# chapter1-1 - 1 Cryptography: Theory and Practice 3nd...

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Unformatted text preview: 1 Cryptography: Theory and Practice 3nd edition Douglas R. Stinson Homework:30% Midterm:35% Final-term:35% 2 A cryptosystem is a five-tuple ( P, C, K, E, D ), s. t.: 1. P is a finite set of possible plaintexts 2. C is a finite set of possible ciphertexts 3. K , the keyspace, is the set of possible keys 4. For each k  K , there are- encryption rule e k , e k : P  C ,- decryption rule d k , d k : C  P ,- s.t. d k (e k (x)) = x- e k must be an injective function for unambiguous decryption 2 hours 3 4 Modular arithmetic (1) • Suppose a , b : integers, m : positive integer. – a  b (mod m ) if m divides a – b . • i.e., a mod m = b mod m – It is called that a is congruent to b modulo m , and m is called the modulus. – Using a mod m >0 to denote the remainder when a is divided by m • e.g., 101 mod 7=3, -101 mod 7=4 (not –3) 5 Modular arithmetic (2) • Given any m , define Z m = {0,1,…, m- 1}, equipped with two operation + and  with modulo m , – ( Z m , +) is an abelian group . – ( Z m , +,  ) is a ring . – + is closed, commutative, associative, 0 is additive identity, additive inverse exist –  is closed, commutative, associative, 1 is multiplicative identity, distributive property is satisfied Group and Ring 6 Shift cipher--example • Suppose a plaintext word: cryptography • Change each letter by shifting the letter three position rightward • The cipherword is: FUBSWRJUDSKB Question: if given the above cipherword, how to get original word? Change each letter by shifting the letter three position leftward. This kind of cryptosystem is called ― Caesar Cipher ‖ Convention: plaintext by small letter but ciphertext by CAPITAL LETTER . For encryption and decryption: change letters a — z to number 0 – 25. 7 Shift cipher—formal definition • Let P = C = K, = Z 26 , for 0  K  25, define e K ( x ) = x + K mod 26 and d K ( y ) = y- K mod 26 ( x , y  Z 26 ) 8 Shift cipher -- security Two basic properties for a cryptosystem: 1. Each encryption function e K and each decryption d K should be efficiently computable. 2. An opponent upon seeing a ciphertext string y , should be unable to determine the key K that was used, or the plaintext string x . Question: is shift cipher secure? Of course NOT, since there are only 26 possible keys, it is easy to be broken by exhaustive key search. 9 Shift cipher -- security Example: page 6 from textbook: JBCRCLQRWCRVNBJENBWRWN a stitch in time saves nine On average, a plaintext will be computed after trying 26/2=13 times. 10 Shift cipher—generic form • Let P = C = K, = Z m , for 0  K  m –1, define e K ( x ) = x + K mod m and d K ( y ) = y- K mod m ( x , y  Z m and m is a positive integer) 11 Substitution cipher—formal definition • Let P = C = Z 26 , K, consists of all possible permutations of the 26 symbols 0,1, …, 25 (or a,b,…,z). For each permutation   K, , define e  ( x ) =  ( x ) and d  ( y ) = -1 ( y ) ( -1 is the inverse permutation of  ) 12...
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## This note was uploaded on 12/25/2010 for the course ALL 0204 taught by Professor 79979 during the Spring '10 term at National Chiao Tung University.

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chapter1-1 - 1 Cryptography: Theory and Practice 3nd...

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