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s-cse_s

# s-cse_s - CLASSICAL ENCRYPTION 1 50 Syntax A symmetric...

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CLASSICAL ENCRYPTION 1/50

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Syntax A symmetric encryption scheme SE = ( K , E , D ) consists of three algorithms: (Adversary) 2/50
Correct decryption requirement For all K , M we have D K ( E K ( M )) = M 3/50

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Terminology recall Alphabets: Σ 1 = { A , B , C , . . . , Z } Σ 2 = { A , B , C , . . . , Z }∪{⊔ , . , ? , . . . } Σ 3 = { 0 , 1 } Strings: Over Σ 1 : HELLO , BZYK , ... Over Σ 2 : HOW ARE YOU ? Over Σ 3 : 01101 Denote by Σ * the set of all strings over alphabet Σ: { A , B , . . . , Z } * { 0 , 1 } * 4/50
Length and size If s is a string then | s | is the number of symbols in it: | HELLO | = 5 | HOW ARE YOU ? | = 12 | 01101 | = 5 We denote by s [ i ] the i -th symbol of string s : s [3] = L if s = HELLO s [5] = A if s = HOW ARE YOU ? s [2] = 1 if s = 01101 If S is a set then | S | is its size: |{ A , B , . . . , Z }| = 26 |{ 0 , 1 } 8 | = 2 8 5/50

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Functions Then notation π : D R means π is a map (function) with inputs drawn from the set D (the domain) outputs falling in the set R (the range) Example : Define π : { 1 , 4 , 6 }→{ 0 , 1 } by x 1 4 6 π ( x ) 1 1 0 Functions can be specified as above or sometimes by code. Example : The above can also be specified by Alg π ( x ) Return x mod 3 6/50
Permutations A map (function) π : S S is a permutation if it is one-to-one. Equivalently, it has an inverse map π - 1 : S S . Example: S = { A , B , C } A permutation and its inverse: x A B C π ( x ) C A B y A B C π - 1 ( x ) B C A Not a permutation: x A B C π ( x ) C B B 7/50

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Counting permutations There are many different possible permutations π : S S on a given set S . How many? To be specific: How many permutations π : S S are there on the set S = { A , B , C } ? Answer: 3! = 3 2 1 = 6 x π ( x ) A ←− 3 choices: A,B,C B ←− 2 choices: not π ( A ) C ←− 1 choice: not π ( A ), π ( B ) In general there are | S | ! permutations π : S S . We let Perm( S ) denote the set of all these permutations. 8/50
Substitution ciphers Alphabet Σ Key is a permutation π : Σ Σ definining the encoding rule Plaintext M Σ * is a string over Σ Encryption of M = M [1] ··· M [ n ] is C = π ( M [1]) ··· π ( M [ n ]) Decryption of C = C [1] ··· C [ n ] is M = π - 1 ( C [1]) ··· π - 1 ( C [ n ]) 9/50

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Substitution ciphers A substitution cipher over alphabet Σ is a symmetric encryption scheme SE = ( K , E , D ) in which the key output by K is a permutation π : Σ Σ, and Algorithm E π ( M ) For i = 1 , . . . , | M | do C [ i ] π ( M [ i ]) Return C Algorithm D π ( C ) For i = 1 , . . . , | C | do M [ i ] π - 1 ( C [ i ]) Return M 10/50
Setup for Examples Σ = { A , B , . . . , Z }∪{⊔ , . , ? , ! , . . . } Plaintexts are members of Σ * , which means any English text (sequence of sentences) is a plaintext. For simplicity we only consider permutations that are punctuation respecting: π ( ) = , π ( . ) = . , π (?) =? , . . . so punctuation is left unchanged by encryption. 11/50

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Example σ A B C D E F G H I J K L M π ( σ ) B U P W I Z L A F N S G K σ N O P Q R S T U V W X Y Z π ( σ ) D H T J X C M Y O V E Q R Then encryption of plaintext M = HI THERE is C = π ( H ) π ( I ) π ( ) π ( T ) π ( H ) π ( E ) π ( R ) π ( E ) = AF MAIXI τ A B C D E F G H I J K L M π - 1 ( τ ) H A S N X I L O E Q M G T τ N O P Q R S T U V W X Y Z π - 1 ( τ ) J V C Y Z K P B W D R U F Decryption of ciphertext C = AF MAIXI is π - 1 ( A ) π - 1 ( F ) π - 1 ( ) π - 1 ( M ) π - 1 ( A ) π - 1 ( I ) π - 1 ( X ) π - 1 ( I ) = HI THERE 12/50
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s-cse_s - CLASSICAL ENCRYPTION 1 50 Syntax A symmetric...

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