T215B Communication and information technologies (II) Session 3

T215B Communication and information technologies (II) Session 3
Arab Open University - Spring 2013 Block 4 Protecting and prying

Session Outline Part 5: Encryption Introduction
Encryption: basic concepts Breaking a cipher Building stronger ciphers Arab Open University - Spring 2013

1. Introduction [1] This part of the block is about the encryption of data and how it can be used to prevent unauthorised people from having access to private information  so both the ‘protecting’ and ‘prying’ themes of this block are featured here. The data that we need to protect and the opportunities available for prying into that data have already undergone a dramatic change over the last three or four decades and are likely to continue to do so at an accelerated rate. We live in a world where it becomes increasingly difficult to conduct many legal, financial or commercial transactions without sending personal details over electronic communication links. Arab Open University - Spring 2013

1. Introduction [2] ONE SUCH MEASURE IS ENCRYPTION.
For many people online transactions are an increasingly important means of conducting normal ‘citizenship’ activities such as renewing a television or motor vehicle licence and transferring bank or building society funds. It’s also become much quicker, easier and cheaper to collect, store, analyse and transmit data, but it’s all too easy to allow this data to ‘leak’ – sometimes with serious consequences. High-profile security breaches that may occur could be prevented by implementing appropriate protection measures. ONE SUCH MEASURE IS ENCRYPTION. Encryption is a method of altering data in a systematic way such that it can be restored to its original form by those ‘in the know’. Arab Open University - Spring 2013

1. Introduction [3] Some encryption techniques have been around for hundreds, even thousands, of years  the Caesar cipher But the real shift in encryption techniques over the last few decades has come about because now we have computers that can do the hard work of trying to break a code and, of course, they can do it much more quickly than a human can. This means that the encryption techniques employed have had to become far more complex and sophisticated. Arab Open University - Spring 2013

1. Introduction [4] The fundamental building block of all modern security systems is encryption. Encryption provides mechanisms for: confidentiality – keeping things secret authentication – ensuring that the identities of people and things are correct integrity – ensuring that data has not been tampered with This part of the block is designed to give an insight into encryption methods. Arab Open University - Spring 2013

2. Encryption: basic concepts [1]
Reminder: Encryption is a process by which information is changed in some systematic way so as to hide its content from everyone except its intended recipient. The branch of science concerned with this concealment of information is known as cryptology, a word that has its roots in Greek from kryptos (hidden) and logos (word). Cryptology is the study of codes and ciphers, and divides into two branches: cryptography, the science of creating codes and ciphers, and cryptanalysis, the science of breaking them. Cryptographers make a distinction between the terms ‘codes’ and ‘ciphers’, though in practice the two are often used interchangeably. Arab Open University - Spring 2013

In its pure sense, a code replaces whole words, phrases or groups of symbols with alternatives (or code words). The purpose of creating a code is not always for secrecy. Often a code is used simply as an abbreviation or used to provide an alternative way of communicating information. Two examples are ASCII and Morse code: ASCII: (American Standard Code for Information Interchange) This is used when storing and transmitting data, and uses only two different coding symbols (usually referred to as 1 and 0). Morse code: a standard for substituting groups of long and short pulses (or groups of dots and dashes) for letters It has been used extensively in telegraphy because of its resistance to corruption from other signals during transmission, and because of its efficiency. Arab Open University - Spring 2013

A code is the output of an encoding process (the reverse is decoding) and generally relies on sets of look-up tables (codebooks) for the conversion processes. When used for secrecy, the code becomes useless if the look-up tables fall into the wrong hands. Arab Open University - Spring 2013

A cipher, on the other hand, is the output of an operation that either replaces data symbols with alternative symbols, or rearranges existing symbols. In both cases the operation is done in a systematic way, following some set rules. A cipher is almost always created for reasons of secrecy. Encryption is the process of transforming data (known as plaintext) into a cipher (known as ciphertext). Decryption reverses the process by transforming ciphertext back into plaintext. Arab Open University - Spring 2013

There are two basic methods for creating a cipher: One is to take a symbol (or a group of symbols) in the plaintext and manipulate it in a systematic way to produce a different symbol (or group of symbols), which becomes the ciphertext. The substituted symbols in the ciphertext appear in exactly the same order as the original versions in the plaintext.  A cipher created using this approach is known as a substitution cipher. The second method is to ‘scramble’ the order of the symbols in some systematic way. Using this approach, the symbols remain unchanged between plaintext and ciphertext, but the ordering of those symbols changes.  A cipher created using this approach is known as a transposition cipher. In effect, the ciphertext is an anagram of the plaintext. Arab Open University - Spring 2013

2.1 A simple substitution cipher: the Caesar cipher [1]
One of the earliest recorded and best known ciphers was used by Julius Caesar in the 1st century BC and has since become known as the Caesar cipher. This is also one of the simplest of substitution ciphers. One of the methods Caesar used to preserve the confidentiality of a message was to substitute each letter in his message with the letter three places further forward in the alphabet. This is an example of the systematic manipulation. Thus the letter ‘a’ would be substituted by the letter ‘d’, the letter ‘b’ by the letter ‘e’, and so on. Arab Open University - Spring 2013

Example: using this method, the word ‘acme’ becomes DFPH. But what if I wanted to encrypt the word ‘zenith’ using the Caesar cipher? The letter ‘z’ is the final letter of the alphabet. The solution is to jump back to the letter ‘a’ and continue the count as if the letters of the alphabet were arranged in a circle  ‘zenith’ then becomes CHQLWK. Study note: When giving examples of encryption, a convention often used is to show plaintext in lower case and ciphertext in UPPER CASE. Arab Open University - Spring 2013

When Augustus Caesar succeeded Julius Caesar, he changed the shift from 3 to 2, producing different ciphertext from a given plaintext. Indeed the choice of the shift is arbitrary; any shift of 1 to 25 would work equally as well, though of course the intended recipient for the encrypted text would need to know the choice in order to carry out the decryption process. The circular nature of the Caesar cipher can be exploited to produce a simple encryption tool known as a cipher wheel. Arab Open University - Spring 2013

Arab Open University - Spring 2013

The wheel is made up of two discs, one slightly smaller than the other. The alphabet is written around the circumference of both discs and the discs are fitted together at their centres in such a way that one can be rotated relative to the other, so any letter on the outer wheel can be aligned with any letter on the inner wheel. Both the sender and the recipient need their own cipher wheel. The starting point for its use is with the wheels set so that each letter on the outer wheel is aligned with the corresponding letter on the inner wheel. The sender and recipient first agree on the number of shifts. Arab Open University - Spring 2013

When an encryption method can be carried out systematically by following some sort of set pattern or procedure, such a procedure is known as an algorithm. When the algorithm includes a variable that can be altered to produce a different outcome, the variable is called a key. So here we can say that Julius Caesar used a key of 3 and Augustus Caesar a key of 2. Figure 5.3 (next slide) gives a graphical representation of the use of the encryption algorithm and the encryption key in the encryption process. Arab Open University - Spring 2013

2.2 Mathematical representation [1]
Modern communication systems use computers to process messages and computers do not work with letters but with numbers. In this section, we show how the Caesar cipher can be represented as a numerical algorithm that can be processed by a computer. Another way of looking at Caesar cipher (cipher wheel) is that the alphabet on each disc is arranged rather like the numbers on a clock face but using letters instead. In fact, if we were to represent each letter of the alphabet as a number, it would look a little like a clock with 26 different numbers rather than the 12 we’re used to. Arab Open University - Spring 2013

To represent this mathematically we use “Modular arithmetic” Modular arithmetic operates with a limited set of integers (integers are all the positive and negative whole numbers, including zero). The number of integers in the set is known as the modulus. Using the clock example, with a conventional 12-hour clock the modulus is 12; for a 24-hour clock, the modulus would be 24; in our alphabet example for the Caesar cipher, the modulus is 26. Whatever mathematical operation we perform on these integers, the result must always be less than the modulus. Arab Open University - Spring 2013

Explanation of Modular Arithmetic: Let’s assume we want to move forward eight hours from ten o’clock using the 12-hour clock, how can we implement this? Mathematicians have a special way of expressing a calculation like this by saying that: modulus 12 is congruent to 6 modulus 12. In other words: First add the two left-hand integers together in the conventional way: = 18 If the result is equal to or greater than the modulus, subtract the modulus from the result, repeating the subtraction as necessary until the result is less than the modulus  18 − 12 = 6 Now express the answer as a congruence modulus 12: ≡ 6 mod 12 Arab Open University - Spring 2013

Now we apply the same method to calculations for encryption using the Caesar cipher. First we convert the letters of the alphabet to numbers so that we can operate on them mathematically. We convert ‘a’ to 0, ‘b’ to 1, ‘c’ to 2 right through to ‘z’ to 25 as shown below. Arab Open University - Spring 2013

Why we have chosen to set ‘a’ to 0 rather than to 1? This is because the result of any calculation in modular arithmetic must always be less than the modulus. So if we had set ‘a’ to 1 and therefore ‘z’ to 26, 26 would be an invalid result. NOW, to encrypt the letter ‘z’ with a Caesar cipher using a key (shift) of 3 would give: ≡ 2 mod 26 The letter ‘c’ is represented by the number 2, so ‘z’ encrypts to C. Arab Open University - Spring 2013

Modular arithmetic can be expressed in general terms by using letters in place of numbers. Conventionally the modulus is expressed as n, and within the context of encryption: p is used to represent the plaintext (the unencrypted text) c is used to represent the ciphertext (the encrypted text) K is used to represent the key. So, the general algorithm for the encryption process using modular addition becomes: p + K ≡ c mod n Arab Open University - Spring 2013

2.3 Decrypting the Caesar cipher [1]
Using the cipher wheel the decryption process simply involves displacing the outer wheel clockwise a number of places corresponding to the agreed key and translating each ciphertext letter shown on the inner wheel to its equivalent plaintext letter on the outer wheel. Using Julius Caesar’s version of the cipher this would require a clockwise displacement of three places. This would be just the same as displacing the outer wheel 23 places in an anticlockwise direction. an anticlockwise displacement of 23 (or 26 − 3) is the equivalent of a clockwise displacement of 3. Thus 3 and 23 form a complementary pair Arab Open University - Spring 2013

In general terms, keys are signified by K for the encryption key and 𝐾 (read as K bar) for the decryption key. Mathematically the decryption algorithm would be expressed as: p ≡ c + 𝐾 mod 26 Activity 5.3 (self-assessment): What are the decryption keys for the Caesar cipher with encryption keys of: 10 15 7 Arab Open University - Spring 2013

Note that in practice one key is so easy to derive from the other that effectively they can be regarded as a single key. So if we know the encryption key we also know the decryption key, or we can decrypt the ciphertext by reversing the encryption algorithm. Encryption systems like this are known as symmetric key systems because effectively only a single key is involved in the encryption and decryption processes. Arab Open University - Spring 2013

2.4 A simple transposition cipher [1]
A transposition cipher is, in effect, an anagram of the plaintext. But for an anagram to be classed as a cipher, it must have been created in some systematic way using a method that can be shared with the intended recipient so that it can be decrypted. There are many ways this systematic process can be done. One way to create the transposition is to use a matrix of cells and to write the message a letter at a time in sequential cells across the matrix. Encryption is performed by reordering the columns of the matrix in some systematic way and then reading off the result to produce the ciphertext. This kind of cipher is known as a columnar transposition cipher. Arab Open University - Spring 2013

A possible approach to this task is for the sender and receiver to agree on a codeword and a way to reorder the letters in the keyword into an anagram. Let’s say that the codeword is Tuesday and the agreed transposition is to reverse the order of the letters (YADSEUT) and then swap pairs of letters, starting at the right-hand end to produce the anagram YDAESTU. The number of letters in the keyword dictates the number of columns in the matrix, and the plaintext is entered into each of the columns (with the keyword at the top) a letter at a time working across the rows. Any empty places in a row can be padded with redundant letters (the ‘x’ in my example). Arab Open University - Spring 2013

Columnar Transposition Cipher: Arab Open University - Spring 2013

The columns are then reordered according to the keyword anagram. The ciphertext is given by reading back the letters from the reordered matrix. Arab Open University - Spring 2013

Thus, in example, the message: “Mary had a little lamb its fleece was white as snow” is encrypted as: “DHARYMAETLITALSITMBLAWCEEEFLEITWHASXOWSNAS” Arab Open University - Spring 2013

There are many variations of transposition ciphers. One of the earliest recorded originated in Sparta in the 5th century BC. It used a wooden pole (or staff) known as a “scytale” A strip of parchment or leather was wound around the pole so that it formed a sleeve. The message was written in rows along the length of the sleeve so that when it was unwound the letters of the message were transposed into a different order. To reconstruct the original message a pole of the correct diameter was needed. Arab Open University - Spring 2013

3. Breaking a cipher [1] The act of breaking (or cracking) a cipher is to derive the plaintext from the ciphertext without knowledge of the key (and often without knowledge of the encryption algorithm). The strength of a cipher is measured by how long it takes to break it. Notice that we said ‘how long it takes to break it’ and NOT ‘whether it can be broken’. Potentially all known ciphers except one are thought to be breakable ! (We will talk about the one exception later) Often, though, the time and effort required to break a cipher is not justified by the value of the information retrieved. Also, the cipher may take so long to break that by the time the information is retrieved it has lost its value, for example by being out of date. Arab Open University - Spring 2013

3. Breaking a cipher [2] Ciphers that are described as strong are those that take a long time to break, but they also tend to be more difficult to use. Weak ciphers, on the other hand, are quicker to break but are usually also quite easy to use. It’s worth bearing in mind that the use of any cipher has an overhead in terms of time and processing demands, so the choice of cipher will usually be determined by the value of the information it is designed to protect. In this section, we will describe two of the main approaches to breaking a cipher: Brute force attack Linguistic analysis Arab Open University - Spring 2013

3.1 Brute force attack [1] Imagine that you have a bunch of keys and you know that one of them (but not which one) will unlock the door to a room you wish to enter. The obvious thing to do is to try every key in the lock in turn. If you are lucky, the first one you try will open the door. If you are unlucky it may be the last one.  The probability is that you will only have to try half of them before you find one that fits. A similar method to this can be used to break a cipher using a known algorithm. For example, if you have a ciphertext message that you know has been encrypted using the simple Caesar cipher described earlier, how many keys would you need to try before you could be certain of finding the right one?  The answer is 26, since there are 26 possible keys that could be used with this algorithm. Arab Open University - Spring 2013

3.1 Brute force attack [2] The number of possible key combinations for a particular algorithm is known as its key space. This method of trying all possible combinations in a key space is known as a brute force attack. Clearly the time taken to break a cipher by this method alone is directly proportional to the key space. The Caesar cipher has a very small key space and so can be broken very quickly. This idea of a brute force attack can be applied to transposition ciphers as well as substitution ciphers.  This might require testing every permutation of the possible transpositions until the correct one is found. Arab Open University - Spring 2013

3.1 Brute force attack [3] Activity 5.5: How many different arrangements would be possible using the seven letters of the word ‘article’? Sol: Each letter in the word ‘article’ appears only once. Taking one letter at a time, the first can appear in any of the seven positions; the second in any of the 6 remaining positions; the third in any of the five remaining positions; and so on. This gives a total possible number of combinations of 7! = 7 × 6 × 5 × 4 × 3 × 2 × 1 = 5040 Arab Open University - Spring 2013

3.1 Brute force attack [4] From Activity 5.5, working through all possible 5040 permutations using a pencil and paper would take quite a long time to do. However, a computer would be able to yield the correct answer in a fraction of a second! For example: Let’s estimate this for a computer that could perform one thousand billion calculations every second, that is 1000 × or 1 × So to perform 5000 = 5 × calculations would take roughly: 5 × × = 5 × 10 −9 Seconds = 5 nanoseconds. Arab Open University - Spring 2013

3.1 Brute force attack [5] Activity 5.6: Using a computer that can perform calculations a second, roughly how long would it take to try all possible permutations of: (a) 10 different letters (b) 15 different letters (c) 20 different letters. Arab Open University - Spring 2013

3.1 Brute force attack [6] Activity 5.6 – Sol. :

3.1 Brute force attack [7] If we were to repeat the calculation for a transposition cipher of 26 different letters , we would find that it would take the computer some 12.7million years to try all the possible combinations! This is not to say that it would be necessary to work through all the possible permutations until the correct one was found. The probability of getting the correct plaintext at the first try is 1 in 26! (or 1 in about 4 × ) Important note: the number of possible permutations to crack a transposition cipher depends on the content of the message. Arab Open University - Spring 2013

3.1 Brute force attack [8] A transposition cipher is incapable of encrypting a string of identical characters and weak when there are long blocks of identical characters within the string. This is quite a serious flaw in situations where there is a requirement to encrypt long strings of machine code, or binary representations of non-text data (such as pictures) that can have long blocks of identical symbols. However, given text with normal language characteristics, a transposition cipher can be strong against a brute force attack. Arab Open University - Spring 2013

3.2 Linguistic analysis [1]
A second approach to cipher breaking is to attempt to exploit any linguistic patterns inherited from the plaintext. (This, of course, is only applicable to encrypted messages with a textual content.) All written languages exhibit characteristic patterns. For example, in written English the letter ‘q’ is almost always followed by a ‘u’ and certain pairs of letters (known as digraphs) are more likely to appear together than others: ‘th’ is common as are ‘ea’, ‘of’, and ‘st’. Some identical letters can often appear together, for example ‘ee’, ‘oo’, ‘tt’, but seldom ‘uu’ or ‘hh’. Furthermore, there is a higher probability of certain letters appearing than other letters The most common letter is ‘e’, followed by ‘t’, ‘a’ and ‘o’. Sources disagree on the exact ordering of probability, but most will identify ‘etaoin’ as the order of the six most frequent letters. Arab Open University - Spring 2013

A ciphertext sequence can also be analysed for letter frequency. If the analysis shows a curve similar to Figure 5.8 or Figure 5.9 (depending on how the results are ordered) then there is a strong possibility that the most frequently occurring ciphertext letters will correspond to their equivalents in standard English text. Activity 5.7 (exploratory): Figure 5.10(a) (Next Slide) shows the results of letter frequency analysis of a sample of ciphertext. For comparison, Figure 5.10(b) shows the standard written English letter frequency. Does the ciphertext exhibit a pattern similar to standard English letter frequencies? If so, what assumptions could you make about the cipher? Arab Open University - Spring 2013

Activity 5.7 – Comment: We can see some very clear similarities between the two graphs. The plots between R and Z in Figure 5.10(a) show a similar pattern to the plots between A and I in Figure 5.10(b). Likewise there are similarities in the sections between B and G and I and P in Figure 5.10(a) with the corresponding sections between K and P and R and Y in Figure 5.10(b). In fact, if the top plot was shifted to the left by 17 places the result would look very similar to the standard English plot. Since the top plot exhibits a similar letter frequency pattern to standard English, it is reasonable to assume that the cipher was created using a simple substitution cipher where the ordering of the letters was preserved in the ciphertext but their identity was changed.  The guess would be that the cipher used was a simple Caesar cipher with a key of 17. Arab Open University - Spring 2013

We have focused on exploiting the linguistic patterns in written text to break substitution ciphers. But the same linguistic patterns can prove useful for breaking simple transposition ciphers too. In a transposition cipher, the positions of the letters change but their identity remains the same so, for example, there will be the same frequency of the letter “E” in the ciphertext as there is in the plaintext. Given a long enough sample of transposition ciphertext, frequency analysis can provide a useful starting point for the cryptanalyst. If the results indicate a match with standard letter frequencies then the ciphertext is most likely the result of a transposition cipher. This knowledge points to various analytical techniques to help break the cipher, exploiting the kind of patterns we discussed. Arab Open University - Spring 2013

4. Building stronger ciphers [1]
From the discussion above, two desirable characteristics emerge for strong ciphers: a very large key space a weak association with the linguistic patterns in the plaintext. This section looks at how ciphers can be designed to address these requirements. Arab Open University - Spring 2013

4.1 Increasing the key space [1]
Using the simple Caesar cipher, the maximum key length is 26. But what if instead we modified the encryption algorithm to encrypt letters as pairs (digraphs) instead of singly? Since each single letter represents one of 26 possibilities, each pair of letters would represent one of 26 × 26 = 26 2  giving 676 different possibilities; So the use of digraphs provides a means of increasing the key space. Arab Open University - Spring 2013

This is how it works: The numerical equivalent of the first letter in the pair is multiplied by 26 and then the numerical equivalent of the second letter in the pair is added to it. Thus, using the coding scheme given below, where p is set to 15 and b is set to 1, the digraph pb would be encoded as (15 × 26) + 1 = 391 Arab Open University - Spring 2013

Activity 5.10: Using the method described above, calculate the coded value for the first letter pair in the word ‘zenith’. Sol.: Using digraphs the word ‘zenith’ would be treated as three separate letter pairs: ze ni th. The first of these is the pair ‘ze’. Using the scheme in Figure 5.11, ‘z’ has the code 25 and ‘e’ has the code 4. So the letter pair ‘ze’ is coded as (25 × 26) + 4 = 654 Arab Open University - Spring 2013

Process of encrypting the digraph ‘ze’ using modular arithmetic: Using digraphs for coding and encrypting alphabetic symbols: (a) what value would be used for the modulus? SOL. - (a): As you saw earlier, a coded digraph can take one of a possible 26 × 26 = 676 different combinations. So the modulus would be 676. (b) what values could be chosen for the key? SOL. -(b): The key can be any number from 0 to 675 Arab Open University - Spring 2013

Let’s select the key 347 so for the encryption algorithm: the plaintext p = 654 (this is the numerical value of the digraph ‘ze’) the modulus n = 676. So: c ≡ = 1001, but this result is not permitted in modular arithmetic with a modulus of 676. So we need to subtract 676 from 1001 and express the result as the remainder: c ≡ 325 mod 676 The next step is to convert 325 into the equivalent digraph to derive the ciphertext. First we divide the numerical value for the ciphertext by 26: 325 ÷ 26 = 12.5 The whole number part of this is 12 which, according to the coding scheme of Figure 5.11, equates to the letter M. The 12 accounts for 12 × 26 = 312 of the original code of 325, leaving a remainder of 13, which equates to the letter N. So the plaintext digraph ‘ze’ is encrypted as MN. Arab Open University - Spring 2013

4.2 Weakening the linguistic association [1]
The Caesar cipher uses a monoalphabetic substitution. This means that the key remains constant so the plaintext letter ‘e’, for example, will always be encrypted as the same ciphertext symbol. This means that the ciphertext inherits the linguistic patterns of the plaintext, making it susceptible to letter frequency and other linguistic analysis. One way to decouple from the linguistic patterns of the plaintext is to encrypt with a cipher that uses a succession of different keys. The Vigenère cipher provides an example of how this can be achieved. Arab Open University - Spring 2013

The Vigenère cipher uses a key known as a running key, which is generated by a keyword. Example: Let’s use the keyword ‘jupiter’. Using the convention of assigning a numerical value to each letter following the pattern ‘a’ = 0, ‘b’ = 1, ‘c’ = 2, and so on (see Figures 5.5 and 5.11), ‘jupiter’ would produce a key sequence of 9, 20, 15, 8, 19, 4, 17. This provides a succession of different keys that can be used over and over again in the same sequence. Each symbol in the plaintext is encrypted using the next key in the sequence. Here we will encrypt a fragment of the proverb ‘a stitch in time saves nine’ using the simple Caesar cipher with a running key derived from the keyword ‘jupiter’ Arab Open University - Spring 2013

4.3 An unbreakable cipher [1]
A solution that addresses the running key length vulnerability of the Vigenère cipher is to use a key that is identical in length to the plaintext, and to use it only once. Encryption based on this method is known as one-time pad encryption. This is how the method works: A random key (the pad) is generated that is at least as long as the plaintext message to be encrypted Two copies of the pad are required – one is used by the sender to encrypt the message and one by the recipient to decrypt it. Once it has been used both copies must be destroyed. When properly applied, the one-time pad is the only known truly unbreakable cipher. The random nature of the pad (key) means that the ciphertext has no linguistic association whatsoever with the original plaintext and so cannot be broken by analysis or brute force. Arab Open University - Spring 2013

4.3 An unbreakable cipher [2]
One-time pads have been successfully used in the past, mostly for high-level diplomatic exchanges. But a one-time pad is not a practical cipher for use in modern communication technologies. Arab Open University - Spring 2013

T215B Communication and information technologies (II) Session 3

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