Math236 Discrete Maths with Applications P. Ittmann UKZN, Pietermaritzburg Semester 1, 2012 Ittmann (UKZN PMB) Math236 2012 1 / 18
Cryptology Cryptology is usually understood to consist of two related disciplines Cryptography and Cryptanalysis Ittmann (UKZN PMB) Math236 2012 2 / 18
Cryptology (cont.) Cryptography is the study of mathematical techniques to provide information security Menezes, Oorschot, and Vanstone identify four cryptographic goals: Confidentiality : Ensuring that only the intended recipient of a message 1 is able to understand it. Data integrity : Preventing the unauthorized alteration of data 2 Authentication : Providing assurance that (i) both sender and recipient 3 are who they say are, and, (ii) that the message comes from where it’s supposed to and goes where it’s supposed to Non-repudiation : Preventing parties from denying previously made 4 commitments Ittmann (UKZN PMB) Math236 2012 3 / 18
Cryptology Cryptanalysis is the study of mathematical techniques to defeat information security The word cryptology was used for the first time by John Wilkins in 1641 The subject itself is much older A very primitive form of cryptology was used by the Egyptians 4000 years ago The Spartans used cryptographic devices in approximately 400 BC About forty years later, Tacitus included in his military manual a chapter headed On secret messages Ittmann (UKZN PMB) Math236 2012 4 / 18
Cryptology (cont.) For most of its history, cryptology has been the province of governments and the military With the increasing use of computer networks, commercial entities have become strongly interested in information security Today, the feasibility of internet commerce rests on the ability to conduct secure electronic transactions Ittmann (UKZN PMB) Math236 2012 5 / 18
Cryptology (cont.) We wish to send a message in such a way that it is unintelligible to all unauthorized persons, but can be understood by the intended recipient The plaintext M is a finite string of symbols from a finite alphabet Σ M is converted, by the process of encryption into an enciphered text called the ciphertext , C The person who enciphers M is called the sender and uses a set of rules (or algorithm ) to encrypt M He sends the ciphertext, C , to the (intended) recipient Ittmann (UKZN PMB) Math236 2012 6 / 18
Cryptology (cont.) Normally the operation of the algorithm involves the use of a key K which is known to both the sender and the receiver The receiver uses an algorithm (involving the key) to obtain M from C ; this is known as decryption Note that the ciphertext C and the key K must determine the plaintext M uniquely Ittmann (UKZN PMB) Math236 2012 7 / 18
Cryptology (cont.) We shall adopt the convention that plaintext is written lowercase and ciphertext uppercase For example, we might encrypt the word goodbye as AHYEKVA Any person who intercepts the message is called an interceptor In general, an interceptor will not know the key The methods used in the encryption/decryption form the subject of cryptography The methods used by the interceptor to derive M from C without having access to the key are studied in cryptanalysis Ittmann (UKZN PMB) Math236 2012 8 / 18
Types of ciphers We now study two classes of encryption schemes In a monoalphabetic cipher , each letter in the plaintext alphabet is always encrypted as the same letter in the ciphertext alphabet For example, if in the word banana the first a is encrypted as F , then the second and third letters a will be encrypted as F as well In a polyalphabetic cipher , a letter in the plaintext alphabet might be encrypted as several different letters in the ciphertext alphabet For example, the first a in the word banana might be encrypted as F while the second and third letters a are encrypted as Z and B Ittmann (UKZN PMB) Math236 2012 9 / 18
Types of ciphers (cont.) Monoalphabetic ciphers are cryptographically weak because they preserve the relative frequency with which each letter occurs in the plaintext language For example, if the plaintext language is English, an interceptor could guess that whichever letter occurs the most frequently in the ciphertext corresponds to the letter e in the plaintext In a simple substitution cipher, 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 Ittmann (UKZN PMB) Math236 2012 10 / 18
Substitution ciphers Example Suppose that the following set of substitutions (the key ) is used Both the encipherer and the decipherer have a copy of this key, which is simply a permutation of the letters of the alphabet Plaintext a b c d e f t u v w · · · Ciphertext D X W E G A B F R C · · · Then cat is enciphered as WDB and AGC is deciphered as few In a simple substitution cipher like this one, the re-ordered alphabet D X W E G A · · · B F R C · · · is called the substitution alphabet Ittmann (UKZN PMB) Math236 2012 11 / 18
Substitution ciphers (cont.) This is a poor system, as it is possible to cryptanalyze it in many cases Memorizing the key is difficult If the key is kept for reference, it can be lost or stolen Ittmann (UKZN PMB) Math236 2012 12 / 18
Shift ciphers In the Gallic wars Julius Caesar used a cipher in which each of the letters a,b, . . . ,z is replaced by the letter which occurs three places after it in the alphabet We can represent this with the following permutation Plaintext a b c d e w x y z · · · Ciphertext D E F G H Z A B C · · · We call this a shift cipher or additive cipher or translation cipher with shift (or key ) Ittmann (UKZN PMB) Math236 2012 13 / 18
Shift ciphers(cont.) More generally, in a shift cipher with shift d , each letter in the plaintext alphabet is encrypted as the letter that occurs d places further on in the alphabet Each letter in the ciphertext alphabet is decrypted by replacing each letter by one that occurs d places earlier on in the alphabet (or 26 − d places further on) As before, z is followed by a b c · · · Note that a shift cipher just a special case of the simple substitution cipher Ittmann (UKZN PMB) Math236 2012 14 / 18
Shift ciphers(cont.) Encryption can be done mechanically by means of a simple device consisting of a large disc on which there is a smaller disc (with the same centre) which can be rotated d places forward for encryption or d places back for decryption The key is easily remembered, but the cipher is so insecure that it is of no practical use, as an interceptor has to test at most 25 possible values of d to find the key Ittmann (UKZN PMB) Math236 2012 15 / 18
Shift ciphers(cont.) Example Suppose that the adversary, who knows that a shift cipher is being employed, intercepts the following ciphertext AOPZ TLZZHNL PZ H MHRL He tests values of d on the word MHRL and finds that d = 7 yields a plaintext of fake Additionally, d = 19 yields toys All other shifts (values of d ) result in unintelligible plaintext Ittmann (UKZN PMB) Math236 2012 16 / 18
Shift ciphers(cont.) Example He now turns his attention to the other words in the ciphertext If he decrypts PZ with d = 7, he gets is , while d = 19 yields wg So he chooses d = 7 and decrypts the ciphertext to find the plaintext message In this example, the words fake and toys are translates of one another Ittmann (UKZN PMB) Math236 2012 17 / 18
Shift ciphers(cont.) We know of no pairs of English words of length six or more that are translates of each other, and only a few of length four or five As we have mentioned, monoalphabetic ciphers are vulnerable to attack by a frequency analysis of letters, pairs of letters ( digrams ), triples of letters ( trigrams ), and so on Hence, if we seek a system which is secure against attack, it must be polyalphabetic Ittmann (UKZN PMB) Math236 2012 18 / 18
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