people Alice Bob Carol Dan

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Solution 8 One mailbox for one person Make a SLOT in the mailbox Everyone (including Oscar) can deposit messages in the mailbox Only the owner of the mailbox can recover the messages So now for n users we only need n mailboxes and n keys

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Why Public Key Cryptography? 9 Developed to address two key issues: Key distribution – how to have secure communications in general without having to trust a KDC with your key (Confidentiality) Digital signatures – how to verify a message comes intact from the claimed sender (non-repudiation)

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Public Key Cryptography 10 Pioneered by Whitfield Diffie and Martin Hellman in 1976 Public-key / two-key / asymmetric cryptography involves the use of two keys: Public-key (KU) Is known to everyone, used to encrypt messages and verify signatures (Slot in the mailbox) Private-key (KR) known only to the recipient, used to decrypt messages and sign (create signatures) (Actual key to open the mailbox) Public Key Cryptography is asymmetric because Those who encrypt messages or verify signatures cannot decrypt messages or create signatures

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Public Key Encryption Model 11 EncryptDecrypt Insecure channel Alice Bob y xx y = e ku (x) : Ciphertext x = d kr (y) : Plaintext ku bob kr bob Oscar knows ku bob

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Requirements 12 It is easy to encrypt using the public key KU It is easy to decrypt using the private key KR It is computationally infeasible to determine the private key given the public key It is computationally infeasible to determine the plaintext x given the ciphertext y and the public key KU It should be easy to generate a public key-private key pair Encryption and decryption should be inverse functions d KR (e KU (x)) = x

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What can satisfy these requirements? 13 There is a need for a mathematical function unlike secret key cryptosystems One way functions: Every function value has a unique inverse Calculating y = f (x) is easy Calculating x = f -1 (y) is not feasible Examples: Integer factorization Discrete logarithms

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Integer Factorization 14 Multiplication is easy 7 17 109 151 = Integer factorization is difficult = ? ? ? ? Answer: 47 59 61 181 Used in RSA

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Discrete Logarithm 15 EASY: Modular exponentiation 2 23 mod 109 = ? 2 23 = 77 mod 109 DIFFICULT: Discrete logarithm 2 x mod 109 = 68 : Find x x = log 2 68 mod 109 One way to solve it: Brute Force Answer: x = 15 Used in Diffie-Hellman Key Exchange, ElGamal Encryption Scheme, and Elliptic Curves

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Trapdoor One-Way Functions 16 A special kind of one-way function that is hard to invert unless some secret information, called the trapdoor, is known Every function value has a unique inverse There are two related keys k 1 and k 2 Calculating y = f (k 1, x) is easy Calculating x = f -1 (k 2, y) is easy if k 2 is known. It is infeasible if k 2 is not known and only k 1 is known Finding k 2 given k 1 is very hard

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Providing Confidentiality 17 plaintext message, m ciphertext encryption algorithm decryption algorithm Bob’s public key plaintext message e KU (m) KU B Bob’s private key KR B m = d KR ( e KU (m) ) B BB

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Providing Authentication 18 plaintext message, m ciphertext encryption algorithm decryption algorithm plaintext message Alice’s public key KU A Alice’s private key KR A Bob’s public key KU B Bob’s private key KR B m = d KR ( e KU (m) ) BB e KR (m) B

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Providing Authentication & Confidentiality 19 plaintext message, m encryption algorithm encryption algorithm decryption algorithm C decryption algorithm plaintext message C ’C e KR (m) A e KU ( e KR (m) ) BA d KR ( e KU ( e KR (m) ) ) BA B d KU ( e KR (m) ) AA

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Remarks 20 Single most major advance in cryptography Much slower than private key cryptosystems Used primarily for signatures and key exchange rather than bulk data encryption Vulnerable to brute force attacks Vulnerable to mathematical analysis Note that KU and KR are related Key sizes are much larger than those in secret key algorithms Probable message attack KU is known If the number of messages is small, Oscar can encrypt all possible messages to break the system

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Public Key Algorithms and Security 21 Three different popular algorithms RSA (integer factorization) ElGamal (discrete logarithms on prime number fields) Menezes-Vanstone (discrete logarithms on elliptic curves) Keys sizes for security 1024 bits for RSA and ElGamal 160 bits for Menezes-Vanstone 80 bits for block ciphers

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