MONOALPHABETIC CIPHERS AND THEIR MATHEMATICS. CIS 400/628 Spring 2005 Introduction to Cryptography

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1 MONOALPHABETIC CIPHERS AND THEIR MATHEMATICS CIS 400/628 Spring 2005 Introduction to Cryptography This is based on Chapter 1 of Lewand and Chapter 1 of Garrett.

2 MONOALPHABETIC SUBSTITUTION CIPHERS These are the ciphers you ld find explained in Donald Duck comic books. Examples: Caesar Cipher (shift cipher) Cryptograms from the newspaper and puzzle books Tic-tac-toe (Masonic) cipher Every occurence of a symbol in the plaintext is replaced by the same symbol in the ciphertext. 1

3 BASIC MATH FOR MSC Example: a X b Y c Z d A... z W starbucks at three PQXOYRZHP XQ QEOBB But we need some math background to discuss even these simple schemes Some Definitions Z = def {..., 3, 2, 1, 0, 1, 2, 3,... } N = def { 0, 1, 2, 3,... } Z + = def { 1, 2, 3,... } 2

4 TWO KEY TOOLS THE WELL-ORDERING AXIOM Let S N be nonempty. Then S contains at least one element. MATHEMATICAL INDUCTION If A(0) and ( k)[ A(k)= A(k + 1) ], Then ( k)a(k). These two principles are equivalent. (Why?) 3

5 PRIME NUMBERS, THE DIVISION ALG. & GCD DEFINITION Suppose a, d Z with d 0. We say that d is a divisor of a (written d a) iff there is a b Z with a = b d. EXAMPLES 2 8 since 8 = since 5 = ( 1)

6 A UTILITY THEOREM THEOREM 1.1 a. d a and a b = d b. b. d a iff d a c. d a iff d a d. ±1 a for any a Z e. d 0 for any d (Z { 0 }) f. a 0 & d a = d a g. a 0 & d a & a ±1 = d < a h. d ± 1 = d = ±1 i. a b & b a = a = ±b j. d a & d b = ( x, y)[ d (ax + by) ] sample proofs 5

7 THE DIVISION ALGORITHM DEFINITION A nonzero p other than ±1 is called prime iff p s only divisors are: ±1, ±p. THEOREM (THE DIVISION ALGORITHM) Suppose a, b Z with b > 0. Then there are unique q and r with proof on board a = q b + r and 0 r < b. DEFINITION The greatest common divisor of a and b (written gcd(a, b)) is the d Z + (a) d a and d b (b) c a & c b = c d 6

8 MORE ON GREATEST COMMON DIVISORS THEOREM 1.3 proof on board Suppose a, b Z +. Then if gcd(a, b) exists, then it is unique. DEFINITION Suppose a, b Z. A number of the form a x + b y is called a linear combination of a and b. THEOREM 1.4 proof in a moment Suppose a, b Z +. Then gcd(a, b) = min{ ax + by x, y Z & ax + by > 0 }. COROLLARY Suppose a, b Z +. Then { ax + by x, y Z } = { z gcd(a, b) z Z }. 7

9 GCDS CONTINUED THEOREM 1.4 Suppose a, b Z + and S = { ax + by x, y Z & ax + by > 0 }. Then gcd(a, b) = min(s). Proof Note: S Z + and a a + b b = a 2 + b 2 > 0. So S. So by the WOP, S has a least element d. Suppose x 1, y 1 Z d = ax 1 + by 1. Claim 1. d a and d b. proof on board Claim 2. c a & c b = c d. proof on board d satisfies the two conditions for gcd(a, b). d must be the g.c.d. of a and b. QED 8

10 TOWARDS A GCD ALGORITHM Suppose a, b Z + with a > b. Define { b, if a = q b + 0; f(a, b) = f(b, r), if a = q b + r with r > 0. Claim. a, b Z + with a > b, f(a, b) = gcd(a, b). Proof by induction on b. Let A(b) def ( a > b)[ f(a, b) = gcd(a, b) ]. Base case: b = 1. Then a = a b & gcd(a, b) = 1 & f(a, b) = 1. A(1) 9

11 TOWARDS A GCD ALGORITHM Suppose a, b Z + with a > b. Define { b, if a = q b + 0; f(a, b) = f(b, r), if a = q b + r with r > 0. Claim. a, b Z + with a > b, f(a, b) = gcd(a, b). Proof by induction of b. A(b) def ( a > b)[ f(a, b) = gcd(a, b) ]. Induction step: Suppose A(1), A(2),..., A(b 1) are true. Case 1: a = q b. Similar to the base case. Case 2: a = q b + r with 0 < r < b. By the I.H., d = gcd(b, r). (Why?) Then since a = q b + r, we have d gcd(a, b). Also since d = x b + y r = x b + y(a q b) = y a + (x y q) b for some x and y Let d = f(b, r). (Why?) gcd(a, b) d (Why?) d = gcd(a, b) (Why?) QED 10

12 RELATIVELY PRIME INTEGERS DEFINITION Suppose a, b Z +. We say that a and b are relatively prime iff gcd(a, b) = 1. THEOREM 1.5 proof on board Suppose a, b Z + and p is a positive prime. Then p a b implies that (p a or p b). THEOREM 1.6 Suppose a 1,..., a n, p Z + and p is prime. If p a 1... a n, then for some i, p a i. proof on board 11

13 THE FUNDAMENTAL THEOREM THE FUNDAMENTAL THEOREM OF ARITHMETIC Every integer a > 1 can be expressed as the product of one or more positive primes uniquely, except for the order of factors. PROOF K = Let { x Z + : x > 1 and x cannot be expressed as a product of positive primes Claim 1. K =. proof on board Claim 2. Suppose proof on board a = p 1 p 2... p s with p 1 p 2... p s. = q 1 q 2... q t with q 1 q 2... q t. } Then s = t and p 1 = q 1 and... and... p s = q s. QED 12

14 MODULAR ARITHMETIC DEFINITION a + n b = def (a + b) mod n. Example 22: hours = 04:00 is just = 4. DEFINITION Suppose n Z +. The group of integers modulo n is the set { 0,..., n 1 } with the operation + n. Example DEFINITION Suppose n > 0. x and y are congruent modulo n (written x = y (mod n)) iff n (x y). Examples 3 = 8 (mod 5). 17 = 5 (mod 3). 3 = 5 (mod 2). 13

15 THEOREM 1.8 MORE ON MODULAR ARITHMETIC Suppose n Z + and w, x, y, z Z. a. x = x (mod n). b. x = y (mod n) y = x (mod n). proof on board c. x = y (mod n) & y = z (mod n) = x = z (mod n). d. x = y (mod n) & w = z (mod n) = x + w = y + z (mod n) & x w = y z (mod n) COROLLARY = is an equivalence relation. COROLLARY For all x, y, z, and k, x = y (mod n) = x z = y z (mod n). x = y (mod n) = x k = y k (mod n). 14

16 MORE ON CONGRUENCES DEFINITION Suppose n Z + and a Z. [a] n = def { x x Z & x = a (mod n) }. (Usually we write [a] for [a] n when n is understood.) DEFINITION Suppose n Z + Z n = def { [0], [1],..., [n 1] } = the sets of ints mod n. DEFINITION [a] + n [b] = def [a + b]. Is this sensible? THEOREM Suppose n Z + and a, a, b, b Z. If [a] = [a ] and [b] = [b ], then [a] + n [b] = [a ] + n [b ]. 15

17 BACK TO CRYPTOSYSTEMS (FINALLY) Monoalphabetic substitution ciphers E: { a, b,..., z } { A, B,..., Z } with E 1 1 & onto. Lewand s Convention a b c... x y z So a n = the nth letter. Standard Convention a b c... x y z So a n = the (n + 1)st letter. Additive (shift) cipher with key k a n a m where m = (m + k) mod 26. (Only 26 keys!) To decrypt use the key ( k) mod 26. Multiplicative cipher with key k a n a m where m = (m k) mod 26. To decrypt use the key (k 1 ) mod 26. (Problem!) (Consider k = 4, and mapping a and n) 16

18 MORE ON MONOALPHABETIC SUBSTITUTION CIPHERS THEOREM 1.11 Suppose k, n Z +. If n and k are relatively prime, then k (mod n), 2k (mod n),..., n k (mod n) are all distinct. Does k 1 (mod n) always exist? How do you find k 1 (mod n)? Will k 1 (mod n) always work as a decryption key? 17

19 AFFINE CIPHERS a n a m where m = c(n + b) (mod 26) and c 1 (mod 26) exists. This is essentially an additive cipher followed by a multiplicative cipher. To decipher, do the inverse! 1. invert the multiplicative cipher (c 1 (mod 26)) 2. invert the additive cipher (( k) mod 26) 18

20 KEYWORD CIPHERS Need two items: a keyword, and a letter. For example, cryptography and X. 1. Write down the alphabet. a b c d e f g h i j k l m n o p q r s t u v w x y z 2. Under that, starting at the chosen letter, write the keyword with duplicate letters removed. Wrap around if necesary. a b c d e f g h i j k l m n o p q r s t u v w x y z P T O G A H C R Y 3. Then write down the rest of the alphabet, in order, except for those letters in the keyword, and wrap around. a b c d e f g h i j k l m n o p q r s t u v w x y z P T O G A H B D E F G I J K L M N Q S U V W X C R Y 19

21 CRYPTANALYSIS OF MONOALPHABETIC SUBSTITUTION CIPHERS There are 26! many monoalphabetic substitution ciphers. (Why?) However, frequencies of characters stay the same modulo a permutation. E.g., If e X and e is 11% of the plaintext, then X is 11% of the cipher text. So cryptanalysis of these ciphers is relatively easy. 20

22 LETTER FREQUENCY IN ENGLISH (SORTED BY FREQUENCY) Letter Freq. % Letter Freq. % Letter Freq. % E T A O I N S H R D L C U M W F G Y P B V K J X Q Z