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# slides4 - CS 531 Fall 2007 Involutions Involutions are...

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T 1 Involutions CS 531, Fall 2007 Copyright © William C. Cheng Involutions are their own inverses let S be a finite and let f be a bijection from S to S (i.e., p: S S ), then f is an involution if f=f -1 if j is the image of i , then i is the image of j ex: S S f 1 2 3 1 2 3 4 4 5 5

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T 2 Basic Terminology and Concepts CS 531, Fall 2007 Copyright © William C. Cheng A denotes a finite set called the alphabet of definition ex: A={0,1} , the binary alphabet any alphabet can be encoded in terms of the binary alphabet, e.g., each letter of the English alphabet can be assigned a unique binary string of length 5 M denotes a set called the message space M consists of strings of symbols from an alphabet of definition an element of M is called a plaintext message (or plaintext ) C denotes a set called the ciphertext space C consists of strings of symbols from an alphabet of definition (may differ from that for M ) an element of C is called a ciphertext
T 3 Encryption and Decryption Transformations CS 531, Fall 2007 Copyright © William C. Cheng an element of K is called a key K denotes a set called the key space each element e K uniquely determines a bijection from M to C , denoted by E e E e is called an encryption function (or an encryption transformation ) E e must be a bijection if the process is to be reversed for each d K , D d denotes a bijection from C to M , (i.e., D d : C M ) D d is called a decryption function (or an decryption transformation ) encrypting message m: c=E e (m) decrypting ciphertext c: D d (c) and a unique plaintext message recovered for each distinct ciphertext

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T 4 Encryption and Decryption (Cont...) CS 531, Fall 2007 Copyright © William C. Cheng a set {E e : e K} of encryption transformations An encryption scheme consists of a corresponding set {D d : d K} of decryption transformations for each e K , there is a unique key d K such that D d =E -1 e D d (E e (m)) = m for all m M an encryption scheme is sometimes referred to as a cipher the keys e and d above are referred to as a key pair sometimes denoted by (e,d) e and d could be the same
T 5 Encryption and Decryption (Cont...) CS 531, Fall 2007 Copyright © William C. Cheng a message space M To construct an encryption scheme requires one to select a ciphertext space C a key space K a set of encryption transformations {E e : e K} a corresponding set of decryption transformations {D d : d K}

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structure of lock is available to anyone Ex: two parties, Alice and Bob Alice and Bob choose or secretly exchange a key pair (e,d)
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