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Some concept of Cryptography are Block Ciphers, Classical Cryptography, Computational, Cryptanalysis, Digital Signatures, Knowledge Proofs, Number Theory, One Way Functions, Perfect Secrecy, Perfect Secrecy. Main points of this lecture are: Classical Cryptography, Classical Cryptosystem, Possible Plaintexts, Possible Ciphertexts, Encryption Rules, Decryption Rules, Plaintext Blocks, Ciphertext String, Shift Cipher Disk, Prime Factor
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(P , C , K , E , D ) satisfying
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Letters are identified with numbers: A=0, B=1, .... , Z= Z 26 P = C = K = Z 26 eK ( x ) = ( x + K ) mod 26 dK ( y ) = ( y − K ) mod 26
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Taking any group G (even non-commutative) can generalize shift cipher by using G ’s “addition” rule:
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DEF: The ring is the set with addition and multiplication taken mod m. LEMMA: x is invertible in iff gcd(x, m) = 1. COR: is a field iff m is prime. DEF:. In other words, is the number of elements in which are relatively prime to m.
!( m ) = |Z ∗ m | Z m !( m ) Z m { 0 , 1 ,... , m − 1 } Z m Z m
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Can define a general cipher over any ring, even non-commutative rings (where multiplication non-commutative, but addition commutative).
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Shift and Affine cipher have limited key spaces. Better to use all possible permutations of :
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THM: Let A be a square matrix whose entries are in. A is invertible modulo m iff det( A ) is invertible in. Furthermore the inverse of A mod- m is given by where is the inverse of A over.
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**>> A = [ [21 10] ; [18 5] ];
d = det(A) d = - [g, e, t] = gcd(d,26) g = 1 e = 9 t = 26 B = A^- B = -0.0667 0. 0.2400 -0. B = Bde B = 45.0000 -90. -162.0000 189. B = round(B) B = 45 - -162 189 B = mod(B,26) B = 19 14 20 7 C = AB C = 599 364 442 287 mod(C,26) ans = 1 0 0 1*
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