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…

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 = 2*320 = 4*57 = 1*7 –Definition: A natural number p is said to be prime if p > 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 2*3 2 *5 –Example 2: Factor 935 Answer: 5 * 11 * 17…

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…

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 Z m a number x has a multiplicative inverse if there is a number y such that xy mod m = 1. –Consider the set Z 6. 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 26Table of multiplicative inverses Example 10: Use the table to find 7 -1 MOD 26.table –Answer: 15…

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 is 19.table 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.)…

Affine Ciphers; More Modular Arithmetic Mathematical Description of Affine Ciphers –Given a and b in Z 26 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 26Example 13 –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…Example 14

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…Example 15

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…Frequency Analysis

Affine Ciphers; More Modular Arithmetic Additional Exercises: Factor the following into a product of prime numbers: 120, 715, Find the following gcd: gcd(30, 40), gcd(150, 500), gcd(187, 455)…!