Welcome to CS 490B Cryptography

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Welcome to CS 490BCryptography

• David Mutchler
• Spring term, 1998-99
• Lecture 1

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Today

• Once upon a time (The Debate in Sign Language)
• Course mechanics
• This course involves lots of abstract
  mathematics.
• Compilers and Systems Programming are great!
• www.rose-hulman.edu/Class/cs/cryptography
• Grading, schedule (especially Week 9)
• Some classical cryptosystems
• Zm, arithmetic modulo m.

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What is a cryptosystem? (informal)

• Alice, Bob and Oscar
• Alice wants to send a message to Bob.
• She uses a key to encrypt her plaintextto obtain
  ciphertext.
• She sends the ciphertext to Bobover an insecure
  channel.
• Oscar, eavesdropping, sees the ciphertext.But he
  cannot determine the plaintext.
• Bob, who knows the key,decrypts the ciphertext
  to obtain the plaintext.

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What is a cryptosystem?

• Three finite sets
• P set of possible plaintexts
• C set of possible ciphertexts
• K set of possible keys
• Encryption and decryption functions e and d.For
  each k in K
• ek P? C dk C ? P
• Main property for any plaintext x and key k
  dk(ek(x)) x

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Private key protocol

• Alice and Bob pick a key k in K
• At the same place (not observed by Oscar).
• Or, over a secure channel.
• Later, if Alice wants to send Bob x x1x2
  ... xn where each xi is in P,she
  uses the encryption rule ek specified by key
  kand sends y y1y2 ... yn where yi
  ek(xi)
• When Bob receives y1y2 ... yn , he decrypts each
  yi using the decryption key dk, obtaining x1x2
  ... xn

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Some important properties

• Main property decrypt and encrypt are inverses
• Each encryption function ek is one-to-one.
  (Why?)
• Must be able to decrypt and encrypt efficiently
• But details vary wildly with the application.
• Must be hard, given the ciphertext, to determine
• The key.
• The plaintext.
• Usually, only the key is kept secret.The
  cryptographic system is assumed to be public.

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Shift cipher

• Encrypt by shifting each letter by k.
• For example, if k 3 (Caesars cipher)
• e3(a) d
• e3(b) e
• ... e3(z) c
• Exercise secret encrypts into ___________
• For a formal definition,we need arithmetic
  modulo m

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Arithmetic modulo m

• Suppose a and b are integersand m is a positive
  integer.
• a mod m is the remainder (residue)when a is
  divided by the modulus m
• a ? b mod m iff a mod m b mod n
• a is congruent to b modulo m
• a is reduced modulo m means replace a by a mod
  m
• Examples
• 38 mod 5 is ?? 38 ? ?? mod 5
• 142 is reduced modulo 32 to ??

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Arithmetic modulo m Zm

• Zm is the set 0, 1, ... m-1 (called
  residues)plus two operations, ? and ?
• Do addition and multiplication just like in the
  real numbers, except reduce results modulo m.
• Zm forms a ring. Exercises
• What are the properties of a ring?
• What is the additive inverse of a in Zm?
• What does a ? b mean in Zm?

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Shift cipher

• P C K Z26
• For k between 0 and 25,and for x and y in Z26,
  define
• ek(x) x ? k mod 26
• dk(y) y ? k mod 26

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Shift cipher encryption

• Exercise (from Stinson) use k 11 to encrypt
  we will meet at midnight

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Shift cipher cryptanalysis

• Exercise (from Stinson)
• Do an exhaustive key search to decrypt
  HBCRCLQRWCRVNBJENBWRWN
• How many keys must be tried, at worst? On average?

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Substitution cipher

• P C a, b, ... z
• K is the set of all permutations of P.
• Encrypt by applying a permutation.
• Decrypt by applying the inverse permutation.

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Substitution cipher

• Example (from Stinson) use the permutation
  below to decrypt MGZVYZLGHCMHJMY
  XSSFMNHAHYCDLMHA

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Substitution cipher

• Exhaustive key search
• How many keys?
• 26! ? 4?1026
• At 1,000,000 keys per second,thats over 100
  billion centuries!
• But there are other methods of attack!
• Shift cipher is a special case of substitution

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Affine cipher

• Another special case of substitution
• P C Z26
• K (a,b) in Z26 ? Z26 ____________
• ek(x) ax b mod 26
• Exercises
• What is dk(x)?
• Does (4, 7) work for this method?
• How many (a,b) pairs work in Z26? In Zm?

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Affine cipher

• Reasoning as if it were real arithmetic
• y ek(x) ax b mod 26
• Solve for x
• y ? b ax mod 26
• a-1(y ? b) x mod 26
• So dk(y) a-1(y ? b)
• But if the key is (4, 7)
• e(4,7)(10) 47 mod 26 21
• e(4,7)(23) 99 mod 26 21

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Affine cipher

• Definitions
• gcd(a, b) is the greatest common divisor of a and
  b
• a and b are relatively prime iff gcd(a, b) 1.
• Suppose a?Zm. The multiplicative inverse of a is
  a number a-1?Zm such that aa-1 1 mod m.
• Theorem
• An element a in Zm has a multiplicative inverse
  a-1 in Zm iff gcd(a, m) 1
• If gcd(a, m) 1, then dk(ek(x)) x