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Handout 1 I. Historical Overview Steganography Code Secret Writing Substitution Cryptography Cipher Transposition

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Handout 1 I. Historical Overview Steganography Code Secret Writing Substitution Cryptography Cipher Transposition 06-20008 Cryptography The University of Birmingham Autumn Semester 2012 School of Computer Science Eike Ritter 24 September, 2012 Handout 1 Summary of this handout: Overview of historical cryptographic techniques — Definition of some im- portant cryptographic concepts — Math: Permutations and Residue Classes I. Historical Overview 1. Secret Writing Communicating secretly in writing is probably as old as writing itself. Already the Egyptians are known to have included secret hieroglyphs in their writing. In ancient times secret writing has mainly been used for religious purposes and mysticism. In the classical antiquity are the first accounts of secret writing being used for military purposes by the ancient Greeks. From there on the ability to secretly pass messages was a major factor for military and diplomatic communication. It is therefore little surprising that many of the techniques for secret communication known today have been developed by government agencies and often only made known to the public long after their invention. Since the ancient Greeks many methods of exchanging secret messages have been developed (and successfully broken). This handout gives an overview over some of the historical techniques and also introduces some basic and simple mathematical concepts used. The Science of Secret Writing can be roughly divided into the following branches: Steganography Code Secret Writing Substitution Cryptography Cipher Transposition 2. Hiding Messages The best form of sending a secret message is by concealing that a message has been sent. The study of how to effectively hide messages such that only the intended receiver can find them is called Steganog- raphy. Traditionally secret messages used to be hidden physically, for instance by using invisible ink, or within unobtrusive communications. Nowadays there are many more ways of sending concealed mes- sages, for instance by hiding them in unsuspicious file formats, such as pictures or sound files, or in propaganda videos. In this course we will be not concerned with techniques to hide messages but only with hiding the information contained in a message. 3. Scrambling Messages In many situations it is obvious that messages are sent. For instance, it is clear that during a phone conversation some information is communicated, thus “hiding the message” is essentially impossible. To guard against interception or eavesdropping, messages can be scrambled in a way that they only make sense to the receiver but not to an uninitiated eavesdropper. The scientific study of scrambling messages is the mathematical discipline of Cryptography. In this handout I will give a short overview over some historical cryptographic techniques and their underlying mathematics. 4. Cryptography can be divided into two main branches: Transposition techniques encrypt messages by permuting the components of a message within itself. Substitution techniques encrypt messages by replacing the components of a message. There are two main branches for substitution techniques: 4 Codes are algorithms that substitute entire words or phrases of a message, i.e. they work on a level of meaning. Ciphers are algorithms that substitute the single letters or characters of a message, i.e. they work on a level of individual letters, small groups of letters, or individual bits. 5. Some basic definitions Let’s first define some more concepts that will be useful throughout the lecture. Encrypt The process of scrambling a message. Decrypt The process of unscrambling a message. Encode The process of encrypting a message with respect to some code. Decode The process of decrypting a message with respect to some code. Encipher The process of encrypting a message with respect to some cipher. Decipher The process of decrypting a message with respect to some cipher. Plaintext A message before it is encrypted. Codetext An encoded message. Since we will concentrate on ciphers in this lecture we will rarely use this term. Ciphertext An enciphered message. Cryptography The science of keeping messages secure. Cryptographer A practitioner of cryptography. Cryptanalysis The science of breaking ciphertexts. While we will mainly concentrate on cryptography in this course I will also discuss some cryptanalytic techniques. Cryptanalyst A practitioner of cryptanalysis. Cryptology The branch of mathematics encompassing both cryptography and cryptanalysis Cryptologist A practitioner of cryptology. 6. Sending Secret Messages We can describe the process of sending encrypted messages with the following schematic view: Plaintext Encryption Cipher-/Codetext Decryption Orig. Plaintext From a practical point of view, codes are relatively useless as they require far too much overhead to be effective. In theory, one would have to design an entire dictionary of code words before commencing communication. Obviously, this makes it not only difficult to work with a code, but also insecure as a code book can not easily be hidden or memorised, as well as very costly to change a code. While codes have a certain usefulness for ad hoc communication, they are not relevant for the type of efficient cryptography we are interested in. With ciphers on the other hand one only has to know how to encipher and decipher the basic set of symbols, for example the 26 letters of the alphabet. Therefore, ciphers are far easier to apply, to keep secret and to change in order to guarantee prolonged secure communication. In this lecture we will therefore discuss cryptographic methods based on ciphers and will start with a look at some of the classical ciphers. 5 I.1 Transposition Ciphers The basic idea of a transposition cipher is to scramble a message by transposing the single letters of a message in a particular order that is only known to the sender and receiver. Transposition ciphers are the oldest form of enciphering messages. They can be traced back to the ancient Egyptians where they were mainly used for religious purposes. The order of hieroglyphs of single words would be changed probably for mystical purposes. Similarly the Jewish mystics used temurah to rearrange words and sentences to find hidden meanings in the Bible. 7. Rail Fence Cipher The first known military use of a transposition cipher was by the Spartans. They invented the rail fence cipher to send messages between troops. We write the plain text message in columns of a particular height and then compose the ciphertext from the resulting rows. To decipher we divide the length of the overall message by the height of the columns and break the ciphertext into rows whose length corresponds to the quotient. Writing the rows again one upon the other, we can read off the original plaintext. Example: We take the plaintext: When shall we three meet again in thunder lightning or in rain We then write it as a rail fence of height 5: WHEEAINIIIN HATMGNDGNNE ELHEATEHGRU NLREIHRTOAK SWETNULNRIE Observe that the original message had 51 characters. In order to complete the rail fence I added four nonsense characters. These characters are meaningless and can be discarded after deciphering. We call meaningless characters added to the ciphertext nulls. Reading off row by row gives the following ciphertext: WHEEAINIIINHATMGNDGNNEELHEAREHGRUNLREIHRTOAKSWETNUL NRIE In order to decrypt the message the receiver, who knows the height of the rail fence, divides the length of the message by the height and thus gets the length of the single rows. In our case we get 55/5 = 11: WHEEAINIIINHATMGNDGNNEELHEAREHGRUNLREIHRTOAKSWETNULNRIE Adding nulls is not strictly necessary, we could have just combined the rows of different lengths and nevertheless could have decrypted the message without problems. In fact not adding nulls can make the work for a cryptanalyst more difficult. But in either case the rail fence cipher is very easy to break as we only have to test all possible divisors up to half the length of the text. 8. Permutation Cipher We can obtain a more elaborate cipher by transposing the characters in the plaintext in a less regular fashion using permutations. Suppose we take a permutation s ∈ Sn, we then divide the plaintext into chunks of length n and permute the letters in each chunk according to our chosen s, which results in the −1 ciphertext. To decipher the text we simply take the inverse of s, s ∈ Sn and apply it to the ciphertext divided into chunks of length n. 123456 Example: We take s = = (1 2 3)(5 6) ∈ S6 as permutation. 231465 We divide our plaintext from above into chunks of length 6. WHENSHALLWETHREEMEETAGAININTHUNDERLIGHTNINGORINRAINEUK Applying the permutation yields EWHNHSLALWTEEHREEMAETGIANNITUHENDRILTGHNNIRGOIRNNAIEKU and thus the ciphertext EWHNHSLALWTEEHREEMAETGIANNITUHENDRILTGHNNIRGOIRNNAIEKU In order to decrypt the message we use the inverse of s, which is (1 3 2)(5 6). 6 I.2 Substitution Ciphers A substitution cipher is a method to replace units of the plaintext with parts of the ciphertext. Substitution ciphers are nearly as old as transposition ciphers. One of the earliest examples is the Atbash, used in Jewish mysticism, a cipher that substitutes the first letter of the Hebrew alphabet with the last, the second letter with the one before last, and so on. For simplicity, we will for now assume that the units of our plaintext are always the 26 letters of the alphabet only, without spaces, punctuation, etc. 9. Shift Cipher and Caesar’s Cipher The oldest form of substitution ciphers is the shift cipher. Its idea is to shift every letter in the message by a set number of places in the alphabet. For example a shift cipher with a shift of 1 simply replaces every letter by its successor and Z by A. More mathematically, if we enumerate all letters of the alphabet from 0 to 25 then we can express the shift cipher as an operation in modular arithmetic as c ≡ p + s(mod 26), where p is the plaintext letter, s is the shift, and c is the enciphered letter. Decryption can be performed similarly with the reverse operation: p ≡ c − s(mod 26).
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