Crypto2

Crypto2

• 1.
  Cryptography: Basics(2)
• 2.
  Outline Classical CryptographyCaesar cipher Vigenère cipher DES Public Key Cryptography Diffie-Hellman RSA Cryptographic Checksums HMAC
• 3.
  Public Key CryptographyTwo keys Private key known only to individual Public key available to anyone Public key, private key inverses Idea Confidentiality: encipher using public key, decipher using private key Integrity/authentication: encipher using private key, decipher using public one
• 4.
  Requirements It mustbe computationally easy to encipher or decipher a message given the appropriate key It must be computationally infeasible to derive the private key from the public key It must be computationally infeasible to determine the private key from a chosen plaintext attack
• 5.
  Diffie-Hellman First publickey cryptosystem proposed Compute a common, shared key Called a symmetric key exchange protocol Based on discrete logarithm problem Given integers n and g and prime number p , compute k such that n = g k mod p Solutions known for small p Solutions computationally infeasible as p grows large
• 6.
  Algorithm Constants Knownto all participants: a prime p , an integer g ≠ 0, 1, p –1 Anne chooses private key kAnne , computes public key KAnne = g kAnne mod p // kAnne : Anne’s private key; KAnne : Anne’s public key To communicate with Bob, Anne computes Kshared = KBob kAnne mod p To communicate with Anne, Bob computes Kshared = KAnne kBob mod p It can be shown these keys are equal
• 7.
  Example Assume p = 53 and g = 17 Alice chooses kAlice = 5 Then KAlice = 17 5 mod 53 = 40 Bob chooses kBob = 7 Then KBob = 17 7 mod 53 = 6 Shared key: KBob kAlice mod p = 6 5 mod 53 = 38 KAlice kBob mod p = 40 7 mod 53 = 38 Question: What keys are exchanged between Alice and Bob?
• 8.
  RSA Exponentiation cipherRelies on the difficulty of determining ‘the number of numbers relatively prime to a large integer n ’ (i.e., totient(n) ) An integer i is relatively prime to n when i and n have no common factors.
• 9.
  Background Totient function  (n) Number of positive integers less than n and relatively prime to n Relatively prime means with no factors in common with n Example:  (10) = 4 1, 3, 7, 9 are relatively prime to 10 Example:  (21) = 12 1, 2, 4, 5, 8, 10, 11, 13, 16, 17, 19, 20 are relatively prime to 21
• 10.
  Algorithm Choose twolarge prime numbers p, q Let n = pq ; then  ( n ) = ( p –1)( q –1) Choose e < n such that e is relatively prime to  ( n ). Compute d such that ed mod  ( n ) = 1 Public key: ( e , n ); private key: d Encipher: c = m e mod n Decipher: m = c d mod n
• 11.
  Example: Confidentiality Take p = 7, q = 11, so n = 77 and  ( n ) = 60 Alice chooses e = 17, making d = 53 Bob wants to send Alice secret message HELLO (07 04 11 11 14) 07 17 mod 77 = 28 04 17 mod 77 = 16 11 17 mod 77 = 44 11 17 mod 77 = 44 14 17 mod 77 = 42 Bob sends 28 16 44 44 42
• 12.
  Example Alice receives28 16 44 44 42 Alice uses private key, d = 53, to decrypt message: 28 53 mod 77 = 07 16 53 mod 77 = 04 44 53 mod 77 = 11 44 53 mod 77 = 11 42 53 mod 77 = 14 Alice translates message to letters to read HELLO No one else could read it, as only Alice knows her private key and that is needed for decryption
• 13.
  Example: Integrity/Authentication Take p = 7, q = 11, so n = 77 and  ( n ) = 60 Alice chooses e = 17, making d = 53 Alice wants to send Bob message HELLO (07 04 11 11 14) so Bob knows it is what Alice sent (no changes in transit, and authenticated) 07 53 mod 77 = 35 04 53 mod 77 = 09 11 53 mod 77 = 44 11 53 mod 77 = 44 14 53 mod 77 = 49 Alice sends 35 09 44 44 49
• 14.
  Example Bob receives35 09 44 44 49 Bob uses Alice’s public key, e = 17, n = 77, to decrypt message: 35 17 mod 77 = 07 09 17 mod 77 = 04 44 17 mod 77 = 11 44 17 mod 77 = 11 49 17 mod 77 = 14 Bob translates message to letters to read HELLO Alice sent it as only she knows her private key, so no one else could have enciphered it If (enciphered) message’s blocks (letters) altered in transit, would not decrypt properly
• 15.
  Example: Both Alicewants to send Bob message HELLO both enciphered and authenticated (integrity-checked) Alice’s keys: public (17, 77); private: 53 Bob’s keys: public: (37, 77); private: 13 Alice enciphers HELLO (07 04 11 11 14): (07 53 mod 77) 37 mod 77 = 07 (04 53 mod 77) 37 mod 77 = 37 (11 53 mod 77) 37 mod 77 = 44 (11 53 mod 77) 37 mod 77 = 44 (14 53 mod 77) 37 mod 77 = 14 Alice sends 07 37 44 44 14
• 16.
  Security Services ConfidentialityOnly the owner of the private key knows it, so text enciphered with public key cannot be read by anyone except the owner of the private key Authentication Only the owner of the private key knows it, so text enciphered with private key must have been generated by the owner
• 17.
  More Security ServicesIntegrity Enciphered letters cannot be changed undetectably without knowing private key Non-Repudiation Message enciphered with private key came from someone who knew it
• 18.
  Warnings Encipher messagein blocks considerably larger than the examples here If 1 character per block, RSA can be broken using statistical attacks (just like classical cryptosystems) Attacker cannot alter letters, but can rearrange them and alter message meaning Example: reverse enciphered message of text ON to get NO
• 19.
  Cryptographic Checksums Mathematicalfunction to generate a set of k bits from a set of n bits (where k ≤ n ). k is smaller then n except in unusual circumstances Example: ASCII parity bit ASCII has 7 bits; 8th bit is “parity” Even parity: even number of 1 bits Odd parity: odd number of 1 bits
• 20.
  Example Use Bobreceives “10111101” as bits. Sender is using even parity; 6 1 bits, so character was received correctly Note: could be garbled, but 2 bits would need to have been changed to preserve parity Sender is using odd parity; even number of 1 bits, so character was not received correctly
• 21.
  Definition Cryptographic checksumfunction h : A  B : For any x IN A, h(x) is easy to compute For any y IN B, it is computationally infeasible to find x IN A such that h(x) = y It is computationally infeasible to find x, x´ IN A such that x ≠ x´ and h(x) = h(x´) Alternate form (Stronger): Given any x IN A, it is computationally infeasible to find a different x´ IN A such that h(x) = h(x´).
• 22.
  Collisions If x≠ x´ and h(x) = h(x´), x and x´ are a collision Pigeonhole principle: if there are n containers for n +1 objects, then at least one container will have 2 objects in it. Application: suppose n = 5 and k = 3. Then there are 32 elements of A and 8 elements of B, so at least one element of B has at least 4 corresponding elements of A
• 23.
  Keys Keyed cryptographicchecksum: requires cryptographic key DES in chaining mode: encipher message, use last n bits. Requires a key to encipher, so it is a keyed cryptographic checksum. Keyless cryptographic checksum: requires no cryptographic key MD5 and SHA-1 are best known; others include MD4, HAVAL, and Snefru
• 24.
  HMAC Make keyedcryptographic checksums from keyless cryptographic checksums h keyless cryptographic checksum function that takes data in blocks of b bytes and outputs blocks of l bytes. k´ is cryptographic key of length b bytes If short, pad with 0 bytes; if long, hash to length b ipad is 00110110 repeated b times opad is 01011100 repeated b times HMAC- h ( k , m ) = h ( k ´  opad || h ( k ´  ipad || m ))  exclusive or, || concatenation Correction: H(K XOR opad, H(K XOR ipad, text))
• 25.
  Key Points Twomain types of cryptosystems: classical and public key Classical cryptosystems encipher and decipher using the same key Or one key is easily derived from the other Public key cryptosystems encipher and decipher using different keys Computationally infeasible to derive one from the other Cryptographic checksums provide a check on integrity