Discrete Mathematics with Applications MATH236 Dr. Hung P. Tong-Viet School of Mathematics, Statistics and Computer Science University of KwaZulu-Natal Pietermaritzburg Campus Semester 1, 2013 Tong-Viet (UKZN) MATH236 Semester 1, 2013 1 / 23
Table of contents Finding generators in Z ∗ 1 p Review of Chapter 3 2 Chapter 4. Fundamentals of cryptopology 3 Introduction Monoalphabetic and Polyalphabetic ciphers Tong-Viet (UKZN) MATH236 Semester 1, 2013 2 / 23
Finding generators in Z ∗ p The multiplicative group For a positive integer n , the multiplicative group of Z n is Z ∗ n = { a ∈ Z n : gcd( a , n ) = 1 } The group operation is multiplication modulo n The identity in Z ∗ n is the number 1 Every element a ∈ Z ∗ n has an inverse The order of Z ∗ n is hi ( n ) If p is a prime, then Z ∗ p = Z p − { 0 } = { 1 , 2 , · · · , p − 1 } The order of a ∈ Z ∗ n is the smallest positive integer k such that a k = 1 . We write | a | = k . Tong-Viet (UKZN) MATH236 Semester 1, 2013 3 / 23
Finding generators in Z ∗ p The multiplicative group Example Consider the group Z ∗ 15 We have Z ∗ 15 = { 1 , 2 , 4 , 7 , 8 , 11 , 13 , 14 } 15 | = 8 = hi (15) = 15(1 − 1 3 )(1 − 1 | Z ∗ 5 ) Order of 2 ∈ Z ∗ 15 2 k mod 15 k 1 2 2 4 3 8 4 1 Thus | 2 | = 4 in Z ∗ 15 . Tong-Viet (UKZN) MATH236 Semester 1, 2013 4 / 23
Finding generators in Z ∗ p Finding generators Theorem Suppose that p is a prime and α ∈ Z ∗ p . Then α is a generator of Z ∗ p if and only if α ( p − 1) / q �≡ 1 (mod p) for all primes q such that q | ( p − 1) . Tong-Viet (UKZN) MATH236 Semester 1, 2013 5 / 23
Finding generators in Z ∗ p Finding generators Example 37 . We have 37 − 1 = 36 = 2 2 · 3 2 . Consider the group Z ∗ For α ∈ Z ∗ 37 , we need to compute α 36 / 2 (mod 37) α 36 / 3 (mod 37) If all the results are not trivial, then α is a generator of Z 37 . We have 2 18 ≡ 36 and 2 12 ≡ 26 (mod 37), so 2 is a generator of Z ∗ 37 However 4 18 ≡ 1 and 4 12 ≡ 10 (mod 37), so 4 is NOT a generator of Z ∗ 37 Is 31 a generator of Z ∗ 37 ? Tong-Viet (UKZN) MATH236 Semester 1, 2013 6 / 23
Review of Chapter 3 Elementary number theory The Division Algorithm: Find gcd( a , b ) , with a , b ∈ Z The Extended Division Algorithm: Find s , t ∈ Z such that gcd( a , b ) = as + bt Study the proofs of Lemma 24 and Theorem 25 Find the multiplicative inverses (using the Extended Division Algorithm) Study Theorems 26 and 27 . (Existence and Uniqueness) Square and multiply in Z m Tong-Viet (UKZN) MATH236 Semester 1, 2013 7 / 23
Review of Chapter 3 Elementary number theory (cont.) Prime numbers Euler’s hi -function Definition and how to compute hi ( n ) for n ∈ Z Theorems 30-32 and Theorem 33 (Formula for hi ( n )) Fermat and Euler Theorems Find remainders and inverses using these theorems Definition of groups, order of elements and how to find a generator for Z ∗ p . Tong-Viet (UKZN) MATH236 Semester 1, 2013 8 / 23
Chapter 4. Fundamentals of cryptopology Introduction Introduction Further reading: Handbook of Applied Cryptography by Menezes, Oorschot and Vanstone Available at www . cacr . math . uwaterloo . ca / hac The word cryptopology was used for the first time by John Wilkins in 1641 This word comes from Greek words krypte : to hide and logos : word Cryptopology consists of two related disciplines: cryptography (graphein: to write) and cryptanalysis Cryptography was used by the Egyptians as early as 1900 BC Classical ciphers are simple substitutions (shift ciphers, block ciphers) with a shared private key If we know how to encrypt, we can decrypt the message easily. Tong-Viet (UKZN) MATH236 Semester 1, 2013 9 / 23
Chapter 4. Fundamentals of cryptopology Introduction Introduction In modern times, cryptography has been used by the governments, military and now by commercial entities Public key cryptograph, invented in 1976, is the modern cryptograph and the most widely used public key system is the RSA cryptosystem In RSA crypto system, we encrypt the message using modular exponentiation, where the modulus is the product of two large primes To decrypt the message, we need to know the prime factors of the modulus. However, the factorisation is a difficult problem. Tong-Viet (UKZN) MATH236 Semester 1, 2013 10 / 23
Chapter 4. Fundamentals of cryptopology Introduction Definition of cryptograph Definition Cryptography is the study of mathematical techniques to provide information security such as Confidentiality: Ensuring that only the intended recipient of the message is able to understand it Data integrity: Preventing the unauthorized alteration of data Authentication: Providing assurance that both sender and recipient are who they say they are, and that the message comes from where it is supposed to and goes where it is supposed to Non-repudiation: Preventing parties from denying previously made commitments Tong-Viet (UKZN) MATH236 Semester 1, 2013 11 / 23
Chapter 4. Fundamentals of cryptopology Introduction Definition of cryptanalysis Definition Cryptanalysis is the study of mathematical techniques to defeat information security. Tong-Viet (UKZN) MATH236 Semester 1, 2013 12 / 23
Chapter 4. Fundamentals of cryptopology Introduction Definitions and Terminology Definition plaintext (message) M is a finite string of symbols from a finite alphabet Σ (Latin alphabet, binary alphabet) M is converted, by the process of encryption (enciphering) into an enciphered text called the ciphertext (cryptogram) C The person who enciphered M is called the sender or encipherer. He used a set of rules or algorithm to encrypt M The sender sends the ciphertext C to the intended recipient (receiver) The algorithm involves the use of a key K which is known to both sender and receiver Tong-Viet (UKZN) MATH236 Semester 1, 2013 13 / 23
Chapter 4. Fundamentals of cryptopology Introduction Definitions and Terminology Definition The receiver uses an algorithm (involving the key) to obtain M from C . This is known as decryption (deciphering) The ciphered text C and the key K must determine the plaintext M uniquely. The plaintext will be written in lowercase and ciphertext in uppercase Any person who intercepts the message is called an inceptor The methods used in the encryption/decryption above form the subject of cryptography The methods used by the inceptor to derive M from C without having access to the key are studies in cryptanalysis. Tong-Viet (UKZN) MATH236 Semester 1, 2013 14 / 23
Chapter 4. Fundamentals of cryptopology Introduction Principle of Cryptography Tong-Viet (UKZN) MATH236 Semester 1, 2013 15 / 23
Chapter 4. Fundamentals of cryptopology Monoalphabetic and Polyalphabetic ciphers Encryption schemes There are two classes of encryption schemes Monoalphabetic cipher: each letter in the plaintext alphabet is always encrypted as the same letter in the ciphertext alphabet. Polyalphabetic cipher: a letter in the plaintext alphabet might be encrypted as several different letters in the ciphertext. Tong-Viet (UKZN) MATH236 Semester 1, 2013 16 / 23
Chapter 4. Fundamentals of cryptopology Monoalphabetic and Polyalphabetic ciphers Monoalphabetic ciphers Simple substitution ciphers: we replace each letter of the alphabet by another. In other words, a simple substitution cipher is a permutation of the letters of the alphabet Shift ciphers: (used by Julius Caesar) each of the letters a , b , · · · z is replaced by the letter which occurs three places after it in the alphabet. Tong-Viet (UKZN) MATH236 Semester 1, 2013 17 / 23
Chapter 4. Fundamentals of cryptopology Monoalphabetic and Polyalphabetic ciphers Simple substitution ciphers Example Suppose that the following key is used: Plaintext a b c d e f · · · t u v w · · · Ciphertext D X W E G A · · · B F R C · · · Both the encipherer and decipherer have a copy of this key The plaintext ‘fat’ is enciphered as ‘ADB’ The ciphertext ‘WDB’ is deciphered as ‘cat’ The reordered alphabet ( DXWEGA · · · BFRC · · · ) is called the substitution alphabet This is a very poor system. It is easy to cryptanalyze. Memorizing the key is difficult. If the key is kept, it can be lost or stolen. Tong-Viet (UKZN) MATH236 Semester 1, 2013 18 / 23
Chapter 4. Fundamentals of cryptopology Monoalphabetic and Polyalphabetic ciphers Shift ciphers Example The key of Caesar shift cipher is represented by the following permutation Plaintext a b c d e f · · · w x y z Ciphertext D E F G H I · · · Z A B C We call this a shifter cipher , or additive cipher or translation cipher with shift (or key) 3 In general, we can use a shift cipher with key d This is a special case of simple substitution cipher The key is easily remember but the cipher is insecure Tong-Viet (UKZN) MATH236 Semester 1, 2013 19 / 23
Chapter 4. Fundamentals of cryptopology Monoalphabetic and Polyalphabetic ciphers Polyalphabetic ciphers a specific ciphertext letter can represent more than one plaintext each plaintext letter can be encrypted in more than one way There are several ways to do this but we must be sure that whatever we do, we can still decipher the message. We will look at ‘ n -gram substitution’ and ‘permutation cipher’ Tong-Viet (UKZN) MATH236 Semester 1, 2013 20 / 23
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