Section 2.2: Affine Ciphers; More Modular Arithmetic

1
Section 2.2 Affine Ciphers More Modular
Arithmetic

• Shift ciphers use an additive key. To increase
  security, we can add a multiplicative parameter.
• For affine ciphers we use both a multiplicative
  and an additive parameter

2
Affine Ciphers More Modular Arithmetic

• Mathematics Background for Affine Ciphers
• A natural number is a number in the set 1, 2, 3,
  .
• Any natural number can always be written as the
  product of two other natural numbers.
• Ex 6 23 20 45 7 17
• Definition A natural number p is said to be
  prime if p gt 1 and its only divisors are 1 and p.
  A natural number that is not prime is said to be
  composite.
• Question Are there an infinite number of primes?
  Yes. This can be proven but is not the focus of
  this course. The idea though is to assume not
  and show that this leads to a contradiction.
• The primes are 2, 3, 5, 7, 11, 13, 17, 19, 23,
  29, where I have listed the first ten.
• Theorem The Fundamental Theorem of Arithmetic.
  Every natural number larger then 1 is a product
  of primes. This factorization can be done in
  only one way if the order is disregarded.
• Example 1 Factor 90
• Answer 2325
• Example 2 Factor 935
• Answer 5 11 17

3
Affine Ciphers More Modular Arithmetic

• The common prime factors of two numbers can be
  used to find the greatest common divisor of two
  numbers (gcd).
• Definition The gcd of two natural numbers a and
  b, denoted gcd(a, b), is the largest natural
  number that divides both a and b with no
  remainder.
• Elementary method for computing the gcd of two
  numbers
• Decompose the two numbers into prime
  factorization.
• The common factors make up the gcd.
• If there are no common factors then the gcd 1.
  (The two numbers are said to be relatively prime)
• Example 3 Find the gcd(20, 30).
• Answer 10
• Example 4 Find the gcd(1190, 935).
• Answer 85
• Example 5 Find the gcd(15, 26).
• Answer 1

4
Affine Ciphers More Modular Arithmetic

• Multiplicative Inverses For a number x in some
  set S, the multiplicative inverse of x is a
  number y, also in S, such that xy 1.
• In the real numbers, every nonzero number has a
  multiplicative inverse.
• Consider the set Z (the integers). Only 1 and -1
  have multiplicative inverses.
• In Zm a number x has a multiplicative inverse if
  there is a number y such that xy mod m 1.
• Consider the set Z6. The numbers 1, 5 have
  inverses. The inverse of 1 is itself. The
  inverse of 5 is itself.
• Fact If the gcd(b, m) 1, then b has a
  multiplicative inverse. the inverse of b is
  denoted by b-1.
• Example 6 In the set R, find the inverses of 2,
  10, a.
• Answer ½, 1/10,1 / a
• Example 8 Does 8 have an inverse modulo 26?
• Answer No. gcd(8, 26) 2.
• Example 9 Does 9 have an inverse modulo 26?
• Answer yes. gcd(9, 26) 1.
• Table of multiplicative inverses modulo 26
• Example 10 Use the table to find 7-1 MOD 26.
• Answer 15

5
Affine Ciphers More Modular Arithmetic

• Multiplicative Property for Modular Arithmetic
  If a b mod m, then for any number k, ka kb
  mod m.
• Example 11 Solve 11x 1 5 mod 26 (Note the
  inverse of 11 is not 1 / 11. From our table we
  see that 11-1 is 19.
• Answer x 114 MOD 26 10 MOD 26 10
• Example 12 Solve (Note -9 17 mod 26)
• 8a b 18 mod 26
• 17a b 11 mod 26 (subtract the second equation
  from the first)
• Answer a 5, b 4 (Note that a 31 and b 30
  are also answers, etc.)

6
Affine Ciphers More Modular Arithmetic

• Mathematical Description of Affine Ciphers
• Given a and b in Z26 where gcd(a, 26) 1. We
  encipher a plaintext letter x to obtain a
  ciphertext letter y, by computing y (ax b)
  MOD 26.
• Example 13 Encipher RADFORD using the affine
  cipher y (5x 4) MOD 26
• Answer LETDWLE
• The following example illustrates the need for a
  and b to be relatively prime.
• Example 14 Encipher AN using the affine cipher
  y (2x 1) mod 26

7
Affine Ciphers More Modular Arithmetic

• Deciphering an Affine Cipher
• For an affine cipher y (ax b) mod 26 where
  gcd(a, 26) 1, the decipherment is unique. The
  formula for the decipherment is
• y ax b
• y b ax
• a-1(y b) x
• x a-1(y b) mod 26
• Example 15 Decipher the message ARMMVKARER
  which was encrypted using the affine cipher y
  (3x 5) MOD 26

8
Affine Ciphers More Modular Arithmetic

• Cryptanalysis of Affine Ciphers
• For an affine cipher to be deciphered the enemy
  must know a and b.
• There are two methods to decipher an affine
  cipher
• Brute force or Exhaustion method (Try all
  possible values for a and b). Since there are 12
  values for a and 26 values for b, then there are
  12 26 312 total (a, b) pair tests.
• Frequency analysis Uses the fact that the most
  frequently occurring letters in ciphertext
  produced by shift cipher has a good chance of
  corresponding to the most frequently occurring
  letters in the standard English alphabet. The
  most frequently occurring letters in English are
  E, T, A, O, I, N, S.
• Example 16 Frequency Analysis. Suppose that the
  two most frequently occurring letters in a
  ciphertext message are W and H. Assuming that
  these correspond to E and T, find a and b

9
Affine Ciphers More Modular Arithmetic

• Additional Exercises
• Factor the following into a product of prime
  numbers 120, 715, 594473
• Find the following gcd gcd(30, 40), gcd(150,
  500), gcd(187, 455)!