# 8. Encryption coding¶

## 8.1. What’s the big picture?¶

Encryption is used to keep data secret. In its simplest form, a file or data transmission is garbled so that only authorised people with a secret key can unlock the original text. If you’re using digital devices then you’ll be using systems based on encryption all the time: when you use online banking, when you access data through through wifi, when your web browser remembers your password, when you pay for something with a credit card (either by swiping, inserting or tapping), in fact, nearly every activity will involve layers of encryption. Without encryption, your information would be wide open to the world — anyone could pull up outside a house and read all the data going over your wifi, and stolen laptops, hard disks and SIM cards would yield all sorts of information about you — so encryption is critical to having computer systems work at all.

Of course, we wouldn’t need encryption if we lived in a world where everyone was honest and could be trusted, and it was ok for anyone to have access to all your personal information such as health records, online discussions, bank accounts and so on, and if you knew that no-one would interfere with things like aircraft control systems and computer controlled weapons. However, information is worth money, people value their privacy, and safety is important, so encryption has become fundamental to the design of computer systems. Even breaking the security on a traffic light system could be used to personal advantage!

Curiosity

An interesting example of the value of using encryption outside of secret messages is the two engineers who were convicted of changing traffic light patterns to cause chaos during a strike http://latimesblogs.latimes.com/lanow/2009/12/engineers-who-hacked-in-la-traffic-signal-computers-jamming-traffic-sentenced.html. A related problem in the US was traffic signals that could respond to codes from emergency vehicles to change to green; originally these didn’t use encryption, and people could figure out how to trigger them to their own advantage.

Like all technologies, encryption can be used for good and bad purposes. A human rights organisation might use encryption to secretly send photographs of human rights abuse to the media, while drug traffickers might use it to avoid having their plans read by investigators. Understanding how encryption works and what is possible can help to make informed decisions around things like freedom of speech, human rights, tracking criminal activity, personal privacy, identity theft, online banking and payments, and the safety of systems that might be taken over if they were “hacked into”.

An encryption system is really two programs: one to encrypt some data (referred to as plaintext) into a form that looks like nonsense (the ciphertext), and a second program that can decrypt the ciphertext back into the plaintext form.

A big issue with encryption systems is people who want to break into them and decrypt messages without the key. Some systems that were used many years ago were discovered to be insecure because of attacks, so could no longer be used. It is possible that somebody will find an effective way of breaking into the widespread systems we use these days, which would result in a lot of chaos!

Jargon Buster

There are various words that can be used to refer to trying to get the plaintext from a ciphertext, including decipher, decrypt, crack, and cryptanalysis. Often the process of trying to break cryptography is referred to as an “attack”. The term “hack” is also sometimes used, but it has other connotations, and is only used informally.

Of course, encryption doesn’t fix all our security problems, and because we have such good encryption systems available, information thieves must turn to other approaches, especially social engineering. The easiest way to get a user’s password is to ask them! A phishing attack does just that, and there are estimates that as many as 1 in 20 computer users have given out secret information this way at some stage.

Other social engineering approaches that can be used include bribing or blackmailing people who have access to a system, or simply looking for a password written on a sticky note on someone’s monitor! Gaining access to someone’s email account is a particularly easy way to get lots of passwords, because many “lost password” systems will send a new password to their email account.

Jargon Buster

When describing an encryption scenario, cryptographers often use the fictitious characters “Alice” and “Bob”, with a message being sent from Alice to Bob (A to B). We always assume that someone is eavesdropping on the conversation (in fact, if you’re using a wireless connection, it’s trivial to pick up the transmissions between Alice and Bob as long as you’re in reach of the wireless network that one of them is using). The fictitious name for the eavesdropper is usually Eve.

People who try to decrypt messages are called cryptanalysts; more informal terms like hackers and crackers are sometimes used, generally with the implication that they have bad intentions. Being a cryptanalyst is generally a good thing to do though: people who use encryption systems actually want to know if they have weaknesses, and don’t want to wait until the bad guys find out for them. It’s like a security guard checking doors on a building; the guard hopes that they can’t get in, but if a door is found unlocked, they can do something about it to make sure the bad guys can’t get in. Of course, if a security guard finds an open door, and takes advantage of that to steal something for themselves, they’re no longer doing their job properly!

Curiosity

There are several other characters used to describe activities around encryption protocols: for example Mallory (a malicious attacker) and Trudy (an intruder). Wikipedia has a list of Alice and Bob’s friends.

## 8.2. Substitution ciphers¶

For these activities, you will need to have pen and paper in front of you to figure out the answers in this section.

Working in a group with 1 or 2 of your classmates (or by yourself if nobody else is around), can you figure out what the following message, encrypted with a simple cipher, says?

DRO BOCMEO WSCCSYX GSVV ECO K ROVSMYZDOB,
KBBSFSXQ KD XYYX DYWYBBYG.
LO BOKNI DY LBOKU YED KC CYYX
KC IYE ROKB DRBOO
LVKCDC YX K GRSCDVO.
S'VV LO GOKBSXQ K BON KBWLKXN.

• What techniques did your group use to decrypt the message?
• If you haven’t already, write out each letter in the alphabet, and then the letter that it corresponds to in the cipher (for the ones that are known, i.e. actually were in the cipher). Can you see a pattern?
• If you were going to make a secret message of your own using this same cipher, how would you go about it?
• What would be wrong with using this cipher method for a secret you’d never want anybody else finding out?

You may have realised that there was a pattern in how letters from the original message corresponded to letters in the decoded one: a letter in the original message is decoded to the letter that is 10 places before it in the alphabet. The conversion table you drew should have highlighted this. Here’s the table for the letter correspondences, where the letter “K” translates to an “A”

The same idea can generate other codes, such as the following one where each letter is replaced with the one that is 8 places earlier.

We sometimes say that the alphabet has been rotated by 8. This system of rotating each letter in a piece of text by a certain amount in order to encrypt it is called Caesar Cipher, named after Julius Caesar, who used it with a rotation of 3 to disguise messages.

You can experiment with this cipher using this interactive.

#### Ciphertext

In this system, the amount of rotation is referred to as a key, since you can unlock the message if you know the key. Normally the sender and receiver would agree on a key in advance (and in person), so that the receiver can easily unlock the message.

However, this encryption method isn’t very secure, and you’ve probably already figured out how to crack a coded message. You actually only need to work out what one of the corresponding letters is, and then use that to calculate what the rotation is, which immediately gives you the key.

If for example you identify that the letter “Y” in the encrypted message is in place of the letter “R”, you can calculate the rotation by working out how many places R is before Y in the alphabet (it might help to write the alphabet out on a piece of paper so that you can count the places, as saying the alphabet backwards is quite challenging for most people!) As R is 7 places before Y, this means that the rotation for this cipher must be 7, and you should be able to convert all letters in the encrypted message to an understandable message by subtracting 7 from them. The quickest way of going about this though would be to write out a conversion table like the ones above.

Curiosity

The Caesar cipher with a key of 13 is the same as an approach called ROT13 (rotate 13 characters), which is sometimes used to obscure things like the punchline of a joke, a spoiler for a story, the answer to a question, or text that might be offensive. It is easy to decode (and there are plenty of automatic systems for doing so), but the user has to deliberately ask to see the deciphered version. A key of 13 for a Caesar cipher has the interesting property that the encryption method is identical to the decryption method i.e. the same program can be used for both. Actually, many strong encryption methods try to make the encryption and decryption processes as similar as possible so that the same software and/or hardware can be used for both parts of the task, perhaps with only minor adaptions.

Taking advantage of the idea that you only need to figure out 1 letter to decide, can you figure out what the following message says? Which letter is the best one to try and guess? Why? What was the rotation? You may make a few incorrect guesses before figuring it out, so be prepared for that! Once you think you know what one of the letters in the ciphertext might correspond to in the plaintext, work out what the rotation is, and then write out the conversions for that rotation and decode the start of the message using that conversion table to see whether or not it makes sense. If the first few words seem to be meaningless, then that rotation is probably not the correct one.

P  OVWL  AOPZ  TLZZHNL  DHZ  UVA  AVV  KPMMPJBSA  MVY  FVB!


Now that you know how to decipher a message that is using Caesar Cipher without actually knowing the key, you should be able to see that it would be very easy to decipher a message if you know the key. The following message was encrypted using a rotation of 6. Generate the conversion table for a rotation of 6. This should allow you to easily decipher the following message. What does the message say? (Use only the conversion table to figure it out!)

ZNK WAOIQ HXUCT LUD PASVY UBKX ZNK RGFE JUM.


It shouldn’t be too difficult to see how a message can be encrypted using Caesar Cipher. Previously, you were generating conversion tables that converted from the ciphertext to the plaintext. In a very similar way, you can generate conversion tables that convert from the plaintext to the ciphertext. The only difference is that instead of subtracting the rotation, you are adding it. i.e. if the rotation was 5, then the letter “H” in the plaintext would go to the letter that is 5 places forward in the alphabet, which is “M”.

Using a rotation of 3, generate a conversion table, and then the ciphertext, for the following message.

HOW ARE YOU


Now that all that is out of the way, you can encrypt your own messages (assuming it doesn’t matter too much if somebody deciphers them — as you saw above, this is not a very secure cipher!). Decide on a message to encrypt, and a rotation key. Generate a conversion table, and then encrypt your message.

If a friend is also doing this activity, once you have your encrypted message you could give them the ciphertext and the rotation key (and get them to give you theirs), and see if you can decrypt one another’s messages (remember to generate a conversion table).

Jargon Buster

The Caesar is an example of a substitution cipher, where each letter is substituted for another one. Other substitution ciphers improve on the Caesar cipher by not having all the letters in order, and some older written ciphers use different symbols for each symbol. However, substitution ciphers are easy to attack because a statistical attack is so easy: you just look for a few common letters and sequences of letters, and match that to common patterns in the language.

## 8.3. Problems with Substitution Ciphers¶

We have looked at one way of cracking Caesar cipher: using patterns in the text. By looking for patterns such as one letter words, other short words, double letter patterns, apostrophe positions, and knowing rules such as all words (excluding some acronyms and words written in txt language of course) must contain at least one of a, e, i, o, u, or y, you were probably able to decipher the messages in the book with little difficulty.

There are many other ways of cracking Caeser cipher which we will look at in this section. Understanding various common attacks on ciphers is important when looking at the security of more sophisticated ciphers.

### 8.3.1. Frequency Analysis Attacks¶

Frequency analysis means looking at how many times each letter appears in the encrypted message, and using this information to crack the message. A letter that appears many times in a message is far more likely to be “T” than “Z”!

For example, try copying and pasting the following text into the statistical analyser at http://www.richkni.co.uk/php/crypta/freq.php. What is the most common letter in the code? Which English letter is that likely to be?

F QTSL RJXXFLJ HTSYFNSX QTYX TK XYFYNXYNHFQ HQZJX YMFY HFS GJ ZXJI YT FSFQDXJ BMFY YMJ RTXY KWJVZJSY QJYYJWX FWJ, FSI JAJS YMJ RTXY HTRRTS UFNWX TW YWNUQJX TK QJYYJWX HFS MJQU YT GWJFP YMJ HTIJ


The most common letter in most English text is the letter E, so it makes sense to try to decrypt the message guessing that the most common letter in the ciphertext corresponds to E.

If that doesn’t work, you could see if the second most common letter in the ciphertext is E, and so on.

Curiosity

Although in almost all English texts the letter E is the most common letter, it isn’t always. For example, the 1939 novel *Gadsby* by Ernest Vincent Wright doesn’t contain a single letter E (this is called a lipogram). Furthermore, the text you’re attacking may not be English. During World War 1 and 2, the US military had many Native American Code talkers translate messages into their own language, which provided a strong layer of security at the time.

Curiosity

A slightly stronger cipher than the Caesar cipher is the Vigenere cipher, which is created by using multiple Caesar ciphers, where there is a key phrase (e.g. “acb”), and each letter in the key gives the offset (in the example this would be 1, 3, 2). These offsets are repeated to give the offset for encoding each character in the plaintext.

By having multiple caesar ciphers, common letters such as E will no longer stand out as much, making frequency analysis a lot more challenging. The following website shows the effect on the distribution. http://www.simonsingh.net/The_Black_Chamber/vigenere_strength.html

However, while this makes the Vigenere cipher more challenging to crack than the Caeser cipher, ways have been found to crack it. In fact, once you know the key length, it just breaks down to cracking several Caesar ciphers (which you have seen is straightforward!). Several statistical methods have been devised for working out the key length.

A brute force attack is harder for the Vigenere cipher because there are a lot more possible keys. In principle there isn’t a limit to the number of key phrases possible, although if the phrase is too long then keeping track of the key would be difficult.

The Vigenere cipher is known as a polyalphabetic substitution cipher, since it is uses multiple substitution rules.

### 8.3.2. Plain Text Attacks¶

Another kind of attack is the known plaintext attack, where you know part or all of the solution. For example, if you know that I start all my messages with “HI THERE”, what is the key for the following message?

AB MAXKX LXVKXM FXXMBGZ TM MPH TF MANKLWTR


Even if you did not know the key was a rotation, you have learnt that A->H, B->I, M->T, X->E, and K->R. This goes a long way towards deciphering the message!

A known plaintext attack is trivial for a Caesar cipher, but a good code shouldn’t have this vulnerability because there it can be surprisingly easy for someone to know that a particular message is being sent. For example, a common message might be “Nothing to report”, or in online banking there are likely to be common messages like headings in a bank account or parts of the web page that always appear.

Even worse is a chosen plaintext attack, where you trick someone into sending your chosen message through their system.

For this reason, it is essential for any good cryptosystem to not be breakable, even if the attacker has pieces of plaintext along with their corresponding ciphertext to work with.

Also, the cryptosystem should not give different ciphertext each time the same plaintext message is encrypted. It may initially sound impossible to achieve this, although there are several clever techniques used by real cryptosystems.

### 8.3.3. Brute force Attacks to Guess the Key¶

Another approach to cracking a ciphertext is a brute force attack, which involves trying out all possible keys, and seeing if any of them produce intelligible text. This is easy for a Caesar cipher because there are only 25 possible keys. For example, the following ciphertext is a single word, but is too short for a statistical attack. Try putting it into the decoder above, and trying keys until you decipher it.

EIJUDJQJYEKI


These days encryption keys are normally numbers that are 128 bits or longer. You could calculate how long it would take to try out every possible 128 bit number if a computer could test a million every second (including testing if each decoded text contains English words). It will eventually crack the message, but would it be any use after that amount of time?

Key size in general substition ciphers

While Caesar cipher has a key specifying a rotation, a more general substitution cipher could randomly scramble the entire alphabet. This requires a key consisting of a sequence of 26 letters or numbers, specifying which letter maps onto each other one. For example, the first part of the key could be “D, Z, E”, which would mean D -> A, Z -> B, E ->C. The key would have to have another 23 letters in order to specify the rest of the mapping.

This increases the number of possible keys, and thus reduces the risk of a brute force attack. A can be substituted for any of the 26 letters in the alphabet, B can then be substituted for any of the 25 remaining letters (26 minus the letter already substituted for A), C can then be substituted for any of the 24 remaining letters…

This gives us 26 possibilities for A times 25 possibilities for B times 24 possibilities for C.. all the way down to 2 possibilities for Y and 1 possibility for Z. 26*25*24*23*22*21*20*19*18*17*16*15*14*13*12*11*10*9*8*7*6*5*4*3*2*1 = 26! Representing each of these possibilites requires around 88 bits, making the cipher’s key size around 88 bits!

However, this only solves one of the problems. The other techniques for breaking caeser cipher we have looked at are still highly effective on all substitution ciphers. For this reason, we need better ciphers in practice, which we will look at shortly!

Jargon Buster

The main terminology you should be familiar with now is that a plaintext is encrypted by to create a ciphertext using an encryption key. Someone without the encryption key who wants to attack the cipher could try various approaches, including a brute force attack (trying out all possible keys), a frequency analysis attack (looking for statistical patterns), and a known plaintext attack (matching some known text with the cipher to work out the key).

Steganography

Cryptography is about hiding the content of a message, but sometimes it’s important to hide the existence of the message. Otherwise an enemy might figure out that something is being planned just because a lot more messages are being sent, even though they can’t read them. One way to achieve this is via steganography, where a secret message is hidden inside another message that seems innocuous. A classic scenario would be to publish a message in the public notices of a newspaper or send a letter from prison where the number of letters in each word represent a code. To a casual reader, the message might seem unimportant (and even say the opposite of the hidden one), but someone who knows the code could work it out. Messages can be hidden in digital images by making unnoticable changes to pixels so that they store some information. You can find out more about steganography on Wikipedia or in this lecture on steganography.

Two fun uses of steganography that you can try to decode yourself are a film about ciphers that contains hidden ciphers (called “The Thomas Beale Cipher”), and an activity that has five-bit text codes hidden in music.

## 8.4. Public Key Systems¶

### 8.4.1. The Key Distribution Problem¶

Alice sending an encrypted message to Bob raises an interesting problem in encryption. The ciphertext itself can safely be sent across an “unsafe” network (one that Eve is listening on), but the key cannot. How can Alice get the key to Bob? Remember the key is the thing that tells Bob how to convert the ciphertext back to plaintext. So Alice can’t include it in the encrypted message, because then Bob would be unable to access it! Alice can’t just include it as plaintext either, because then Eve will be able to get ahold of it and use it to decrypt any messages that come through using it! You might ask why Alice doesn’t just encrypt the key using a different encryption scheme, but then how will Bob know the new key? Alice would need to tell Bob the key that was used to encrypt it... and so on... this idea is definitely out!

Remember that Alice and Bob might be in different countries, and can only communicate through the internet. This also rules out Alice simply passing Bob the key in person.

Distributing keys physically is very expensive, and up to the 1970s large sums of money were spent physically sending keys internationally. Systems like this are call symmetric encryption, because Alice and Bob both need an identical copy of the key. The breakthrough was the realisation that you could make a system that used different keys for encoding and decoding!

### 8.4.2. Solving the problem with Public Key Systems¶

One of the remarkable discoveries in computer science in the 1970s was a method called public key encryption, where it’s fine to tell everyone what the key is to encrypt any messages, but you need a special private key to decrypt it. Because Alice and Bob use different keys, this is called an asymmetric encryption system.

It’s like giving out padlocks to all your friends, so anyone can lock a box and send it to you, but if you have the only (private) key, then you are the only person who can open the boxes. Once your friend locks a box, even they can’t unlock it. It’s really easy to distribute the padlocks. Public keys are the same — you can make them completely public — often people put them on their website or attach them to all emails they send. That’s quite different to having to hire a security firm to deliver them to your colleague.

Public key encryption is very heavily used for online commerce (such as internet banking and credit card payment) because your computer can set up a connection with the business or bank automatically using a public key system without you having to get together in advance to set up a key. Public key systems are generally slower than symmetric systems, so the public key system is often used to then send a new key for a symmetric system just once per session, and the symmetric key can be used from then on with a faster symmetric encryption system.

A very popular public key system is RSA. The following interactives use RSA so that you can try using a public key system for yourself.

Firstly, you will need to generate a pair of keys using the key generator interactive. Note that each key consists of two numbers and the interactive separates them with a “+” (this does not mean addition). You should keep the private key secret, and publicly announce the public key so that your friends can send you messages (e.g. put it on the whiteboard, or email it to some friends). Make sure you save your keys somewhere so you don’t forget them — a text file would be best.

RSA Key Generator

The second interactive is the encrypter, and it is used to encrypt or decrypt messages with the keys. In order to encrypt messages for you, your friends should use your public key (and select the “encrypt” button on the interactive). In order to decrypt the messages your friends have sent you, you should use your private key (and select the “decrypt” button on the interactive).

RSA Encrypter & Decrypter

Despite even your enemies knowing your public key (as you publically announced it), they cannot use it to decrypt your messages which were encrypted using the public key.

Digital Signatures In order to encrypt a message, the public key is used. In order to decrypt it, the corresponding private key must be used. But what would happen if the message was encrypted using the private key? Could you then decrypt it with the public key? Initially this might sound like a silly thing to do, as why would you encrypt a message which can be decrypted using a key that everybody in the world can access!?! It turns out that indeed, encrypting a message with the private key and then decrypting it with the public key works, and it has a very useful application! The only person who is able to encrypt the message using the private key is the person who owns the private key. The public key will only decrypt the message if the private key that was used to encrypt it actually is the public key’s corresponding private key! If the message can’t be decrypted, then it could not have been encrypted with that private key. This allows the sender to prove that the message actually is from them, and is known as a digital signature.

You could check that someone is the authentic private key holder by giving them a phrase to encrypt with their private key. You then decrypt it with the public key to check that they encrypted the phrase you gave them.

Note that this interactive’s implementation of RSA only uses around 50 bits of encryption and has other weaknesses. It is just for demonstrating the concepts here and is not quite the same as the implementations used in live encryption systems. In the RSA chapter, we will look at a more realistic implementation.

Jargon Buster

Public key systems rely on one way functions, which are mathematical functions where it’s easy to calculate the output for a particular input, but very hard to work out the input given the output. In the physical world a telephone book is a one-way function: if you’re given a name, it’s easy to work out the number, but given someone’s phone number, it’s difficult to work out their name from the phone book. In cryptography a widely used one-way function is multiplying numbers. Given two large numbers, you can multiply them very quickly, but given the result of the multiplication, it is very difficult to find out what its factors are. For example, see if you can work out which numbers multiply to give 806,849,546,124,373,268,247,678,601. You could try writing a program to try every combination of factors, but you’ll probably find it takes too long. Although this particular number can be factorised by modern software, larger numbers can’t be, and this is a problem that mathematicians and computer scientists can’t find an efficient solution to.

Jargon Buster

The methods that we considered at the start of this chapter are symmetric key systems, which just means that you use the same key to encode and decode the text. Public key systems are often called asymmetric key systems, where the sender and receiver have different keys. An asymmetric system can make it a lot easier to distribute the encryption key, because if the eavesdropper gets hold of it, all they can do is encrypt messages, not decrypt them, so they are no use for attacking messages.

## 8.5. The RSA Cryptosystem¶

The RSA cryptosystem is a widely used algorithm for public key systems. Many real world cryptosystems are based on RSA. Because it is a public key system, this means that keys are in pairs; a private key and a public key. A message that was encrypted using the public key can only be decrypted using the private key. This means that the key owner is able to keep their private key secret, and distribute their public key to the world.

In a nutshell, the RSA cryptosystem uses some clever math based on the unsolved mathematical problem of efficiently factoring a number which is the product of two prime numbers. If you need a reminder of what prime numbers and factoring a number are, read the Jargon Buster just below.

Jargon Buster

Remember that the factors of a number are all the numbers which divide into it without giving a remainder. For example: The factors of 12 are 1, 2, 3, 4, 6, and 12. Numbers such as 5 and 7 are NOT factors of 12, because 12 divided by 7 gives 1 remainder 5.

If a number only has 1 and itself as factors (i.e. all other numbers would give a remainder if divided into it), it is a prime number (For example, the factors of 37 are only 1 and 37, making it a prime number).

Factoring means to find all the factors of a number.

Currently the only known algorithm to find what the two primes that were multiplied are is a brute force one that has to try most of the possibilities that are less the the number itself. If the number is massive, then that is a huge number of possibilities that need to be checked, and it cannot be done before the sun is expected to burn out, even with huge amounts of computing power! This problem is known as the factoring problem. A public and private key pair has a mathematical relationship to the primes that were used.

If somebody was to find a good solution to this problem that could work on massive numbers, then RSA would no longer be secure, unlocking many important computer systems around the world including banks! Mathematicians are quite confident no such solution exists though. We aren’t going to go any further into the mathematical details here.

The following interactive provides a key generator and encrypter/decrypter for RSA. http://travistidwell.com/jsencrypt/demo/

Keys of various sizes can be generated, and then used to encrypt a message. Encryption is carried out using the public key. Decryption is carried out using the private key

Easy vs Difficult problems in Computer Science

If you were asked to multiply the following two big prime numbers, you might find it a bit tiring to do by hand (although it is definitely doable!), but could get an answer in milliseconds by putting it in the big numbers calculator! (included just below this box).

97394932817749829874327374574392098938789384897239489848732984239898983986969870902045828438234520989483483889389687489677903899

34983724732345498523673948934032028984850938689489896586772739002430884920489508348988329829389860884285043580020020020348508591


If on the other hand you were asked what two prime numbers were multiplied to get the following big number, you’d have a lot more trouble! (If you do find the answer, let us know! We’d be very interested to hear about it!)

3944604857329435839271430640488525351249090163937027434471421629606310815805347209533599007494460218504338388671352356418243687636083829002413783556850951365164889819793107893590524915235738706932817035504589460835204107542076784924507795112716034134062407


So why is it that despite these two problems being similar, one of them is “easy” and the other one is “hard”? Well, it comes down to the algorithms we have to solve each of the problems. You have probably done long multiplication in school by making one line for each digit in the second number and then adding all the rows together. We can analyse the speed of this algorithm, much like we did in the algorithms chapter for sorting and searching. Assuming that each number has the same number of digits, which we will call n (“Number of digits”), we need to write n rows. For each of those n rows, we will need to do around n multiplications. That gives us n*n little multiplications. We need to add the n rows together at the end as well, but that doesn’t take long so lets ignore that part. We have determined that the number of small multiplications needed to multiply two big numbers is approximately the square of the number of digits. So for two numbers with 1000 digits, that’s 1,000,000 little multiplication operations. A computer can do that in less than a second! If you know about Big-O notation, this is a O(n^2) algorithm, where n is the number of digits! Note that some slightly better algorithms have been designed, but this estimate is good enough for our purposes.

For the second problem, we’d need an algorithm that could find the two numbers that were multiplied together. You might initially say, why can’t we just reverse the multiplication? The reverse of multiplication is division, so can’t we just divide to get the two numbers? It’s a good idea, but it won’t work. For division we need to know the big number, and one of the small numbers we want to divide into it, and that will give us the other small number. But in this case, we only know the big number. So it isn’t a straightforward long division problem at all! It turns out that there is no known good algorithm to solve the problem. One way is to just try every number that is less than the number (well, we only need to go up to the square root, but that doesn’t help much!) There is still billions of billions of billions of numbers we need to check. Even a computer that could check 1 billion possibilities a second isn’t going to help us much with this! If you know about Big-O notation, this is an O(10^n) algorithm, where n is the number of digits – even small numbers of digits are just too much to deal with! There are slightly better solutions, but none of them shave off enough time to actually be useful for problems of the size of the one above!

The chapter on Complexity and Tractability looks at more computer science problems which are surprisingly challenging to solve. If you found this stuff interesting, do read about Complexity and Tractability when you are finished here!

What isn’t known though, is whether or not the general problem of breaking RSA is actually as difficult as the factoring problem. In other words, is there a completely different way of breaking it that does not involve factoring numbers?

What happens when we try the tricks that we could use to break Caesar cipher?

### 8.5.1. Does RSA reveal any patterns in the ciphertext?¶

You will need to scroll up to the Caesar cipher interactive for this exercise.

Using a Caesar cipher key of your choice, encrypt a short sentence, for example:

“I HAVE A PET CAT”


Now, encrypt a very similar sentence using the same key, for example:

“I HAVE A PET BAT”


Put the ciphertext for each side by side. As you might expect, they look very similar. This is problematic given that if Eve has the first message, she could probably use it to guess the second one! This means Caesar cipher is not semantically secure. It is essential that ciphers used in practice are!

But what about RSA? Do similar plaintext messages lead to similar ciphertext messages when RSA is used?

In order to find out, generate an RSA key and use it to encrypt each of the two above messages. What do you observe?

Because of how RSA encrypts messages, the way each character is jumbled is dependent on all the other characters in the message. This makes many of the analysis techniques we used to break Caeser cipher useless for breaking RSA! Well, nearly...

Jargon Buster: Semantically Secure

Semantically secure means that there is no known efficient algorithm that can use the ciphertext to get any information about the plaintext, other than the length of the message. It is very important that cryptosystems used in practice are semantically secure.

The plain RSA algorithm actually fails one important requirement of a good cryptosystem though! If Eve thinks she knows what message Alice is sending to Bob using public key encryption, she can attempt to encrypt that message using the public key and then see if the ciphertext she gets is the same as what Alice sent. If it is, she now knows what Alice sent Bob!

Luckily, a simple solution to this problem has been found. Alice can add random padding to the plaintext message, which then gets mixed into her message in the ciphertext. As long as Bob’s computer knows how much padding is on the message so that once it decrypts the message, it can throw away the padding, this will work.

For example, assume 5 characters of padding have been added onto the end of Alice’s message. Hi Bob, want to meet for lunch?1si98 Even if Eve knew it was likely Alice was asking Bob to go to lunch with her, she would have no way of knowing what random padding has been used. She might try Hi Bob, want to meet for lunch?72kld

Try encrypting both of these messages using the same public key. Is there any way to know from the ciphertext that they are even the same message?

Cryptosystems which implement RSA use padding in order to counteract this weakness of RSA in practice. This makes them semantically secure

### 8.5.3. Can we just guess the RSA key?¶

You might remember from the Algorithms chapter that problems can have good and bad algorithms to solve them, and that a good algorithm is fast even when the size of the problem is massive. While we have no good algorithms for breaking a message that is encrypted with RSA without the key, we have good algorithms for encrypting or decrypting the message, given the appropriate key. This means that large keys can be used, that will take a long time to guess with brute force.

If we were using a key size of 1024 bits (which is pretty standard), this would mean that there are 2^1024 different possible keys. Even if every computer in the world was working to guess the key and was able to check a million combinations a second, the universe will still end well before the key is guessed!

You might like to calculate how long it would take for various levels of encryption to be broken. e.g. 256, 1024, 2048, and 4096 using a big numbers calculator.

RSA normally uses keys in the size range of 1024 bits to 4096 bits. This makes it incredibly unlikely for somebody to guess the key!

### 8.5.4. A pitfall of RSA¶

So far RSA has held up really well against the potential attacks we have looked at. However, one big problem exists. How can Alice be certain that the public key she is about to use actually is Bob’s? This problem isn’t trivial, as Eve could easily publish that a public key belongs to Bob, when infact it is Eve that has the private key for it! All she has to do is get Alice to encrypt a message with that public key, mistakenly believing it is Bob’s, and she can now intercept and read the message with the private key she holds!

No mathematical solution exists, although there is a practical solution. Public Key Certificates are distributed by Public Key Certificate Authorities (CA’s) in order to prove the ownership of a public key. This now assumes that the CA’s are trustworthy and that they won’t be fooled or compromised. For the most part it works, although there have been some worrying exceptions…

## 8.6. The whole story!¶

Many of the examples in this chapter use very weak encryption methods that were chosen to illustrate concepts, but would never be used for commercial or military systems.

There are many aspects to computer security beyond encryption. For example, access control (such as password systems and security on smart cards) is crucial to keeping a system secure. Another major problem is writing secure software which doesn’t leave ways for a user to get access to information that they shouldn’t (such as typing a database command into a website query and have the system accidentally run it, or overflowing the buffer with a long input, which could accidentally replace parts of the program). Also, systems need to be protected from “denial of service” (DOS) attacks, where they get so overloaded with requests (e.g. to view a web site) that the server can’t cope, and legitimate users get very slow response from the system, or it might even fail completely.

For other kinds of attacks relating to computer security, see the Wikipedia entry on Hackers.

There’s a dark cloud hanging over the security of all current encryption methods: Quantum computing. Quantum computing is in its infancy, but if this approach to computing is successful, it has the potential to run very fast algorithms for attacking our most secure encryption systems (for example, it could be used to factorise numbers very quickly). In fact, the quantum algorithms have already been invented, but we don’t know if quantum computers can be built to run them. Such computers aren’t likely to appear overnight, and if they do become possible, they will also open the possibility for new encryption algorithms. This is yet another mystery in computer science where we don’t know what the future holds, and where there could be major changes in the future. But we’ll need very capable computer scientists around at the time to deal with these sorts of changes!

On the positive side, quantum information transfer protocols exist and are used in practice (using specialised equipment to generate quantum bits); these provide what is in theory a perfect encryption system, and don’t depend on an attacker being unable to solve a particular computational problem. Because of the need for specialised equipment, they are only used in high security environments such as banking.

The Wikipedia entry on cryptography has a fairly approachable entry going over the main terminology used in this chapter (and a lot more)

The encryption methods used these days rely on fairly advanced maths; for this reason books about encryption tend to either be beyond high school level, or else are about codes that aren’t actually used in practice.

There are lots of intriguing stories around encryption, including its use in wartime and for spying e.g.

• How I Discovered World War II’s Greatest Spy and Other Stories of Intelligence and Code (David Kahn)
• Decrypted Secrets: Methods and Maxims of Cryptology (Friedrich L. Bauer)
• Secret History: The Story of Cryptology (Craig Bauer)
• The Codebreakers: The Comprehensive History of Secret Communication from Ancient Times to the Internet (David Kahn) — this book is an older version of his new book, and may be hard to get

The following activities explore cryptographic protocols using an Unplugged approach; these methods aren’t strong enough to use in practice, but provide some insight into what is possible:

War in the fifth domain looks at how encryption and security are key to our defence against a new kind of war.

There are lots of articles in cs4fn on cryptography, including a statistical attack that lead to a beheading.

The book “Hacking Secret Ciphers with Python: A beginner’s guide to cryptography and computer programming with Python” (by Al Sweigart) goes over some simple ciphers including ones mentioned in this chapter, and how they can be programmed (and attacked) using Python programs.