T H E U N I V E R S I T Y O F B R I T I S H C O L U M B I A. Introduction to Cryptography
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1 T H E U N I V E R S I T Y O F B R I T I S H C O L U M B I A Introduction to Cryptography 1
2 Module Outline Historical background Classic ciphers One-time pad The Random Oracle model Random functions: Hash functions Random generators: stream ciphers Random Permutations: block ciphers Public key encryption and trapdoor one-way permutations Digital signatures 2
3 Crypto Cryptology The art and science of making and breaking secret codes Cryptography making secret codes Crypto all of the above (and more) Cryptanalysis breaking secret codes 3
4 How to Speak Crypto A cipher or cryptosystem is used to encrypt the plaintext The result of encryption is ciphertext We decrypt ciphertext to recover plaintext A key is used to configure a cryptosystem A symmetric key cryptosystem uses the same key to encrypt as to decrypt A public key cryptosystem uses a public key to encrypt and a private key to decrypt 4
5 Crypto as Black Box key key plaintext encrypt decrypt plaintext ciphertext A generic use of crypto 5
6 basic assumptions in crypto assumptions 1. The system is completely known to the attacker 2. Only the key is secret Also known as Kerckhoffs Principle Crypto algorithms are not secret 6
7 Kerckhoff s Principle The security of a cryptosystem must not depend on keeping secret the crypto-algorithm. The security depends only on keeping secret the key Auguste Kerckhoff von Nieuwenhof Dutch linguist
8 basic assumptions in crypto assumptions 1. The system is completely known to the attacker 2. Only the key is secret Also known as Kerckhoffs Principle Crypto algorithms are not secret Why do we make this assumption? Experience has shown that secret algorithms are weak when exposed Secret algorithms never remain secret Better to find weaknesses beforehand 8
9 Historical Background To read: Anderson s book 9
10 two types of ciphers substitution transposition 10
11 Letter Indices in English Alphabet A B C D E F G H I J K L M N O P Q R S T U V W X Y z
12 Caesar Cipher Plaintext is HELLO WORLD Change each letter to the third letter following it (X goes to A, Y to B, Z to C) Key is 3, usually written as letter D C = P + K mod 26 Ciphertext: KHOOR ZRUOG Plain HELLOWORLD Key DDDDDDDDDD Cipher KHOORZRUOG 12
13 a simple attack how to attack Caesar Cipher? exhaustive/brute-force (key) search Trudy hours months how to increase key space for substitution cipher? 13
14 Monoalphabetic Substitution Invented by Arabs in 8th or 9th centuries A B C D E F G H I J K L M N.. Z F T W S G M P A Z C L V O D.. B Plain HELLOWORLD Key Cipher Cipher AGVVYEYZVS 14
15 Frequency Analysis of English Letters A B C D E F G H I J K L M N O P Q R S T U V W X Y Z 15
16 Polyalphabetic Vigenère Cipher proposed by Blaise de Vigenere from the court of Henry III of France in the sixteenth century Like Cæsar cipher, but use a phrase Example Message: TO BE OR NOT TO BE THAT IS THE QUESTION Key: RELATIONS Encipher using Cæsar cipher for each letter: Plain TO BE OR NOT TO BE TH AT I S THE QUESTION Key RE LA T I ONS RE LA T I ON SR ELA T I ONSREL Cipher KS ME HZ BBL KS ME MPOG AJ XSE J CSF LZSY 16
17 Double Transposition Permute rows and columns Plaintext: attackxatxdawn Ciphertext: xtawxnattxadakc Key: matrix size and permutations (3,5,1,4,2) and (1,3,2) 17
18 Cryptanalysis: Terminology Cryptosystem is secure if best know attack is to try all keys Cryptosystem is insecure if any shortcut attack is known By this definition, an insecure system might be harder to break than a secure system! 18
19 One-Time Pad A Vigenère cipher with a random key at least as long as the message Provably unbreakable Why? Plain text D O I T D O N T Key A J I Y A J D Y Cipher text D X Q R D X Q R Warning: keys must be random, or you can attack the cipher by trying to regenerate the key 19
20 Little Bit of History 91 years ago, January 19,
21 Literally, a book filled with codewords Zimmerman Telegram encrypted via codebook Februar fest finanzielle folgender Frieden Friedenschluss Codebook : : Modern block ciphers are codebooks! 21
22 Zimmerman Telegram One of most famous codebook ciphers ever Led to US entry in WWI Ciphertext shown here 22
23 Zimmerman Telegram Decrypted British had recovered partial codebook Able to fill in missing parts 23 23
24 T H E U N I V E R S I T Y O F B R I T I S H C O L U M B I A Random Oracle Model Read Anderson
25 What is Random Oracle Model? Queries Responses source of randomness infinite storage 25
26 Random Function as Random Oracle In: string of any length Queries Responses Out: random string of fixed length Applications: One-way functions Hash functions Message digests Time stamping Properties efficiency -- easy to compute h(x) for any x. one-way -- given any y, it s infeasible to find x, s.t., h(x) = y weak collision resistance -- given x and h(x), it s infeasible to y!= x, s.t. h(y) == h(x) strong collision resistance -- infeasible to find any x!= y, s.t., h(x) == h(y) 26
27 Random Generator (Stream Cipher) as Random Oracle In: short string (key) length of the output Queries Responses Out: long random stream of bits (keystream) Properties Should not reuse Use seed Applications: Communications encryption Storage encryption 27
28 Example: A5 stream cipher for GSM From: Alex Biryukov, Adi Shamir, David Wagner 28 Real Time Cryptanalysis of A5/1 on a PC 28
29 Random Permutation (Block Cipher) as Random Oracle In fixed size short string (plaintext) M, DES bits Key K Queries Responses K 1 K 2 Out same fixed size short string (ciphertext) C 29 Notation C = { M } K M = { C } K Properties Invertible 29
30 Public Key Encryption and Trap-door One-Way Permutation as Random Oracle Public Key Encryption Scheme: Key pair (KR, KR -1 ) generation function from random string R KR KR -1 is infeasible C = {M) KR Queries Responses H(K 1 ) K 1 H(K 2 ) K2 M = {C) KR -1 In: fixed size short string (plaintext) M, Key KR Out: fixed size short string (ciphertext) C 30
31 Digital Signature as Random Oracle Public Key Signature Scheme: Key pair (σr, VR) generation function VR σr is infeasible S = Sig σr (M) {True, False} = Ver VR (S) Queries Responses H(K 1 ) K 1 H(K 2 ) K2 Signing Verifying Input Any string M + σr S + VR Output S = hash(m) cipher block True or False 31
32 Summary Historical background Caesar and Vigenère ciphers One-time pad One-way functions Asymmetric cryptosystems The Random Oracle model Random functions: Hash functions Random generators: stream ciphers Random Permutations: block ciphers Queries Responses Public key encryption and trapdoor one-way permutations Digital signatures 32
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