Mathematical Concepts for Cryptography

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Mathematical Concepts for Cryptography
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Mathematical Concepts

• Prime number
• Prime factorization
• Modulo arithmetic
• GCD
• LCM
• Relatively prime
• Congruence

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Mathematical Concepts

• Euler phi function
• Factorial
• Permutations and combinations
• Finite group
• Finite field

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Mathematical Concepts

• Prime number is a positive integer that is
  divisible by itself and 1 only
• E.g., 2, 3, 5, 7, 11, 13, 17, 19,
• Every positive integer can be written as a
  product of primes
• E.g. 24 2 2 2 3
• Modulo arithmetic provides the remainder upon
  division
• E.g. mod(11, 4) 3

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Mathematical Concepts

• GCD (24, 16) denotes the greatest common divisor
  of 24 and 16
• GCD (24, 16) 8
• Euclids algorithm for GCD
• a and b are positive integers
• While a ! 0 set (a, b) (b mod a, a) repeatedly
• When a 0 stop the iteration and the GCD is the
  value of b

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Mathematical Concepts

• LCM (10, 4) denotes the least common multiple of
  10 and 4
• LCM (10, 4) 20
• LCM (a, b) (a b) / GCD (a, b)
• Two numbers m and n are relatively prime if they
  have no common divisor other than 1,
• i.e., GCD (m, n) 1.
• Standard notation for relatively prime numbers
  (m, n)1
• E.g., m 10 and n 9 is written as (10, 9) 1
• Relatively prime does not mean that the
  individual numbers are prime

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Mathematical Concepts

• If m divides (a b) then we say that a is
  congruent to b modulo m and it is denoted as
• a ? b (mod m)
• Euler phi function is denoted by ? (m) where m is
  any positive integer
• The ? function is also called the totient
  function
• ?(m) is the number of positive integers that are
  less than m and relatively prime to m
• By definition ?(1) 1

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Mathematical Concepts

• E. g., ?(2) 1
• ?(3) 2
• ?(4) 2
• ?(5) 4
• ?(6) 2
• ?(7) 6
• ?(8) 4
• ?(9) 6
• ?(10) 4
• ?(11) 10
• ?(12) 4

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Mathematical Concepts

• For any prime p, ?(p) p 1
• E.g., ?(7) 6
• If m pn, then ?(pn) pn (1 1/p)
• E. g., ?(23) 23 (1 ½) 8 ½ 4
• For any positive integer m whose prime factors
  are p1, p2, , pr,
• ?(m) m (1 1/p1) (1 1/p2) (1 1/pr)
• E.g. ?(42) 42 (1 1/2) (1 1/3) (1 1/7)
• 42 1 / 2 2 / 3 6 / 7
• 12

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Mathematical Concepts

• For any positive integer m,
• m ? ?(d)
• d ? m
• for all divisors d of m
• E.g., Take m 42
• Its divisors are 1, 2, 3, 6, 7, 14, 21, 42
• ?(1)?(2)?(3)?(6)?(7)?(14)?(21)?(42)
• 1122661212 42

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Mathematical Concepts

• Euler-Fermat theorem For any two positive
  integers a and m that are relatively prime,
• a?(m) ? 1 mod(m)
• When a and m are relatively prime, the
  inverse of a, denoted by a-1, is given by the
  equation
• a-1 ? a?(m)-1 mod(m)
• If m, n are relatively prime, then
• ?(m n) ?(m) ?(n)

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Mathematical Concepts

• Factorial (n) is denoted by n!
• n! n (n-1) (n-2) 3 2 1
• By convention, 0! 1
• Permutations of a set of numbers takes into
  account the order of the numbers. Thus, (1, 2)
  is different from (2, 1).
• nPr denotes the number of permutations of r items
  from n items
• nPr n! / (n r)!

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Mathematical Concepts

• E.g., 5P3 5! / (5 3)!
• 5 4 3 2! / 2!
• 5 4 3 60
• Number of permutations of the numbers in the set
  1, 2, 3 taken two at a time produces the set
  (1,2), (1,3), (2,1), (2,3), (3,1), (3,2)
• nP0 1 nP1 n nPn n!

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Mathematical Concepts

• In combinations the order of the numbers is
  unimportant
• nCr denotes the number of combinations of r items
  from n items
• nCr n! / r! (n r)!
• 5C3 5! / 3! (5 3)!
• 5 4 3! / 3! 2!
• 5 4 / 2 10

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Mathematical Concepts

• Number of combinations of the numbers in the set
  1, 2, 3 taken two at a time produces the set
  (1,2), (1,3), (2,3)
• nC0 1 nC1 n nCn 1

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Mathematical Concepts

• A finite group G is a finite set with a specific
  operation (such as or ) that satisfies the
  following properties
• a, b ? G ? a b ? G (closure)
• There is an identify 0 ? G (identity)
• a 0 a for all a ? G
• For every a ? G there is an inverse (-a) such
  that
• a (-a) 0
  (inverse)
• (a b) c a (b c)
  (associative)

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Mathematical Concepts

• E.g., Take G to be the set of numbers modulo 3.
  Thus, G 0, 1, 2. The addition operation is
  modulo 3 arithmetic. The set G with this
  operation is a finite group.
• If the group G has the property
• a b b a for every a, b ? G, then the
  group is called an Abelian group. The above
  property is called commutative.

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Mathematical Concepts

• A finite field is a set G with two operations
  and such that (G, ) is a finite group with
  identify 0 and (G-0, ) is a finite group with
  identity 1. The two operations and are
  connected using the distributive property as
  follows
• a (b c) a b a c for all a,b,c ? G

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Mathematical Concepts

• E.g., Take G to be the set of numbers modulo 5.
  Thus, G 0, 1, 2, 3, 4. The addition
  operation is modulo 5 arithmetic. The set G with
  this operation is a finite group. Take G-0
  1, 2, 3, 4. This set with the multiplicative
  operation modulo 5 is a group with identity 1.
  Thus, G is a finite field.

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History

• 50 B.C. Julius Caesar uses cryptographic
• technique
• 400 A.D. Kama Sutra in India mentions
• cryptographic techniques
• 1250 British monk Roger Bacon
• describes simple ciphers
• 1466 Leon Alberti develops a cipher
• disk
• 1861 Union forces used a cipher during
• Civil War

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History

• 1914 World War I British, French, and
• German forces use encryption
• technology
• 1917 William Friedman, Father of U.S.
• encryption efforts starts a school
• for teaching cryptanalysis in
• Illinois
• 1917 ATT employee Gilbert Vernam
• invents polyalphabetic cipher
• 1919 Germans develop the Engima machine
• for encryption

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History

• 1937 Japanese design the Purple
• machine for encryption
• 1942 Navajo windtalkers help with secure
• communication during World War II
• 1948 Claude Shannon develops statistical
• methods for encryption/decryption
• 1976 IBM develops DES
• 1976 Diffie Hellman develop public key /
• private key cryptography
• 1977 Rivest Shamir Adleman develop the
• RSA algorithm for public key / private key

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Basic Terminologies

• Steganography is the method of hiding secret
  messages in an ordinary document
• Steganography does not use encryption
• Steganography does not increase file size for
  hidden messages
• Example select the bit patterns in pixel colors
  to hide the message

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Basic Terminologies

• Cryptography deals with creating documents that
  can be shared secretly over public communication
  channels
• Cryptographic documents are decrypted with the
  key associated with encryption, with the
  knowledge of the encryptor
• The word cryptography comes from the Greek words
  Krypto (secret) and graphein (write)
• Cryptanalysis deals with finding the encryption
  key without the knowledge of the encryptor
• Cryptology deals with cryptography and
  cryptanalysis
• Cryptosystems are computer systems used to
  encrypt data for secure transmission and storage

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Basic Terminologies

• Keys are rules used in algorithms to convert a
  document into a secret document
• Keys are of two types
• Symmetric
• Asymmetric
• A key is symmetric if the same key is used both
  for encryption and decryption
• A key is asymmetric if different keys are used
  for encryption and decryption

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Basic Terminologies

• Examples
• Symmetric key methods
• DES 56-bit
• Triple DES 128-bit
• AES 128-bit and higher
• Blowfish 128-bit and higher
• Asymmetric key methods
• RSA (Rivest-Shamir-Adleman of MIT)
• PGP (Phil Zimmerman of MIT)

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Basic Terminologies

• Plaintext is text that is in readable form
• Ciphertext results from plaintext by applying the
  encryption key
• Notations
• M message, C ciphertext,
• E encryption, D decryption,
• k key
• E(M) C
• E(M, k) C
• Fact D(C) M, D(C, k) M

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Basic Terminologies

• Hash functions generate a digest of the message
• Substitution cipher involves replacing an
  alphabet with another character of the same
  alphabet set
• Mono-alphabetic system uses a single alphabetic
  set for substitutions
• Poly-alphabetic system uses multiple alphabetic
  sets for substitutions
• Caesar cipher is a mono-alphabetic system in
  which each character is replaced by the third
  character in succession. Julius Caesar used this
  method of encryption.

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Context of Cryptography

• Cryptography is not a stand alone method in a
  security system
• Attack on a cryptographic system could be
  traceless (e.g., breaking the encryption key)
• Cryptography is like having a curtain on a window
  use it to protect what is inside but does not
  provide fool-proof protection
• Anyone trying to break-in will not persist on
  attacking only one access point. Protecting all
  access points is not easy.

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Context of Cryptography

• Hackers do not play by the rules
• Every system is vulnerable for attack
• Secure Electronic Transaction (SET) is one tool
  available to protect online transactions. It has
  weaknesses.
• Physical security is one better way to protect
  systems