Cryptography Part 1: Classical Ciphers

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CryptographyPart 1 Classical Ciphers

• Jerzy Wojdylo
• May 4, 2001

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Overview

• Classical Cryptography
• Simple Cryptosystems
• Cryptoanalysis of Simple Cryptosystems
• Shannons Theory of Secrecy
• Modern Encryption Systems
• DES, Rijndel
• RSA
• Signature Schemes

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Cryptosystem

• A cryptosystem is a five-tuple (P,C,K,E,D),
  where the following are satisfied
• P is a finite set of possible plaintexts
• C is a finite set of possible ciphertexts
• K, the keyspace, is a finite set of possible
  keys
• ?K?K, ?eK?E (encryption rule), ?dK?D (decryption
  rule). Each eK P?C and dK C?P are functions
  such that ?x?P, dK(eK(x)) x.

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Notation

• English alphabet
• Lower case a, b, c,, z for plaintext
• Upper case A, B, C,, Z for ciphertext
• For encryption and decryption algorithms, we will
  substitute letters a, b, c,, z with numbers 0,
  1, 2,, 25.

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Classical Cryptography

• Monoalphabetic CiphersOnce a key is chosen, each
  alphabetic character of a plaintext is mapped
  onto a unique alphabetic character of a
  ciphertext.
• The Shift Cipher (Caesar Cipher)
• The Substitution Cipher
• The Affine Cipher

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Classical Cryptography

• Polyalphabetic CiphersEach alphabetic character
  of a plaintext can be mapped onto m alphabetic
  characters of a ciphertext. Usually m is related
  to the encryption key.
• The Vigenère Cipher
• The Hill Cipher
• The Permutation Cipher

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The Shift (Caesar) Cipher

• Let P C K Z26.
• ?x?P, ?y?C, ?K?K, define
• eK(x) x K (mod 26)
• and
• dK(y) y - K (mod 26).
• Example on www.

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The Substitution Cipher

• Let P C Z26, let K S26
• ?x?P, ?y?C, ???K, define
• e?(x) ?(x)
• and
• d?(x) ?-1(x).
• Example on www.

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The Affine Cipher

• Let P C Z26, let
• K (a, b) ? Z26 ? Z26 gcd(a, 26) 1.
• ?x?P, ?y?C, ?K ?K, define
• eK(x) ax b (mod 26)
• and
• dK(y) a-1(y b) (mod 26).
• Example on www.

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The Vigenère Cipher

• Let m ? Z, let P C K (Z26)m. For a key K
  (k1, k2, ,, km),
• we define
• eK (x1, x2, ,, xm) (x1 k1, x2 k2,, xm km)
• and
• dK (x1, x2, ,, xm) (x1 k1, x1 k1,, xm
  km)
• where all operations are modulo 26.
• This is an example (www) of a block cipher.

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The Hill Cipher

• Let m ? Z, let P C (Z26)m, let
• K m?m invertible matrices over Z26.
• For a key K, we define
• eK(x) Kx (mod 26)
• and
• dK(y) K-1y (mod 26).
• Example MATLAB.

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The Permutation Cipher

• Let m ? Z, let P C (Z26)m, let K Sm.
• For a key (i.e. a permutation) p we define
• ep (x1, x2, ,, xm) (xp (1), xp (2),, xp (m))
• and
• dp (y1, y2, ,, ym)(yp-1(1), yp -1 (2),,
  yp-1(m))
• where p-1 is the inverse permutation to p.
• (The Hill Cipher, where K a permutation matrix.)

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Cryptoanalysis

• Kerchkhoffs Principle cryptosystem (the
  algorithm) is NOT secret, the key is secret.
• Common attacks to obtain the key
• Ciphertext-only
• Known plaintext
• Chosen plaintext
• Chosen ciphertext

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Attack on a Shift Cipher

• Ciphertext-only
• Exhaustive search
• 26 cases
• Very insecure cipher

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Cryptoanalysis of a Monoalphabetic Cipher

• Ciphertext-only attack
• Letter frequencies the English language

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Attack on a Substitution Cipher

• Insecure cipher, even though the number of
  possible keys is 26! 40329146112660563558400000
  0(approximately 4.03291026)
• Letter frequencies calculator
• www

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Attack on the Vigenère Cipher

• Kasiski test (m, length of the key)
• Fredrich Wilhelm Kasiski (1863)
• Charles Babbage (1854, result remained secret)
• Two identical segments of plaintext will be
  encrypted to the same ciphertext if their
  occurrence in the plaintext is x position apart,
  where x is a multiple of m.

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Attack on the Vigenère Cipher

• CHREEVOAHMAERATBIAXXWTNXBEEOPHBSBQMQEQERBWRVXUOAK
  XAOSXXWEAHBWGJMMQMNKGRFVGXWTRZXWIAKLXFPSKAUTEMNDCM
  GTSXMXBTUIADNGMGPSRELXNJELXVRVPRTULHDNQWTWDTYGBPHX
  TFALJHASVBFXNGLLCHRZBWELEKMSJIKNBHWRJGNMGJSGLXFEYP
  HAGNRBIEQJTAMRVLCRREMNDGLXRRIMGNSNRWCHRQHAEYEVTAQE
  BBIPEEWEVKAKOEWADREMXMTBHHCHRTKDNVRZCHRCLQOHPWQAII
  WXNRMGWOIIFKEE

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Attack on the Vigenère Cipher

• Positions of CHR 1, 166, 236, 276, 286.
• Differences of positions 166 1 165 236
  1 235276 1 235 286 1 285
• The gcd of these differences is 5, so the key is
  most likely of length m 5.

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Attack on the Vigenère Cipher

• Divide the ciphertext into 5 subsrtings
  (positions 5k, 5k1, 5k2, 5k3, 5k4)
• Analize each substring as a monoalphabetic
  cipher.
• Continue on http//math.ucsd.edu/crypto/java/EARL
  YCIPHERS/Vigenere.html
• Also an insecure cipher

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Cryptonalysis of the Hill Cipher

• Number of keys k number of invertible m?m
  matrices with coefficients from Z26.Does anyone
  know the formula?
• If p is prime, the alphabet is Zp then
• If p 29 and

m 3 4 5 10
k 1.41013 2.41023 3.51036 1.710146
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Cryptonalysis of the Hill Cipher

• Easily broken with known plaintext attack.
• Permutation Cipher Hill Cipher, where the key
  is a permutation matrix.
• Both ciphers are insecure.

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Perfect Secrecy

• A cryptosystem is computationally secure if the
  best algorithm for breaking it requires at least
  N operations, where N is some specified , very
  large number.Problems
• A cryptosystem is unconditionally secure if it
  cannot be broken with infinite computational
  resources.

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Perfect Secrecy

• None of the classical cryptosystems is even
  computationally secure.
• However the Shift Cipher, the Substitution
  Cipher, and the Vigènere Cipher are
  unconditionally secure if only one element of
  plaintext is encrypted with a given key!REALLY???

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Perfect Secrecy

• Claude Shannon Communication Theory of Secrecy
  Systems, Bell Systems Technical Journal, (1949)
  .
• A cryptosystem has perfect secrecy if pP(xy)
  pP(x) for any x?P and y?C. That is the a
  posteriori probability that the plaintext is x,
  given that the ciphertext is y, is identical to
  the a priori probability that the plaintext is x.

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Perfect Secrecy

• Theorem (Shannon). Suppose the 26 keys in the
  Shift Cipher are used with equal probability
  1/26. Then for any plaintext probability
  distribution, the Shift Cipher has perfect
  secrecy.
• Consequences One-time Pad Cryptosystem (Gilbert
  Vernam, 1917). Key, plaintext, and ciphertext
  have the same length. Problems with keys very
  long, distribution. Each key can be used only
  ONCE!

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The EndCryptography, Part 1 Classical Ciphers

• Cryptography
• Part 2 Modern Cryptosystems
• Stay Tuned